# One-phase Free Boundary Problems on RCD Metric Measure Spaces

Chung-Kwong Chan  Department of Mathematics, Sun Yat-sen University, Guangzhou
510275<EMAIL_ADDRESS>,  Hui-Chun Zhang  Department of
Mathematics, Sun Yat-sen University, Guangzhou 510275
<EMAIL_ADDRESS>and  Xi-Ping Zhu  Department of Mathematics, Sun
Yat-sen University, Guangzhou 510275<EMAIL_ADDRESS>
###### Abstract.

In this paper, we consider a vector-valued one-phase Bernoulli-type free
boundary problem on a metric measure space $(X,d,\mu)$ with Riemannian
curvature-dimension condition $RCD(K,N)$. We first prove the existence and the
local Lipschitz regularity of the solutions, provided that the space $X$ is
non-collapsed, i.e. $\mu$ is the $N$-dimensional Hausdorff measure of $X$. And
then we show that the free boundary of the solutions is an $(N-1)$-dimensional
topological manifold away from a relatively closed subset of Hausdorff
dimension $\leqslant N-3$.

The second author was partially supported by NSFC 11521101 and 12025109.

The third author was partially supported by NSFC 11521101.

###### Contents

  1. 1 Introduction
    1. 1.1 The Bernoulli-type free boundary problems on Euclidean spaces
    2. 1.2 Free boundary problems in $RCD$-spaces and the main results
  2. 2 Preliminaries
    1. 2.1 $RCD(K,N)$ metric measure spaces and their calculus
    2. 2.2 Non-collapsed $RCD(K,N)$ metric measure spaces
    3. 2.3 Sets of finite perimeter and the reduced boundary
  3. 3 Existence of a minimizer
  4. 4 Hölder continuity of local minimizers
  5. 5 Lipschitz continuity of local minimizers
    1. 5.1 Mean value inequality
    2. 5.2 Lipschitz continuity of local minimizers of $J_{Q}$
  6. 6 Local finiteness of perimeter for the free boundary
    1. 6.1 Nondegeneracy
    2. 6.2 Density estimates near the free boundary
    3. 6.3 Local finiteness of perimeter
  7. 7 Compactness and the Euler-Lagrange equation
  8. 8 Regularity of the free boundary
  9. A Weiss-type monotonicity on cones

##  1\. Introduction

Since the pioneer work of Alt-Caffarelli [AC81], Dirichlet problems with free
boundary on Euclidean spaces have been extensively studied. Consider the
critical points of the one-phase Bernoulli energy functional:

(1.1) $J(u):=\int_{\Omega}(\left|\nabla
u\right|^{2}+\chi_{\left\\{u>0\right\\}})\mathrm{d}x\text{,}\\\ $

where $\Omega\subset\mathbb{R}^{n}$ is a bounded open set. The domain
$\Omega_{u}:=\\{x\in\Omega\mid u(x)>0\\}$ is a priori unspecified and
$\partial\\{u>0\\}\cap\Omega$ is the _free boundary_. From [AC81], the
fundamental results about existence and regularity of minimizers of $J$ and
regularity of free boundaries were established (see also [Caf87, Caf89, Caf88]
for a view of viscosity solutions). Recently, the studies on free boundary
problems have been extended to the fully nonlinear uniformly elliptic
operators [DSFS15] and uniformly elliptic operators with variable coefficients
[Tre20a, Tre20b]. In the meantime, the vector-valued Bernoulli-type free
boundary problem have been systemically studied by Caffarelli-Shahgholian-
Yeressian [CSY18], Mazzoleni-Terracini-Velichkov [MTV17, MTV20] and
Kriventsov-Lin [KL18]. We refer the readers to surveys [FS15, F18, DSFS19,
CS20] and their references for recent developments of the free boundary
problems in the Euclidean settings.

In this paper, we will extend the study on one-phase Bernoulli-type free
boundary problems from the Euclidean setting to the setting of _non-smooth_
spaces satisfying a synthetic notion of lower bounds of Ricci curvature. More
precisely, letting $(X,d,\mu)$ be a metric measure space (a complete metric
space $(X,d)$ equipped with a Radon measure $\mu$ with ${\rm supp}(\mu)=X$),
we assume that it satisfies the Riemannian curvature-dimension condition
$RCD(K,N)$ for some $K\in\mathbb{R}$ and $N\in[1,+\infty)$. The main examples
in the class of $RCD(K,N)$ spaces include the Ricci limit spaces in the
Cheeger-Colding theory [CC96, CC97, CC00] and finite dimensional Alexandrov
spaces with curvature bounded from below (see [Pet11] and [ZZ10, Appendix A]).
The parameters $K\in\mathbb{R}$ and $N\in[1,+\infty]$ play the role of “Ricci
curvature $\geqslant K$ and dimension $\leqslant N$” in Riemannian geometry.
The theory of $RCD(K,N)$ metric measure spaces and their geometric analysis
have fast and remarkable developments, see [Amb18] for a recent survey on this
topic.

Let $(X,d,\mu)$ be an $RCD(K,N)$ metric measure space for some
$K\in\mathbb{R}$ and $N\in[1,+\infty)$, and let $\Omega\subset X$ be a bounded
domain. The Bernoulli-type energy functional is given by

(1.2) $J_{Q}({\bf u})=\int_{\Omega}(\left|\nabla{\bf
u}\right|^{2}+Q\chi_{\left\\{\left|{\bf u}\right|>0\right\\}})\mathrm{d}\mu,\
\ \ $

where $Q\in L^{\infty}(\Omega)\ (=L^{\infty}(\Omega,\mu))$ with

(1.3) $0<Q_{\min}\leqslant Q(x)\leqslant Q_{\max}<+\infty\quad\mu-{\rm a.e.}\
x\in\Omega$

for two positive real numbers $Q_{\min}$ and $Q_{\max}$. According to [Che99],
it is now known that the Sobolev space $W^{1,2}(\Omega)$ is well-defined.
Given a boundary data $\mathbf{g}\in
W^{1,2}(\Omega,\left[0,+\infty)^{m}\right)$, we consider the minimization
problem:

(1.4) $\min_{{\bf u}\in\mathscr{A}_{\mathbf{g}}}J_{Q}({\bf
u}),\quad\mathscr{A}_{\mathbf{g}}:=\left\\{{\bf u}\in
W^{1,2}(\Omega,\left[0,+\infty\right)^{m})\mid{\bf u}-\mathbf{g}\in
W_{0}^{1,2}(\Omega,\mathbb{R}^{m})\right\\}\text{.}$

It is a cooperative vector-valued one-phase Bernoulli-type free boundary
problem.

######  Definition 1.1.

A map ${\bf u}\in\mathscr{A}_{\mathbf{g}}$ is called a _local minimizer_ of
$J_{Q}$ in (1.2) if there exists some $\varepsilon_{\bf u}>0$ such that
$J_{Q}({\bf u})\leqslant J_{Q}({\bf v})$ for every ${\bf
v}\in\mathscr{A}_{\mathbf{g}}$ with $\operatorname{d}({\bf u},{\bf
v})<\varepsilon_{\bf u}$, where

(1.5) $\operatorname{d}({\bf u},{\bf v}):=\left\|{\bf u}-{\bf
v}\right\|_{W^{1,2}(\Omega,\mathbb{R}^{m})}+\left\|\chi_{\left\\{\left|{\bf
u}\right|>0\right\\}}-\chi_{\left\\{\left|{\bf
v}\right|>0\right\\}}\right\|_{L^{1}(\Omega)}\text{.}$

If $\varepsilon_{\bf u}=+\infty$, we call that it is a _minimizer_ of $J_{Q}$
in (1.2).

The fundamental problems include the existence and regularity of minimizers
(or local minimizers) of $J_{Q}$ and the regularity of the free boundary
$\partial\\{|{\bf u}|>0\\}\cap\Omega$.

###  1.1. The Bernoulli-type free boundary problems on Euclidean spaces

We first recall some classical results on this problem in the Euclidean
setting, i.e., $(X,d,\mu)=(\mathbb{R}^{n},d_{\rm Eucl},\mathcal{L}^{n}).$ In
the seminal work of Alt-Caffarell [AC81], for the scalar case where $m=1$,
they established the following:
$\bullet$ the existence of the minimizer of (1.4),
$\bullet$ Lipschitz continuity of any local minimizer $u$, and
$\bullet$ when $Q\in C^{\alpha}$, the free boundary
$\partial\left\\{u>0\right\\}\cap\Omega$ is a $C^{1,\alpha}$-manifold away
from a relatively closed subset $\mathcal{S}_{u}$ with
$\mathscr{H}^{n-1}(\mathcal{S}_{u})=0$.

Nowadays, it is well-known that the singular set $\mathcal{S}_{u}$ has
$\dim_{\mathcal{H}}(\mathcal{S}_{u})\leqslant n-k^{\ast}$ for some
$k^{\ast}\in\\{5,6,7\\}$ (by Weiss [Wei99], Caffarelli-Jerison-Kenig [CJK04],
De Silva-Jerison [DSJ09] and Jerison-Savin [JS15]). Edelen-Engelstein [EE19]
explored the rectifiable structure of the singular set $\mathcal{S}_{u}$.

Recently, the vector-valued case where $m\geqslant 2$, of Bernoulli-type free
boundary problem for local minimizers of $J({\bf u})$ have been systemically
studied by Caffarelli-Shahgholian-Yeressian [CSY18], Mazzoleni-Terracini-
Velichkov [MTV17, MTV20] and Kriventsov-Lin [KL18]. See also [Tre20a, Tre20b]
for the uniformly elliptic operators with variable coefficients.

######  Theorem 1.2 (Caffarelli-Shahgholian-Yeressian [CSY18]).

Let $m\geqslant 2$ and let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain.
Suppose that $Q\in L^{\infty}(\Omega)$ satisfies (1.3). For each $g\in
W^{1,2}(\Omega,[0,+\infty)^{m})$, there exists a minimizer ${\bf
u}\in\mathscr{A}_{\bf g}$ of $J_{Q}$ in (1.2). Moreover, for any local
minimizer ${\bf u}=(u_{1},\ldots,u_{m})$ of $J_{Q}$, the following properties
hold:

  1. (1)

(Lipschitz regularity) ${\bf u}$ is locally Lipschitz continuous on $\Omega$.

  2. (2)

(Local finiteness of perimeter and Euler-Lagrange equation) If $Q\in
C(\Omega)$, then the free boundary has locally finite perimeter, and hence the
reduced boundary $\partial_{\rm red}\\{|{\bf u}|>0\\}$ is well-defined.
Furthermore, for $\mathscr{H}^{n-1}$-a.e. point $x\in\Omega\cap\partial_{\rm
red}\\{|{\bf u}|>0\\}$, $i=1,2,\cdots,m$, and $\eta>0$, the non-tangential
limit

(1.6) $w_{i}(x):=\lim_{y\in\\{|{\bf u}|>0\\}\cap\\{-(y-x)\cdot\nu_{\\{|{\bf
u}|>0\\}}(x)\geqslant\eta\\},\ y\to x}\frac{u_{i}(y)}{|{\bf u}(y)|}$

(here $\nu_{\\{|{\bf u}|>0\\}}(x)$ is the outer normal to $\\{|{\bf u}|>0\\}$
at $x$) exists, and we have the equations

(1.7) $\Delta
u_{i}=w_{i}\sqrt{Q}\mathscr{H}^{n-1}\llcorner(\Omega\cap\partial_{\rm
red}\\{|{\bf u}|>0\\})\quad{\rm for}\ \ i=1,2,\cdots,m.$

  3. (3)

(Regularity of free boundary) If $Q\in C^{\alpha}(\Omega)$ and ${\bf u}$ is a
minimizer of $J_{Q}$, then the singular part of the free boundary

$\mathcal{S}_{\bf u}:=\big{(}\partial\\{|{\bf u}|>0\\}\backslash\partial_{\rm
red}\\{|{\bf u}|>0\\}\big{)}\cap\Omega$

is a closed set in the relative topology of $\Omega$ with
$\dim_{\mathcal{H}}(\mathcal{S}_{\bf u})\leqslant n-k^{\ast}$ for some
$k^{\ast}\in\\{5,6,7\\}$, and the regular part of the free boundary
$\partial_{\rm red}\left\\{|{\bf u}|>0\right\\}\cap\Omega$ is locally
$C^{1,\beta}$ smooth for some $\beta\in(0,\alpha]$ ($C^{k+1,\beta}$ smooth or
analytic if $Q$ is $C^{k,\alpha}$ smooth or analytic, respectively).

There are many other important developments, for example, two-phase free
boundary problems [ACF84, DPSV21a, DPSV21b], free boundary problems for almost
minimizer [DT15, DET19], for the fully nonlinear uniformly elliptic operators
[DSFS15] and for the fractional $\alpha$-Laplace operator [CRS10].

The theory of free boundary problems was used by Caffarelli-Lin [CL08] to the
study of the nodal sets of harmonic maps into a singular space with non-
positive curvature in the sense of Alexandrov.

In general, the theory of free boundary problems can be divided into two main
steps. The first step is to establish the existence and the Lipschitz
regularity of the solutions. The second step is to explore the structure of
free boundary of these solutions, including the smoothness of its regular part
and the size and structure of its singular part. A blowup argument and an
improved flatness property are applied to analyze the structure of the free
boundary of solutions. One may notice that some basic ideas in the theory of
free boundary problem are similar to the ones in the theory of minimal
surfaces [F69, Giu84] and harmonic maps [Sim96, SU82].

###  1.2. Free boundary problems in $RCD$-spaces and the main results

In this subsection, we state the main results of this paper. Let $(X,d,\mu)$
be an $RCD(K,N)$-space with $K\in\mathbb{R}$ and $N\in(1,+\infty),$ and let
$Q$ be a $\mu$-measurable function on $\Omega$ with (1.3). Given a map
$\mathbf{g}=(g_{1},g_{2},\cdots,g_{m})\in
W^{1,2}(\Omega,\left[0,+\infty\right)^{m})$, we consider the minimization
problem (1.4).

The first result is the existence of a minimizer as following.

######  Proposition 1.3 (Existence of a minimizer).

If ${\rm diam}(\Omega)\leqslant{\rm diam}(X)/3$, then for each ${\bf g}\in
W^{1,2}(\Omega,[0,+\infty)^{m})$, there exists a ${\bf
u}\in\mathscr{A}_{\mathbf{g}}$ such that

(1.8) $J_{Q}({\bf u})=\inf_{{\bf v}\in\mathscr{A}_{\mathbf{g}}}J_{Q}({\bf
v})\text{.}$

This proposition is somewhat known for experts. For the completeness, we
include a proof in Section 3.

We then consider the Lipschitz regularity of a local minimizer ${\bf u}$ in
(1.4). Up to our knowledge, the existing proofs of the Lipschitz regularity of
${\bf u}$ in the Euclidean setting do not work directly in the setting of
$RCD(K,N)$-spaces. In fact, some proofs [AC81, CSY18, Caf87, Caf88, Caf89]
make heavy use of the Poisson formula, which is not clear on
$RCD(K,N)$-spaces. Other proofs [DT15, DSS20] rely on the fact that gradients
of a harmonic function are again harmonic, which fails even on smooth
Riemannian manifolds. In this paper, we will overcome this difficulty by using
the Cheng-Yau gradient estimates for harmonic functions and a mean value
property (see Lemma 5.3), to obtain the following the Lipschitz continuity,
provided that the space is non-collapsed.

######  Theorem 1.4 (Lipschitz regularity).

Let $(X,d,\mu)$ be an $RCD(K,N)$-space with $K\in\mathbb{R}$ and
$N\in(1,\infty)$. Assume that $\mu=\mathscr{H}^{N}$, the $N$-dimensional
Hausdorff measure on $X$. (I.e., $X$ is non-collapsed.) Let $\Omega\subset X$
be a bounded domain. Suppose that ${\bf u}=(u_{1},\ldots,u_{m})$ is a local
minimizer of $J_{Q}$ in (1.2) and that $Q$ satisfies (1.3), then ${\bf u}$ is
locally Lipschitz continuous on $\Omega$. Precisely, for any ball
$B_{R}(x)\subset\Omega$, there exists a constant $L$ depending only on
$N,K,\Omega,R,Q_{\max},\varepsilon_{\bf{u}}$ and $\int_{B_{R}(x)}|{\bf u}|{\rm
d}\mu$, such that

(1.9) $|u_{i}(y)-u_{i}(z)|\leqslant L\cdot d(y,z),\qquad\forall\ y,z\in
B_{R/4}(x),\quad i=1,2,\cdots,m.$

######  Remark 1.5.

The results for the Lipschitz regularity of energy minimizing harmonic maps
from/into/between singular spaces were established in [GS92, KS93, ZZ18,
GJZ19].

Our next result is about the finiteness of the perimeter of the free boundary
of a local minimizer. We will also derive the associated Euler-Lagrange
equation.

######  Theorem 1.6 (Local finiteness of perimeter and Euler-Lagrange
equation).

Let $(X,d,\mu)$, $\Omega$ and ${\bf u}$ be as in the above Theorem 1.4.
Suppose $Q\in C(\Omega)$. Then $\Omega_{\bf u}:=\Omega\cap\\{|{\bf u}|>0\\}$
is a set of locally finite perimeter. Moreover, it holds:

  1. (1)

For all $\Omega^{\prime}\Subset\Omega$,
$\mathscr{H}^{N-1}\big{(}\partial\\{|{\bf
u}|>0\\}\cap\Omega^{\prime}\big{)}<+\infty$;

  2. (2)

There exist nonnegative Borel functions $q_{i},\ i=1,2,\cdots,m,$ such that

${\bf\Delta}u_{i}=q_{i}\cdot\mathscr{H}^{N-1}\llcorner\big{(}\partial\\{|{\bf
u}|>0\\}\cap\Omega\big{)}$

in the sense of distributions (i.e., $-\int_{\Omega}\left\langle{\nabla
u_{i}},{\nabla\phi}\right\rangle{\rm d}\mu=\int_{\partial\\{|{\bf
u}|>0\\}\cap\Omega}\phi q_{i}{\rm d}\mathscr{H}^{N-1}$ for any Lipschitz
continuous $\phi$ with compact support in $\Omega$), and

(1.10) $\sum_{i=1}^{m}q_{i}^{2}(x)=Q(x),\quad{\rm for}\
\mathscr{H}^{N-1}{\rm-a.e.}\ x\in\partial\\{|{\bf u}|>0\\}\cap\Omega.$

######  Remark 1.7.

Recalling in the Euclidean setting, by using the non-tangential limits
$w_{i},i=1,\cdots,m,$ in (1.6), the densities in the Euler-Lagrange equation
(1.7), $q_{i}=w_{i}\sqrt{Q}$, fulfil (1.10). The proof of the existence of
non-tangential limits $w_{i}$ relies heavily on a domain variation formula via
a $C^{1}$ vector field (see [CSY18, Lemma 11]). In the $RCD$ setting, the
notion of “non-tangential limit” is not well-defined at present. In this
paper, we will prove (1.10) by applying a blow-up argument and the theory of
sets of finite perimeter in the setting of [Amb01, Mir03, ABS19] (see
Corollary 7.2 for the details).

To consider the regularity of free boundary of the local minimizers of
$J_{Q}$, let us recall that on the Euclidean space $\mathbb{R}^{n}$, the
dimension of singular part $\dim_{\mathcal{H}}(\mathcal{S}_{\bf u})\leqslant
n-k^{\ast}$ for some $k^{\ast}\in\\{5,6,7\\}$ (see Theorem 1.2(3)). However,
in the non-smooth setting, the singularities of the free boundary may arise
from the singularities of the space itself, see the following example.

######  Example 1.8.

Let $Y$ be the doubling of an equilateral triangle in $\mathbb{R}^{2}$ (gluing
two same equilateral triangles along their boundaries). This is a two-
dimensional Alexandrov space with nonnegative curvature, and thus,
$(Y,d_{Y},\mathscr{H}^{2})$ is an $ncRCD(0,2)$ metric measure space. Let
$X:=\mathbb{R}\times Y$. It is clear that $X$ is an $ncRCD(0,3)$-space. Assume
that $\Omega=(-1,1)\times Y$ and

$f(t,y):=\begin{cases}t&{\rm if}\quad t\geqslant 0\\\ 0&{\rm if}\quad
t<0.\end{cases}$

It is easy to check that $f$ is a minimizer of $J_{Q=1}$ on $\Omega$. The free
boundary $\partial\\{f>0\\}=\\{0\\}\times Y$. It is clear that, assuming that
$y\in Y$ is one of vertexes of the equilateral triangle, the point $x:=(0,y)$
is a singular point of the free boundary.

This example shows that the best expectation of the bound of the singular part
of the free boundary in the $RCD(K,N)$-space without boundary is co-dimension
$\geqslant 3$. We shall prove this bound for non-collapsed $RCD(K,N)$-spaces
without boundary.

Two different notions of boundary of $RCD$-spaces have been introduced in
[DPG18] and [KM19] respectively. Here we will use the one introduced in
[DPG18]. Let $(X,d,\mu:=\mathscr{H}^{N})$ be a non-collapsed $RCD(K,N)$-space.
Recall that [AnBS19, DPG18] the singular part of $X$ has a stratification:

(1.11)
$\mathcal{S}^{0}\subset\mathcal{S}^{1}\subset\cdots\subset\mathcal{S}^{N-1}=\mathcal{S}:=X\backslash\mathcal{R},$

where $\mathcal{R}$ is the regular part of $X$ given by

$\mathcal{R}:=\Big{\\{}x\in X\big{|}\ {\rm each\ tangent\ cone\ at}\ x\ {\rm
is}\ (\mathbb{R}^{N},d_{\rm Eucl})\Big{\\}},$

and, for any $0\leqslant k\leqslant N-1$,

$\mathcal{S}^{k}:=\Big{\\{}x\in X\big{|}\ {\rm no\ tangent\ cone\ at}\ x\ {\rm
splits\ off}\ \mathbb{R}^{k+1}\Big{\\}}.$

It holds

(1.12) $\dim_{\mathscr{H}}(\mathcal{S}^{k})\leqslant k,\quad\forall\
k=1,2,\cdots,N-1.$

(This was first given in [CC97] for non-collapsed Ricci limit spaces.)
According to [DPG18], the boundary of $X$ is defined by

$\partial X:=\overline{\mathcal{S}^{N-1}\backslash\mathcal{S}^{N-2}}.$

Our last result is to show that the free boundary has a manifold structure
away from a subset having co-dimension 3, which is similar to the one in
[MS21] for the boundary of a set minimizing the perimeter in $RCD$-spaces.

######  Theorem 1.9 (Regularity of free boundary).

Let $(X,d,\mu)$ and $\Omega$ be as in the above Theorem 1.4. Suppose that
${\bf u}=(u_{1},\ldots,u_{m})$ is a minimizer of $J_{Q}$ in (1.2) and that $Q$
satisfies (1.3). Assume that $\Omega\cap\partial X=\emptyset$ and $Q\in
C(\Omega)$. Then for any $\varepsilon>0$, there exists a relatively open set
$O_{\varepsilon}\subset\partial\\{|{\bf u}|>0\\}\cap\Omega$ satisfying the
following properties:

  1. (1)

($\varepsilon$-Reifenberg flatness) For any $x\in O_{\varepsilon}$ there
exists a radius $r_{x}>0$ such that for any ball $B_{r}(y)$ with $y\in
B_{r_{x}}(x)\cap\partial\\{|{\bf u}|>0\\}$ and $r\in(0,r_{x})$, it holds that
$B_{r}(y)$ is $\varepsilon r$-closed to $B_{r}(0^{N})$ in the pointed measured
Gromov-Hausdorff topology and that $B_{r}(y)\cap\partial\\{|{\bf u}|>0\\}$ is
$\varepsilon r$-closed to $B_{r}(0^{N-1})$ in the Gromov-Hausdorff topology,
where $B_{r}(0^{N})$ is the ball in $\mathbb{R}^{N}$ with centered at
$0\in\mathbb{R}^{N}$ and radius $r$;

  2. (2)

(The smallnss of the remainder of $O_{\varepsilon}$)

(1.13) $\dim_{\mathscr{H}}\big{(}(\partial\\{|{\bf
u}|>0\\}\cap\Omega)\backslash O_{\varepsilon}\big{)}\leqslant N-3.$

Moreover, if $N=3$, then $(\partial\\{|{\bf u}|>0\\}\cap\Omega)\backslash
O_{\varepsilon}$ is a discrete set of points.

In particular, the relatively open set $O_{\varepsilon}$ is
$C^{\alpha}$-biHölder homeomorphic to an $(N-1)$-dimensional manifold, where
$\alpha=\alpha(\varepsilon)\in(0,1)$ with $\lim_{\varepsilon\to
0}\alpha(\varepsilon)=1$.

######  Remark 1.10.

According to the above Example 1.8, the bound (1.13) is sharp.

As a direct consequence, we have the following result.

######  Corollary 1.11.

Let $(X,d,\mu),\Omega$ and ${\bf u}=(u_{1},\ldots,u_{m})$ be as in the above
Theorem 1.9. Assume that $\Omega\cap\partial X=\emptyset$ and $Q\in
C(\Omega)$. Let

$\partial\\{|{\bf u}|>0\\}\cap\Omega:=\mathcal{R}^{\Omega_{\bf
u}}\cup\mathcal{S}^{\Omega_{\bf u}},$

where $\mathcal{R}^{\Omega_{\bf u}}$ and $\mathcal{S}^{\Omega_{\bf u}}$ are
the regular and singular parts of the free boundary $\partial\\{|{\bf
u}|>0\\}\cap\Omega$. (That is, $x\in\mathcal{R}^{\Omega_{\bf u}}$ means that
each tangent cone at $x$ is $(\mathbb{R}^{N},d_{\rm Eucl})$ and each blow-up
limit of $\partial\\{|{\bf u}|>0\\}\cap\Omega$ at $x$ is an
$(N-1)$-dimensional affine hyperplane in $\mathbb{R}^{N}$. The singular part
$\mathcal{S}^{\Omega_{\bf u}}:=(\partial\\{|{\bf
u}|>0\\}\cap\Omega)\setminus\mathcal{R}^{\Omega_{\bf u}}.$)

Then we have

$\dim_{\mathscr{H}}\big{(}\mathcal{S}^{\Omega_{\bf u}}\big{)}\leqslant N-3.$

######  Remark 1.12.

In general, the regular part $\mathcal{R}^{\Omega_{\bf u}}$ might not form a
manifold. In fact, it might not be relatively open in the free boundary
$\partial\\{|{\bf u}|>0\\}\cap\Omega$. (In the Euclidean case and if $Q\in
C^{\alpha}$, the regular part is relatively open in the free boundary, see
Theorem 1.2(3)). This will be seen by the following simple example. Recall
that Y. Ostu and T. Shioya in [OS94] constructed a two-dimensional Alexandrov
space without boundary, denoted by $Y_{OS}$, such that the singular set
$\mathcal{S}$ of $Y_{OS}$ is dense in $Y_{OS}$. Recalling the above Example
1.8, we replace the space $Y$ in Example 1.8 by $Y_{OS}$. By using the same
construction of $f$, we know that for any singular point $y\in Y_{OS}$, the
point $x:=(0,y)$ is a singular point of the free boundary. Thus, the singular
part $\mathcal{S}^{\Omega_{f}}$ is dense in the free boundary
$\partial\\{f>0\\}=\\{0\\}\times Y_{OS}$.

##  2\. Preliminaries

Let $(X,d)$ be a complete metric space and $\mu$ be a Radon measure on $X$
with ${\rm supp}(\mu)=X.$ The triple $(X,d,\mu)$ is called a metric measure
space. Given any $p\in X$ and $R>0$, we denote by $B_{R}(p)$ the open ball
centered at $p$ with radius $R$.

###  2.1. $RCD(K,N)$ metric measure spaces and their calculus

Let $K\in\mathbb{R}$ and $N\in[1,+\infty]$. The curvature-dimension condition
$CD(K,N)$ for a metric measure space $(X,d,\mu)$ was introduced by Sturm
[Stu06a, Stu06b] and Lott-Villani [LV09, LV07]. The $RCD(K,\infty)$-condition
was introduced by Ambrosio-Gigli-Savaré in [AGS14b]. The finitely dimensional
case, $RCD(K,N)$, was given by Gigli in [Gig13, Gig15]. Erbar-Kuwada-Sturm
[EKS15] and Ambrosio-Mondino-Savaré [AMS16] proved that a weak formulation of
Bochner inequality is equivalent to the (reduced) Riemannian curvature-
dimension condition $RCD(K,N)$. In [CM16, Theorem 1.1], Cavalletti-Milman
showed that the condition $RCD(K,N)$ is equivalent to the condition
$RCD(K,N)$, if the total measure $\mu(X)<+\infty$.

We refer the readers to the survey [Amb18] and its references for the basic
facts of the theory of $RCD(K,N)$ metric measure spaces. Here we only recall
some basic properties [LV09, AGS14b, AGMR15, EKS15] as follows:
$\bullet$ $(X,d)$ is a locally compact length space. In particular, for any
$p,q\in X$, there is a shortest curve connecting them;
$\bullet$ If $N>1$, then the generalized Bishop-Gromov inequality holds. In
particular, it implies a local measure doubling property: for all
$0<r_{1}<r_{2}<R$, we have

(2.1)
$\frac{\mu\big{(}B_{r_{2}}(p)\big{)}}{\mu\big{(}B_{r_{1}}(p)\big{)}}\leqslant
C_{N,K,R}\Big{(}\frac{r_{2}}{r_{1}}\Big{)}^{N}$

for some constant $C_{N,K,R}>0$ depending only on $N,K$ and $R$;
$\bullet$ If $N>1$ and $\Omega\subset X$ is a bounded set, then there exists a
constant $C_{N,K,\Omega}>0$ such that

(2.2) $\frac{d^{+}}{dr}\mu(B_{r}(x)):=\limsup_{\delta\to
0^{+}}\frac{\mu\big{(}B_{r+\delta}(x)\backslash
B_{r}(x)\big{)}}{\delta}\leqslant C_{N,K,\Omega}$

for all $x\in\Omega$ and $r\leqslant 1$. In fact, the generalized Bishop-
Gromov inequality implies

(2.3) $\frac{\mu\big{(}B_{r+\delta}(x)\backslash
B_{r}(x)\big{)}}{\mu\big{(}B_{r}(x)\big{)}}\leqslant\frac{\bar{\mu}\big{(}B_{r+\delta}\backslash
B_{r}\big{)}}{\bar{\mu}\big{(}B_{r}\big{)}},$

where $\bar{\mu}$ is the $N$-dimensional Hausdorff measure on
$\mathbb{M}^{N}_{K/(N-1)}$, the simply connected space form with constant
sectional curvature $K/(N-1)$, and $B_{r}$ is a geodesic ball of radius $r$ in
$\mathbb{M}^{N}_{K/(N-1)}$. It follows

$\limsup_{\delta\to 0^{+}}\frac{\bar{\mu}\left(B_{r+\delta}\backslash
B_{r}\right)}{\delta\cdot\bar{\mu}(B_{r})}\leqslant\frac{C_{N,K}}{r},\quad\forall
r\leqslant 1.$

This gives $\frac{d^{+}}{dr}\mu(B_{r}(x))\leqslant
C_{N,K}\cdot\frac{\mu(B_{r}(x))}{r}$ for all $x\in X$ and $r\leqslant 1$.
Thus, by

$\mu(B_{r}(x))\leqslant C_{N,K,\Omega}\cdot r,\qquad\forall x\in\Omega,\ \
\forall r\leqslant 1,$

(see [CC00, Eq.(4.3)] or [KL16, Corollary 5.5]), we conclude (2.2).

Several different notions of Sobolev spaces for metric measure spaces have
been given in [Che99, Sha00, AGS13, AGS14a, HK00]. They are equivalent to each
other in the setting of $RCD$-metric measure spaces (see, for example,
[AGS14a, AGS13]). Given a continuous function $f$ on $X$, the _pointwise
Lipschitz constant_ ([Che99]) of $f$ at $x$ is defined by

(2.4) $\begin{split}{\rm Lip}f(x):&=\limsup_{y\to
x}\frac{|f(y)-f(x)|}{d(x,y)}\\\ &=\limsup_{r\to 0}\sup_{d(x,y)\leqslant
r}\frac{|f(y)-f(x)|}{r},\end{split}$

and ${\rm Lip}f(x)=0$ if $x$ is isolated. It is clear that ${\rm Lip}f$ is
$\mu$-measurable. Let $\Omega\subset X$ be an open domain and let $1\leqslant
p\leqslant+\infty$. The $W^{1,p}$-norm of a locally Lipschitz function $f\in
Lip_{\mathrm{loc}}(\Omega)$ on $\Omega$ is defined by

$\left\|f\right\|_{W^{1,p}(\Omega)}=\left\|f\right\|_{L^{p}(\Omega)}+\left\|\operatorname{Lip}f\right\|_{L^{p}(\Omega)}\text{.}$

The Sobolev space $W^{1,p}(\Omega)$ is defined by the completion of the set of
locally Lipschitz functions $f$ with
$\left\|f\right\|_{W^{1,p}(\Omega)}<+\infty$. The space $W_{0}^{1,p}(\Omega)$
is defined by the closure of $Lip_{0}(\Omega)$ under the $W^{1,p}$-norm, where
$Lip_{0}(\Omega)$ is the set of Lipschitz continuous functions on $\Omega$
with compact support in $\Omega$. We denote $f\in
W^{1,p}_{\mathrm{loc}}(\Omega)$ if $f\in W^{1,p}(\Omega^{\prime})$ for every
open subset $\Omega^{\prime}\Subset\Omega$, where
“$\Omega^{\prime}\Subset\Omega$” means $\Omega^{\prime}$ is compactly
contained in $\Omega$. We itemize some basic properties of Sobolev functions
as follows.

######  Proposition 2.1.

Let $1<p<\infty$.

  1. (1)

For each $f\in W^{1,p}(\Omega)$, there is a function in $L^{p}(\Omega)$,
denoted by $|\nabla f|$, $($so-called weak upper gradient for $f$, see [Che99,
Sect.2]$,)$ such that $\|f\|_{W^{1,p}(\Omega)}=\|f\|_{L^{p}(\Omega)}+\||\nabla
f|\|_{L^{p}(\Omega)}$. Moreover, if $f\in Lip_{\rm loc}(\Omega)$ then $|\nabla
f|=\operatorname{Lip}f$ holds $\mu$-a.e. in $\Omega$ $($[Che99, Theorem
5.1]$).$

  2. (2)

(Lower semicontinuity of energy.) If $f_{j}\in W^{1,p}(\Omega)$ and $f_{j}\to
f$ in $L^{p}(\Omega)$, then $\liminf_{j\to\infty}\||\nabla
f_{j}|\|_{L^{p}(\Omega)}\geqslant\||\nabla f|\|_{L^{p}(\Omega)}$.

  3. (3)

If $f,g\in W^{1,p}(\Omega)$ and $f|_{A}=g|_{A}$ for some Borel set
$A\subset\Omega$, then $|\nabla f|(x)=|\nabla g|(x)$ at $\mu$-a.e. $x\in A$.

  4. (4)

The $W^{1,2}(\Omega)$ is a Hilbert space, and the inner product
$\left\langle{\nabla f},{\nabla g}\right\rangle\in L^{1}(\Omega)$ for $f,g\in
W^{1,2}(\Omega)$ can be given by the polarization $($see [Gig15]$)$:

(2.5) $\left\langle{\nabla f},{\nabla
g}\right\rangle=\frac{1}{4}(|\nabla(f+g)|^{2}-|\nabla(f-g)|^{2}).$

  5. (5)

(Poincaré inequality, see [BB11, Eq. (2.6)] or [Raj12].) If $\Omega$ is
bounded, then there exists a constant $C_{P}>0$ depending only on $p,K,N$ and
${\rm diam}(\Omega)$, such that for every ball $B_{R}(x)\subset\Omega$ with
$R\leqslant{\rm diam}(X)/3$, it holds

$\int_{B_{R}(x)}f^{p}\leqslant C_{P}\cdot R^{p}\int_{B_{R}(x)}|\nabla
f|^{p},\qquad\forall\ f\in W^{1,p}_{0}(B_{R}(x)).$

The following fact is well-known, and a proof is given here, since we are not
able to find a reference.

######  Proposition 2.2.

Let $D,\Omega$ be two open sets with $\overline{D}\subset\Omega$ and
$\mu(\partial D)=0$. Let $1<p<\infty$ and $u\in W^{1,p}(\Omega)$. We denote
$v\in W^{1,p}_{u}(\Omega)$ whenever $v-u\in W^{1,p}_{0}(\Omega)$.

Assume $g\in W_{u}^{1,p}(D)$ and $h\in
W^{1,p}_{u}(\Omega\backslash\overline{D})$. Then the function

$f(x):=\begin{cases}g(x),&x\in D\\\
h(x),&x\in\Omega\backslash\overline{D},\end{cases}$

has a representative in $W^{1,p}_{u}(\Omega)$.

In particular, if $g\in W^{1,p}_{0}(D)$, then its zero extension $\bar{g}$ on
$\Omega$ (namely, $\bar{g}=g$ on $D$ and $\bar{g}=0$ on $\Omega\backslash D$)
is in $W_{0}^{1,p}(\Omega)$.

###### Proof.

Since $g-u\in W^{1,p}_{0}(D)$, there are $\hat{g}_{j}\in Lip_{0}(D)$ such that
$\hat{g}_{j}\to g-u$ in $W^{1,p}(D)$ as $j\to\infty$. Similarly, there are
$\hat{h}_{j}\in Lip_{0}(\Omega\backslash\overline{D})$ such that
$\hat{h}_{j}\to h-u$ in $W^{1,p}(\Omega\backslash\overline{D})$. Consider the
functions

$\hat{f}_{j}(x):=\begin{cases}\hat{g}_{j}(x),&x\in D\\\ 0,&x\in\partial D\\\
\hat{h}_{j}(x),&x\in\Omega\backslash\overline{D},\end{cases}$

for each $j\in\mathbb{N}$. Then $\hat{f}_{j}\in Lip_{0}(\Omega)$ for each
$j=1,2,\cdots.$ Since $\mu(\partial D)=0$, we have ${\rm Lip}\hat{f}_{j}={\rm
Lip}\hat{g}_{j}$ or ${\rm Lip}\hat{h}_{j}$ $\mu$-a.e. in $\Omega$, and then

$\|\hat{f}_{j}-\hat{f}_{k}\|_{W^{1,p}(\Omega)}=\|\hat{g}_{j}-\hat{g}_{k}\|_{W^{1,p}(D)}+\|\hat{h}_{j}-\hat{h}_{k}\|_{W^{1,p}(\Omega\backslash\overline{D})}$

for all $j,k=1,2,\cdots.$ It follows that $\\{\hat{f}_{j}\\}_{j=1}^{\infty}$
is a Cauchy sequence under $W^{1,p}(\Omega)$-norm. Let $\hat{f}\in
W^{1,p}_{0}(\Omega)$ be the $W^{1,p}(\Omega)$-limit of
$\\{\hat{f}_{j}\\}_{j=1}^{\infty}$. Meanwhile, since $\mu(\partial D)=0$, it
is clear that $\hat{f}_{j}\to(f-u)$ in $L^{p}(\Omega)$, as $j\to\infty$.
Therefore, $\hat{f}=f-u$ in $L^{p}(\Omega)$. It follows that $\hat{f}$ is a
$W_{0}^{1,p}(\Omega)$-representative of $f-u$.

The second assertion follows from the first one, by taking $u=0$ and $h=0$. ∎

######  Definition 2.3 (Distributional Laplacian [Gig15]).

For each function $f\in W_{\mathrm{loc}}^{1,2}(\Omega)$, its _Laplacian_
$\boldsymbol{\Delta}f$ on $\Omega$ is a linear functional acting on
$Lip_{0}(\Omega)$ given by

(2.6) $\boldsymbol{\Delta}f(\phi)=-\int_{\Omega}\left\langle{\nabla
f},{\nabla\phi}\right\rangle\mathrm{d}\mu$

for all $\phi\in Lip_{0}(\Omega)$. If there is a signed Radon measure $\nu$
such that $\boldsymbol{\Delta}f(\phi)=\int_{\Omega}\phi\mathrm{d}\nu$ for all
$\phi\in Lip_{0}(\Omega)$, then we say that $\boldsymbol{\Delta}f=\nu$ in the
sense of distribution.

In general, the measure-valued Laplacian ${\bf\Delta}f$ for a function $f\in
W^{1,2}_{\rm loc}(\Omega)$ may not be absolutely continuous with respect to
$\mu$. We consider its Radon-Nikodym decomposition

${\bf\Delta}f=({\bf\Delta}f)^{\rm ac}\cdot\mu+({\bf\Delta}f)^{\rm sing}.$

This Laplacian on $\Omega$ is linear, and it satisfies the chain rule and the
Leibniz rule [Gig15].

######  Remark 2.4.

(1) When $f\in W^{1,2}(\Omega)$, the test function in (2.6) can be taken any
$\phi\in W^{1,2}_{0}(\Omega)$.

(2) When $\Omega=X$, the inner product (2.5) provides a Dirichlet form
$\mathcal{E}(f,g):=\int_{X}\left\langle{\nabla f},{\nabla g}\right\rangle$ on
$X$. This Dirichlet form $(\mathcal{E},W^{1,2}(X))$ has an infinitesimal
generator $\Delta_{\mathcal{E}}$ with domain $D(\Delta_{\mathcal{E}})\subset
W^{1,2}(X)$, i.e., for any $f\in D(\Delta_{\mathcal{E}})$ and any $g\in
W^{1,2}(X),$ it holds
$(\Delta_{\mathcal{E}}f,g)_{L^{2}(X)}=-\mathcal{E}(f,g)$. In case $f\in
D(\Delta_{\mathcal{E}})$, the measure valued Laplacian ${\bf\Delta}f$ is
absolutely continuous with respect to $\mu$ and
${\bf\Delta}f=\Delta_{\mathcal{E}}f\cdot\mu.$

We recall the following Laplacian comparison theorem for distance functions.

######  Theorem 2.5 (Laplacian comparison theorem, [Gig15, Corollary 5.15]).

Let $(X,d,\mu)$ be an $RCD(K,N)$ space with $K\in\mathbb{R}$ and $N>1$, and
let $p\in X$. Put $\rho(x):=d(x,p)$, then

$\boldsymbol{\Delta}\rho\leqslant(N-1)\cot_{\kappa}(\sqrt{-\kappa}\rho)\cdot\mu$

in the sense of distribution on $X\backslash\left\\{p\right\\}$, where

$\kappa=K/(N-1)\quad{\rm
and}\quad\cot_{\kappa}(s)=\begin{cases}\sqrt{\kappa}\cot(\sqrt{\kappa}s)&{\rm
if}\quad\kappa>0,\\\ 1/s&{\rm if}\quad\kappa=0,\\\
\sqrt{-\kappa}\coth(\sqrt{-\kappa}s)&{\rm if}\quad\kappa<0.\end{cases}$

In particular, if $K<0,\ N>1$, then we have

(2.7) ${\bf\Delta}\phi_{N,K}(\rho)\leqslant 0$

in the sense of distributions, where

$\phi_{N,K}(s)=-\int^{1}_{s}\left(\frac{\sinh(\sqrt{-\kappa}t)}{\sqrt{-\kappa}}\right)^{1-N}\mathrm{d}t,\qquad\kappa=K/(N-1);$

and if $\rho\leqslant 1$ additionally, then we have

(2.8)
$\boldsymbol{\Delta}(\rho^{2})=2|\nabla\rho|^{2}\cdot\mu+2\rho{\bf\Delta}\rho\leqslant
2(N+C\rho^{2})\cdot\mu$

in the sense of distributions on $X$, where $C$ depends only on $N$ and $K$.

A function $f\in W^{1,2}_{\mathrm{loc}}(\Omega)$ is called _subharmonic_ on
$\Omega$ if $\boldsymbol{\Delta}f\geqslant 0$ in the sense of distributions,
that is $\int_{\Omega}\left\langle{\nabla
f},{\nabla\phi}\right\rangle\mathrm{d}\mu\leqslant 0$ for all
$0\leqslant\phi\in Lip_{0}(\Omega).$ A function $f\in
W^{1,2}_{\mathrm{loc}}(\Omega)$ is called _harmonic_ on $\Omega$ if both $f$
and $-f$ are subharmonic on $\Omega$. From [Che99, Theorem 7.17], the maximum
principle holds for subharmonic functions. To be precise, if $f\in
W^{1,2}(\Omega)$ is a subharmonic function such that $f-g\in
W^{1,2}_{0}(\Omega)$ for some $g\in L^{\infty}(\Omega)\cap W^{1,2}(\Omega)$,
then ${\rm esssup}_{\Omega}f\leqslant{\rm esssup}_{\Omega}g.$

According to [Che99, Theorem 7.12], the (relaxed) Dirichlet problem is
solvable: for any ball $B_{R}(z)$ with $B_{3R}(z)\subset\Omega$ and any $f\in
W^{1,2}(B_{R}(z))$, there exists a (unique) solution $f_{H}$ to
${\bf\Delta}f_{H}=0$ in the sense of distributions with boundary data
$f_{H}-f\in W^{1,2}_{0}(B_{R}(z))$. The classical Cheng-Yau’s estimate [CY75]
for harmonic functions has been extended to $RCD$ metric measure spaces
[JKY14, ZZ16].

######  Theorem 2.6 (see [ZZ16, Theorem 1.6]).

Let $(X,d,\mu)$ be an $RCD(K,N)$-space with $K\leqslant 0$ and
$N\in(1,+\infty)$. Then every harmonic function $f$ on a geodesic ball
$B_{R}(x_{0})\subset X$ admits a locally Lipschitz continuous representative.
Moreover, there exists a constant $C_{N}$, depending only on $N$, such that
every positive harmonic function $f$ on $B_{R}(x_{0})$ satisfies

(2.9) $\sup_{B_{R/2}(x_{0})}\frac{|\nabla f|}{f}\leqslant
C_{N}\frac{1+\sqrt{-K}R}{R}.$

Remark that the Cheng-Yau’s estimate implies the Harnack estimate: if $f$ is a
positive harmonic function on $B_{R}(x_{0})$ with $R\leqslant 1$, then

(2.10) $\frac{f(x)}{f(y)}\leqslant C_{K,N},\quad\ \forall\ x,y\in
B_{R/2}(x_{0})$

for some constant $C_{K,N}$ depending only on $K$ and $N$. Indeed, for any two
points $x,y\in B_{R/2}(x_{0})$, one can connect them by a curve
$\gamma(t)\subset B_{R/2}(x_{0})$ with length $L(\gamma)\leqslant R$. By (2.9)
and $R\leqslant 1$, it holds

$\begin{split}|\ln f(x)-\ln f(y)|&\leqslant\int_{0}^{L(\gamma)}|(\ln
f\circ\gamma)^{\prime}|\leqslant\int_{0}^{L(\gamma)}(|\nabla\ln
f|\circ\gamma)\\\ &\leqslant\frac{C_{N}(1+\sqrt{-K}R)}{R}L(\gamma)\leqslant
C_{N,K}.\end{split}$

We recall the notion of the pointed measured Gromov-Hausdorff convergence of a
sequence of metric measure spaces (see [Gro07, GMS15]). We focus on $RCD(K,N)$
metric measure spaces.

######  Definition 2.7.

(1) Let $(Z,d_{Z})$ be a complete metric space. Given $\epsilon>0$ and subsets
$A,B\subset Z$, we say that the Hausdorff distance $d^{Z}_{H}(A,B)<\epsilon$
if

$A\subset B_{\epsilon}\quad{\rm and}\quad B\subset A_{\epsilon},$

where $A_{\epsilon}$ is the $\epsilon$-neighborhood of $A$ by
$A_{\epsilon}:=\\{z\in Z|\ d_{Z}(z,A)<\epsilon\\}$. We denote by
$A_{j}\overset{H}{\longrightarrow}A_{\infty}$ in $Z$ if
$d^{Z}_{H}(A_{j},A_{\infty})\to 0$ as $j\to\infty$.

(2) Let $(X_{j},d_{j})$ be a sequence of compact metric spaces. We say that
$(X_{j},d_{j})$ converge to a metric space $(X_{\infty},d_{\infty})$ in the
Gromov-Hausdorff topology ($GH$ for short), denoted by
$X_{j}\overset{GH}{\longrightarrow}X_{\infty}$ for short, if there exist a
complete metric space $(Z,d_{Z})$ and a sequence of isometric embedding
$\Phi_{j}:X_{j}\to Z$ such that
$\Phi_{j}(X_{j})\overset{H}{\longrightarrow}\Phi_{\infty}(X_{\infty})$ in $Z$.

(3) Let $K\in\mathbb{R}$, $N\in[1,\infty)$ and let
$\\{(X_{j},d_{j},\mu_{j})\\}_{j\in\mathbb{N}}$ be a sequence of $RCD(K,N)$
metric measure spaces with based points $p_{j}\in X_{j}$ for all
$j\in\mathbb{N}$. We say that $(X_{j},d_{j},\mu_{j},p_{j})$ converges to a
pointed metric measure space $(X_{\infty},d_{\infty},\mu_{\infty},p_{\infty})$
in the pointed measured Gromov-Hausdorff topology ($pmGH$ for short), denoted
by

$(X_{j},d_{j},\mu_{j},p_{j})\overset{pmGH}{\longrightarrow}(X_{\infty},d_{\infty},\mu_{\infty},p_{\infty}),$

if there exist a metric space $(Z,d_{Z})$ and a sequence of isometric
embeddings $\Phi_{j}:X_{j}\to Z$, for all $j\in\mathbb{N}$, and
$\Phi_{\infty}:X_{\infty}\to Z$ such that the following hold:

$\bullet\quad\Phi_{j}(p_{j})\to\Phi_{\infty}(p_{\infty})$ in $Z$,

$\bullet\quad$for every $R>0$,
$\Phi_{j}(B_{R}(p_{j}))\overset{H}{\longrightarrow}\Phi_{\infty}(B_{R}(p_{\infty}))$
in $Z$, and

$\bullet\quad(\Phi_{j})_{\\#}\mu_{j}\rightharpoonup(\Phi_{\infty})_{\\#}\mu_{\infty}$
as $j\to\infty$ (as the duality on $C_{bs}(Z)$, the space of continuous
functions on $Z$ with bounded support).

This is the so-called extrinsic approach [GMS15, Gro07], and we will fix the
choice of $(Z,d_{Z})$ and the embeddings $\Phi_{j},\Phi_{\infty}$ in the rest
of this paper. It is well-known that the limit metric measure space
$(X_{\infty},d_{\infty},\mu_{\infty})$ is also of $RCD(K,N)$, and that [GMS15]
the ambient space $(Z,d_{Z})$ can be chosen to be proper. Hence, the weak
convergence of measures
$(\Phi_{j})_{\\#}\mu_{j}\rightharpoonup(\Phi_{\infty})_{\\#}\mu_{\infty}$ can
be also understood in the duality on $C_{0}(Z)$ (the space of continuous
functions on $Z$ with compact support).

Let
$(X_{j},d_{j},\mu_{j},p_{j})\overset{pmGH}{\longrightarrow}(X_{\infty},d_{\infty},\mu_{\infty},p_{\infty})$,
and let $A_{j}\subset X_{j}$, $A_{\infty}\subset X_{\infty}$ be Borel subsets.
We will denote by $A_{j}\overset{GH}{\longrightarrow}A$ if
$\Phi_{j}(A_{j})\overset{H}{\longrightarrow}\Phi_{\infty}(A_{\infty})$ in $Z$,
where the ambient space $Z$ and the embeddings $\Phi_{j},\Phi_{\infty}$ are
given in Definition 2.7(3).

For any $x\in X$ and $r>0$, we consider the rescaled pointed metric measure
space

(2.11) $(X,r^{-1}d,\mu^{x}_{r},x),\quad{\rm where}\ \
\mu^{x}_{r}:=c^{x}_{r}\cdot\mu:=\frac{\mu}{\int_{B_{r}(x)}(1-r^{-1}d(\cdot,x)){\rm
d}\mu}.$

######  Definition 2.8.

A pointed metric measure space $(Y,\rho,\nu,y)$ is called a _tangent cone_ of
$(X,d,\mu)$ at $x$ if there exists $r_{j}\to 0$ such that
$(X,r^{-1}_{j}d,\mu^{x}_{r_{j}},x)\overset{pmGH}{\longrightarrow}(Y,\rho,\nu,y)$.

A point $x$ is called a _$k$ -regular_ if the tangent cone at $x$ is unique
and is isomorphic to

(2.12) $(\mathbb{R}^{k},d_{E},c_{k}\mathscr{L}^{k},0^{k}),\quad{\rm where}\ \
c_{k}:=(\int_{B_{1}(0^{k})}(1-|x|){\rm d}x)^{-1}.$

Remark that, from [MN19, BS19], now it is known that there exists a unique
integer $k\in[1,N]$ such that $\mu(X\backslash\mathcal{R}_{k})=0$, where
$\mathcal{R}_{k}$ is the set of all $k$-regular points of $(X,d,\mu)$.

We also consider the convergence of functions defined on varying spaces. Let

$(X_{j},d_{j},\mu_{j},p_{j})\overset{pmGH}{\longrightarrow}(X_{\infty},d_{\infty},\mu_{\infty},p_{\infty}).$

######  Definition 2.9.

Let $R>0$. Suppose that $\\{f_{j}\\}_{j\in\mathbb{N}\cup\\{\infty\\}}$ is a
sequence of Borel functions on $B_{R}(p_{j})$. It is said that:
$(i)$ $f_{j}\rightarrow f_{\infty}$ over $B_{R}(p_{j})$, if for any
$x_{j}\overset{GH}{\longrightarrow}x_{\infty}\in B_{R}(p_{\infty})$ then
$f_{j}(x_{j})\to f_{\infty}(x_{\infty})$ as $j\to\infty$;
$(ii)$ $f_{j}\rightarrow f_{\infty}$ _uniformly_ over $B_{R}(p_{j})$, if for
any $\varepsilon>0$ there exist $N(\varepsilon)\in\mathbb{N}$ and
$\delta:=\delta(\varepsilon)>0$ such that

$\sup_{x\in B_{R}(p_{j}),\ y\in B_{R}(p_{\infty}),\
d_{Z}(\Phi_{j}(x),\Phi_{\infty}(y))<\delta}|f_{j}(x)-f_{\infty}(y)|<\varepsilon,\quad\forall\
j\geqslant N(\varepsilon),$

where $\Phi_{j},\Phi_{\infty}$ and $Z$ are given in Definition 2.7.

We remark that the Arzela-Ascoli theorem can be generalized to the case where
the functions live on varying spaces (see, for example, [LV09]). We recall the
following Cheeger’s lifting lemma:

######  Lemma 2.10 ([Che99, Lemma 10.7]).

Let $R>0$ and let $(X_{j},d_{j},\mu_{j})$ be a sequence of $RCD(K,N)$ metric
measure spaces and

$(X_{j},d_{j},\mu_{j},p_{j})\overset{pmGH}{\longrightarrow}(X_{\infty},d_{\infty},\mu_{\infty},p_{\infty}).$

Then, given any Lipschitz function $f_{\infty}\in
Lip(\overline{B_{R}(p_{\infty})})$ with a Lipschitz constant $L>0$, there
exist a sequence of Lipschitz functions $f_{j}\in
Lip(\overline{B_{R}(p_{j})})$ such that $f_{j}\rightarrow f_{\infty}$
uniformly over $B_{R}(p_{\infty})$, $\||\nabla
f_{j}|\|_{L^{\infty}(B_{R}(p_{j}))}\leqslant L+1$ for all $j\in\mathbb{N}$,
and that

$\lim_{j\to\infty}\int_{B_{R}(p_{j})}|\nabla f_{j}|^{2}{\rm
d}\mu_{j}=\int_{B_{R}(p_{\infty})}|\nabla f_{\infty}|^{2}{\rm d}\mu_{\infty}.$

We also need a variant of it as follows.

######  Lemma 2.11.

Let $R>0$ and let $(X_{j},d_{j},\mu_{j})$ be as above in Lemma 2.10. Let
$F_{j}\in Lip(\overline{B_{R}(p_{j})})$ be a sequence of Lipschitz functions
with a uniform Lipschitz constant and satisfy that $F_{j}\to F_{\infty}$
uniformly over $B_{R}(p_{j})$. Then, given any function $f_{\infty}\in
Lip_{\rm loc}(B_{R}(p_{\infty}))$ with $f_{\infty}-F_{\infty}\in
W^{1,2}_{0}(B_{R}(p_{\infty}))$, for each $\delta>0$, there exist a sequence
of functions $f_{j}$ on $B_{R}(p_{j})$ such that $f_{j}-F_{j}\in
W^{1,2}_{0}(B_{R}(p_{j}))$, $f_{j}\rightarrow f_{\infty}$ uniformly over
$B_{R-\delta}(p_{j})$, and that

$\lim_{j\to\infty}\int_{B_{R}(p_{j})}|\nabla f_{j}|^{2}{\rm
d}\mu_{j}=\int_{B_{R}(p_{\infty})}|\nabla f_{\infty}|^{2}{\rm d}\mu_{\infty}.$

###### Proof.

Fix any $\delta>0$. Since $f_{\infty}\in
Lip(\overline{B_{R-\delta}(p_{\infty})})$, we can use the above Cheeger’s
lemma, Lemma 2.10, to obtain a sequence of Lipschitz functions $g_{j}\in
Lip(\overline{B_{R-\delta}(p_{j})})$ such that $g_{j}\to
g_{\infty}:=f_{\infty}|_{\overline{B_{R-\delta}(p_{\infty})}}$ uniformly over
$B_{R-\delta}(p_{j})$, $\|\nabla
g_{j}\|_{L^{2}(B_{R-\delta}(p_{j}))}\to\|\nabla
f_{\infty}\|_{L^{2}(B_{R-\delta}(p_{\infty}))}$ as $j\to\infty$, and $|\nabla
g_{j}|_{L^{\infty}(B_{R-\delta}(p_{j}))}\leqslant L_{\delta}$, where
$L_{\delta}>0$ is independent of $j$ (may depend on $\delta$).

Denoted by
$A_{\delta}(p_{j}):=B_{R}(p_{j})\backslash\overline{B_{R-\delta}(p_{j})}$ for
each $j\in\mathbb{N}\cup\\{+\infty\\}.$ Let $G_{j}\in
Lip(\partial{A_{\delta}(p_{j})})$ be defined by $G_{j}=F_{j}$ on $\partial
B_{R}(p_{j})$ and $G_{j}=g_{j}$ on $\partial B_{R-\delta}(p_{j})$. Then, for
each $j\in\mathbb{N}$, we can extend $G_{j}$ to a Lipschitz function
$\bar{G}_{j}\in Lip(\overline{A_{\delta}(p_{j})})$. Since $g_{j}$ and $F_{j}$
have a uniform Lipschitz constant, we can assume that $\bar{G}_{j}$ have a
uniform Lipschitz constant $L^{\prime}_{\delta}.$ Since $G_{j}\to G_{\infty}$
uniformly on $\partial A_{\delta}(p_{j})$, we can also assume that
$\bar{G}_{j}\to\bar{G}_{\infty}$ uniformly (up to a subsequence, by Arzela-
Ascoli theorem). Now we have $f_{\infty}-\bar{G}_{\infty}\in
W^{1,2}_{0}(A_{\delta}(p_{\infty}))$. By [ZZ19, Proposition 3.2(ii)], there
exists a sequence $h_{j}$ on $A_{\delta}(p_{j})$ such that
$h_{j}-\bar{G}_{j}\in W^{1,2}_{0}(A_{\delta}(p_{j}))$ and $|\nabla
h_{j}|_{L^{2}(A_{\delta}(p_{j}))}\to|\nabla
f_{\infty}|_{L^{2}(A_{\delta}(p_{\infty}))}$ as $j\to\infty$.

At last, we define the function $f_{j}$ on $B_{R}(p_{j})$ by $f_{j}:=g_{j}$ on
$\overline{B_{R-\delta}(p_{j})}$ and $f_{j}:=h_{j}$ on $A_{\delta}(p_{j})$.
From Proposition 2.2, we conclude that $f_{j}\in W^{1,2}(B_{R}(p_{j}))$ and
$f_{j}-F_{j}\in W^{1,2}_{0}(B_{R}(p_{j}))$. The functions $f_{j}$ satisfy the
desired assertions. The proof is finished. ∎

###  2.2. Non-collapsed $RCD(K,N)$ metric measure spaces

We will recall the notion of _non-collapsed_ $RCD$-space as introduced in
[DPG18].

######  Definition 2.12.

Let $(X,d,\mu)$ be an $RCD(K,N)$-space with $K\in\mathbb{R}$ and
$N\in[1,+\infty)$. It is called a non-collapsed $RCD(K,N)$-space, denoted by
$ncRCD(K,N)$-space for short, if $\mu=\mathscr{H}^{N}$, the $N$-dimensional
Hausdorff measure on $X$.

The main examples of $ncRCD(K,N)$ metric measure spaces are non-collapsed
Ricci limit spaces [CC97, CC00, CJN21] and $N$-dimensional Alexandrov space
with curvature $\geqslant K/(N-1)$. It was shown [DPG18] that if
$\left\\{(X_{\text{i}},d_{\text{i}},\mu_{\text{i}})\right\\}$ is a sequence of
$ncRCD(K,N)$ metric measure spaces and
$(X_{j},d_{j},\mu_{j},p_{j})\overset{pmGH}{\longrightarrow}(X_{\infty},d_{\infty},\mu_{\infty},p_{\infty})$,
then $(X_{\infty},d_{\infty},\mu_{\infty})$ is of $ncRCD(K,N)$ too.

If $(X,d,\mu)$ is an $ncRCD(K,N)$-space, then $N$ must be an integer, and
there holds (from Corollary 2.14 in [DPG18])

(2.13) $\mu\big{(}B_{r}(x)\big{)}\leqslant\bar{\mu}(B_{r})\leqslant
C_{N,K}\cdot r^{N},\quad\forall\ x\in X\ {\rm and}\ r\leqslant 1,$

for a constant $C_{N,K}>0$, where $\bar{\mu}$ is the $N$-dimensional Hausdorff
measure on $\mathbb{M}^{N}_{K/(N-1)}$, the simply connected space form with
constant sectional curvature $K/(N-1)$, and $B_{r}$ is a geodesic ball of
radius $r$ in $\mathbb{M}^{N}_{K/(N-1)}$. Furthermore, if $N>1$,

(2.14) $\frac{d^{+}}{dr}\mu(B_{r}(x)):=\limsup_{\delta\to
0^{+}}\frac{\mu\big{(}B_{r+\delta}(x)\backslash
B_{r}(x)\big{)}}{\delta}\leqslant C_{N,K}\cdot r^{N-1}$

for all $\ r\leqslant 1$ and $x\in X$. In fact by (2.3) and
$\mu(B_{r}(x))\leqslant\bar{\mu}(B_{r})$, we get
$\mu\left(B_{r+\delta}(x)\backslash
B_{r}(x)\right)\leqslant\bar{\mu}\left(B_{r+\delta}\backslash B_{r}\right).$
It follows (2.14). Remark that it holds only (2.2) in general
$RCD(K,N)$-spaces (without the assumption of non-collapsing).

Let $(X,d,\mu:=\mathscr{H}^{N})$ be an $ncRCD(K,N)$-space. For each point
$x\in X$, any tangent cone is a metric measure cone (a Euclidean cone with a
natural measure). Indeed, the existence of the limit $\lim_{r\to
0}\frac{\mu(B_{r}(x))}{\omega_{N}r^{N}}$ implies that any tangent cone at $x$
is a volume cone, and hence, by [DPG16], it is a metric cone. It was shown
that [DPG18, Corollary 1.7] a point $x\in X$ is regular (i.e. any one tangent
cone is isometric to $\mathbb{R}^{N}$) if and only if

$\lim_{r\to 0}\frac{\mu(B_{r}(x))}{\omega_{N}r^{N}}=1.$

###  2.3. Sets of finite perimeter and the reduced boundary

The theory of Euclidean sets of finite perimeter of De Giorgi has been
extended to $RCD(K,N)$-spaces [Amb01, Mir03, ABS19], and recently [BPS19,
BPS21].

######  Definition 2.13.

A function $f\in L^{1}(X,\mu)$ is called a function of bounded variation,
denoted by $f\in BV(X)$ for short, if there exists a sequence $f_{j}\in
Lip_{\rm loc}(X)$ converging to $f$ in $L^{1}(X)$ such that

$\limsup_{j\to\infty}\int_{X}|\nabla f_{j}|{\rm d}\mu<+\infty.$

Its total variation is a finite Borel measure and denoted by $|Df|$. Moreover,
for any open subset $A\subset X$,

$|Df|(A):=\inf\left\\{\liminf_{j\to\infty}\int_{\Omega}|\nabla f_{j}|{\rm
d}\mu\ \big{|}\ f_{j}\in Lip_{\rm loc}(A),\ \
f_{j}\overset{L^{1}(A)}{\longrightarrow}f\right\\}.$

A function $f\in BV_{\rm loc}(X)$ if $\phi f\in BV(X)$ for any $\phi\in
Lip_{0}(X)$.

######  Definition 2.14.

Let $E\subset X$ be a Borel subset and let $A$ be an open set. The perimeter
$\mathcal{P}(E,A)$ is given by

$\mathcal{P}(E,A):=\inf\left\\{\liminf_{j\to\infty}\int_{A}|\nabla f_{j}|{\rm
d}\mu\ \big{|}\ f_{j}\in Lip_{\rm loc}(A),\ \ f_{j}\overset{L^{1}_{\rm
loc}(A)}{\longrightarrow}\chi_{E}\right\\}.$

A Borel set $E$ is called of finite perimeter in $X$ if
$\mathcal{P}(E,X)<\infty.$ In that case, it is proved [Amb01, Mir03] that the
set function $A\mapsto\mathcal{P}(E,A)$ is the restriction to open sets of a
finite Borel measure $\mathcal{P}(E,\cdot)$ defined by

$\mathcal{P}(E,B):=\inf\\{\mathcal{P}(E,A)|\ B\subset A,\ A\subset X\ \ {\rm
open}\\}.$

A subset $E\subset X$ with $\mu(E)<\infty$ is a set of finite perimeter if and
only if the characteristic function $\chi_{E}\in BV(X)$, and its perimeter
measure is $\mathcal{P}(E,\cdot):=|D\chi_{E}|(\cdot).$ A subset $E\subset X$
is called a set of locally finite perimeter if $\chi_{E}\in BV_{\rm loc}(X)$.

We recall the following basic properties, collected in [Amb01, Mir03, ABS19].

######  Proposition 2.15.

Let $(X,d,\mu)$ be an $RCD(K,N)$ metric measure space with $K\in\mathbb{R}$
and $N\in[1,+\infty)$. Then the followings hold:

  1. (1)

(Lower semicontinuity) $E\mapsto\mathcal{P}(E,X)$ is lower semicontinuous with
respect to the $L^{1}_{\rm loc}(X)$ topology;

  2. (2)

(Coarea formula) Let $v\in BV(X)$. Then $\\{v>r\\}:=\\{x|\ v(x)>r\\}$ has
finite perimeter for $\mathscr{L}^{1}$-a.e. $r\in\mathbb{R}$. Moreover, if
$v\in BV(X)$ is continuous and nonnegative, then, for any Borel function
$f:X\to[0,+\infty]$, it holds

(2.15) $\int_{s\leqslant v<t}f{\rm d}|Dv|=\int_{s}^{t}\int_{X}f{\rm
d}\Big{(}\mathcal{P}(\\{v>r\\},\cdot)\Big{)}{\rm d}r$

for any $0\leqslant s<t<+\infty$ (see [Mir03, Remark 4.3] or [ABS19, Corollary
1.9]).

By applying to distance functions, we get the following weak convergence of
the measures on spheres.

######  Lemma 2.16.

Let $(X_{j},d_{j},\mu_{j})$ be a sequence of $RCD(K,N)$ metric measure spaces
with $K\in\mathbb{R}$ and $N\in[1,+\infty)$. Suppose that
$(X_{j},d_{j},\mu_{j},p_{j})\overset{pmGH}{\longrightarrow}(X_{\infty},d_{\infty},\mu_{\infty},p_{\infty})$.
Then we have, for $\mathscr{L}^{1}$-a.e. $r\in\mathbb{R}^{+}$, the functions
$\chi_{B_{r}(p_{j})}$ is in $BV(X_{j})$, and it holds

(2.16)
$|D\chi_{B_{r}(p_{j})}|\rightharpoonup|D\chi_{B_{r}(p_{\infty})}|,\quad{\rm
as}\ \ j\to\infty$

in duality with $C_{0}(Z)$, where $Z$ is given in Definition 2.7(3).

###### Proof.

By the coarea formula, we know that for $\mathscr{L}^{1}$-a.e.
$r\in\mathbb{R}^{+}$, the functions $\chi_{B_{r}(p_{j})}\in BV(X_{j})$. For
such $r>0$, from the observation that
$\chi_{B_{r}(p_{j})}\to\chi_{B_{r}(p_{\infty})}$ in $L^{1}$-strong and
Proposition 3.6 [ABS19], we conclude that

$\liminf_{j\to\infty}\int_{X_{j}}g{\rm
d}|D\chi_{B_{r}(p_{j})}|(X_{j})\geqslant\int_{X_{\infty}}g{\rm
d}|D\chi_{B_{r}(p_{\infty})}|(X_{\infty}),\quad\forall 0\leqslant g\in
Lip_{0}(Z),$

where $Z$ is given in Definition 2.7 and it is proper. On the other hand, for
any $R>0$,

$\begin{split}\mu(B_{R}(p_{j}))&=\int_{0}^{R}|D\chi_{B_{r}(p_{j})}|(X_{j}){\rm
d}r\\\
\to\mu_{\infty}(B_{R}(p_{\infty}))&=\int_{0}^{R}|D\chi_{B_{r}(p_{\infty})}|(X_{\infty}){\rm
d}r.\end{split}$

Therefore, we have
$\lim_{j\to\infty}|D\chi_{B_{r}(p_{j})}|(X_{j})=|D\chi_{B_{r}(p_{\infty})}|(X_{\infty})$
for $\mathscr{L}^{1}$-a.e. $r\in(0,R)$. At last, the desired assertion (2.16)
follows from Corollary 3.7 in [ABS19], and it completes the proof. ∎

######  Definition 2.17.

Let
$(X_{j},d_{j},\mu_{j},p_{j})\overset{pmGH}{\longrightarrow}(X_{\infty},d_{\infty},\mu_{\infty},p_{\infty})$,
and let $Z,\Phi_{j},\Phi_{\infty}$ be as in Definition 2.7. A sequence of
Borel sets $E_{j}\subset X_{j}$ with $\mu_{j}(E_{j})<\infty$ for all
$j\in\mathbb{N}$ is called to converge in $L^{1}$-strong to a Borel set
$F\subset X_{\infty}$ with $\mu_{\infty}(F)<\infty$ if
$\chi_{E_{j}}\cdot\mu_{j}\rightharpoonup\chi_{F}\cdot\mu_{\infty}$ and
$\mu_{j}(E_{j})\to\mu_{\infty}(F)$ as $j\to\infty$.

A sequence of Borel set $E_{j}\subset X_{j}$ is called to converge in $L_{\rm
loc}^{1}$ to a Borel set $F\subset X_{\infty}$ if $E_{j}\cap B_{R}(p_{j})\to
F\cap B_{R}(p_{\infty})$ in $L^{1}$-strong for every $R>0.$

Now recall the notion of reduced boundary of a set of locally finite perimeter
in [ABS19].

######  Definition 2.18 (Reduced bondary).

Let $E$ be a set of locally finite perimeter in an $ncRCD(K,N)$ metric measure
space $(X,d,\mu)$. A point $x\in X$ is called a reduced boundary point of $E$,
denoted by $x\in\mathcal{F}E$, if it satisfies the following:
(1) it is in ${\rm supp}(\mathcal{P}(E,\cdot))$ and it is a regular point of
$X$. That is, for each $\\{r_{j}\\}$ with $r_{j}\to 0$ the sequence
$(X,r_{j}^{-1}d,\mu^{x}_{r_{j}},x)$ pointed Gromov-Hausdorff converges to
$\mathbb{R}^{N}$ with the Euclidean metric; and
(2) for each $\\{r_{j}\\}$ with $r_{j}\to 0$, the sequence
$E\subset(X,r_{j}^{-1}d,\mu^{x}_{r_{j}},x)$ converges to the upper half space
$\\{x_{N}>0\\}\subset\mathbb{R}^{N}$ in $L^{1}_{\rm loc}$.

We need also the following properties for sets of finite perimeter in non-
collapsed spaces, see [Amb01, ABS19].

######  Proposition 2.19.

Let $(X,d,\mu)$ be an $ncRCD(K,N)$ metric measure space with $K\in\mathbb{R}$
and $N\in[1,+\infty)$. Then the followings hold:

  1. (1)

If $E$ is a set of finite perimeter in $X$, then
$\mathscr{H}^{N-1}(\partial^{*}E)<+\infty$, where $\partial^{*}E$ is the
_essential boundary_ of $E$, defined by

$\partial^{*}E:=\left\\{x\in M\big{|}\ \limsup_{r\to 0}\frac{\mu(B_{r}(x)\cap
E)}{\mu(B_{r}(x))}>0\ \ \&\ \ \limsup_{r\to 0}\frac{\mu(B_{r}(x)\backslash
E)}{\mu(B_{r}(x))}>0\right\\}.$

  2. (2)

If $E$ is a set of locally finite perimeter, then
$\mathscr{H}^{N-1}(\partial^{*}E\backslash\mathcal{F}E)=0$. Moreover, up to an
$\mathscr{H}^{N-1}$-negligible set, it holds

$\mathcal{F}E=\Big{\\{}x\in E\big{|}\ \lim_{r\to
0}\frac{\mathscr{H}^{N}(B_{r}(x)\cap
E)}{\omega_{N}r^{N}}=\frac{1}{2}\Big{\\}},$

and that $\mathcal{P}(E,\cdot)=\mathscr{H}^{N-1}\llcorner\mathcal{F}E,$ (the
De Giorgi’s Theorem, see [ABS19, Corollary 4.7].)

##  3\. Existence of a minimizer

In this section, we will derive the existence of a minimizer of (1.4), where
we always assume that $\Omega$ is a bounded domain in an $RCD(K,N)$-space
$(X,d,\mu)$ with $K\in\mathbb{R}$ and $N\in(1,+\infty)$. Let ${\bf
g}=(g_{1},g_{2},\cdots,g_{m})\in W^{1,2}(\Omega,[0,+\infty)^{m})$ and let
$\mathscr{A}_{\mathbf{g}}$ be given in (1.4).

Now we are ready to prove the existence of a minimizer, which is asserted in
Proposition 1.3.

###### Proof of Proposition 1.3.

Since $\mathbf{g}\in\mathscr{A}_{\mathbf{g}}$ guarantees that
$\mathscr{A}_{\mathbf{g}}\neq\emptyset$, there exists a minimizing sequence
$\left\\{{\bf u}^{k}\right\\}_{k=1}^{\infty}\subseteq\mathscr{A}_{\mathbf{g}}$
such that

(3.1) $\lim_{k\to\infty}J_{Q}({\bf u}^{k})=\inf_{{\bf
v}\in\mathscr{A}_{\mathbf{g}}}J_{Q}({\bf v})\ \ \ (\leqslant J_{Q}({\bf
g}))\text{.}$

From the Poincaré inequality (see Proposition 2.1(5)), we get

$\int_{\Omega}\left|{\bf u}^{k}-{\bf g}\right|^{2}\leqslant
C_{1}\int_{\Omega}\left|\nabla({\bf u}^{k}-{\bf g})\right|^{2},$

where the constant $C_{1}>0$ depends only on $N,K,\Omega$. Thus,

$\displaystyle\left\|{\bf
u}^{k}\right\|_{W^{1,2}(\Omega,\mathbb{R}^{m})}^{2}\leqslant$ $\displaystyle
2\left\|{\bf u}^{k}-{\bf
g}\right\|_{W^{1,2}(\Omega,\mathbb{R}^{m})}^{2}+2\left\|\mathbf{g}\right\|_{W^{1,2}(\Omega,\mathbb{R}^{m})}^{2}$
$\displaystyle\leqslant$ $\displaystyle
2(1+C_{1})\int_{\Omega}\left|\nabla({\bf
u}^{k}-\mathbf{g})\right|^{2}+2\left\|\mathbf{g}\right\|_{W^{1,2}(\Omega,\mathbb{R}^{m})}^{2}$
$\displaystyle\leqslant$ $\displaystyle
C_{2}\Big{(}\int_{\Omega}\left|\nabla{\bf
u}^{k}\right|^{2}+\int_{\Omega}\left|\nabla\mathbf{g}\right|^{2}\Big{)}+2\left\|\mathbf{g}\right\|_{W^{1,2}(\Omega,\mathbb{R}^{m})}^{2}$
$\displaystyle\leqslant$ $\displaystyle C_{2}J_{Q}({\bf
u}^{k})+(C_{2}+2)\left\|\mathbf{g}\right\|_{W^{1,2}(\Omega,\mathbb{R}^{m})}^{2}$
$\displaystyle\leqslant$ $\displaystyle
C_{2}(J_{Q}(\mathbf{g})+1)+(C_{2}+2)\left\|\mathbf{g}\right\|_{W^{1,2}(\Omega,\mathbb{R}^{m})}^{2}$

for all sufficiently large $k$, where $C_{2}=4(1+C_{1})$. Then, the fact that
$W^{1,2}(\Omega,\mathbb{R}^{m})$ is a Hilbert space implies that there exists
a subsequence $\left\\{{\bf u}^{k_{\ell}}\right\\}_{\ell=1}^{\infty}$ of
$\left\\{{\bf u}^{k}\right\\}_{k=1}^{\infty}$ such that $\left\\{{\bf
u}^{k_{\ell}}\right\\}_{\ell=1}^{\infty}$ weakly converges to some ${\bf u}$
in $W^{1,2}(\Omega,\mathbb{R}^{m})$ and converges to ${\bf u}$ almost
everywhere on $\Omega$. Noted that $\mathscr{A}_{\mathbf{g}}$ is a closed
convex subset of $W^{1,2}(\Omega,\mathbb{R}^{m})$, we conclude that ${\bf
u}\in\mathscr{A}_{\mathbf{g}}$.

By noticing that $\left\\{\chi_{\left\\{x\in\Omega\mid\left|{\bf
u}^{k_{\ell}}(x)\right|>0\right\\}}\right\\}_{\ell=1}^{\infty}$ converges to
$1$ almost everywhere on $\left\\{x\in\Omega\mid\left|{\bf
u}(x)\right|>0\right\\}$ and that $Q\geqslant 0$, we have

$Q\chi_{\\{|{\bf u}|>0\\}}\leqslant\liminf_{\ell\to\infty}Q\chi_{\\{|{\bf
u}^{k_{\ell}}|>0\\}}\quad\mu-{\rm a.e.\ in}\ \ \Omega.$

It follows from the Fatou lemma that

$\int_{\Omega}Q\chi_{\\{|{\bf
u}|>0\\}}\leqslant\liminf_{\ell\to\infty}\int_{\Omega}Q\chi_{\\{|{\bf
u}^{k_{\ell}}|>0\\}}.$

By combining this and the lower semicontinuity of energy in Proposition
2.1(2), we obtain

$\displaystyle J_{Q}({\bf u})$ $\displaystyle=\int_{\Omega}(\left|\nabla{\bf
u}\right|^{2}+Q\chi_{\left\\{\left|{\bf u}\right|>0\right\\}})$
$\displaystyle\leqslant\liminf_{\ell\to\infty}\int_{\Omega}\left|\nabla{\bf
u}^{k_{\ell}}\right|^{2}+\liminf_{\ell\to\infty}\int_{\Omega}Q\chi_{\left\\{\left|{\bf
u}^{k_{\ell}}\right|>0\right\\}}$
$\displaystyle\leqslant\liminf_{\ell\to\infty}J_{Q}({\bf u}^{k_{\ell}})$
$\displaystyle=\inf_{{\bf v}\in\mathscr{A}_{\mathbf{g}}}J_{Q}({\bf
v})\text{,}$

where we have used (3.1). Therefore, ${\bf u}$ is a minimizer of $J_{Q}$. The
proof is finished. ∎

######  Remark 3.1.

Here, we only need to assume that $Q\in L^{\infty}(\Omega)$ and $Q\geqslant 0$
almost everywhere on $\Omega.$

##  4\. Hölder continuity of local minimizers

In this section, we will derive the locally Hölder regularity for the local
minimizers of $J_{Q}$ in (1.2). Recall the notations. Let $(X,d,\mu)$ be
$RCD(K,N)$ metric measure space $(X,d,\mu)$ with $K\in\mathbb{R}$ and
$N\in(1,+\infty)$. Let $\Omega\subset X$ be a bounded domain and let $Q\in
L^{\infty}(\Omega)$. Suppose that

${\bf u}:=(u_{1},u_{2},\cdots,u_{m})\in W^{1,2}(\Omega,[0,+\infty)^{m})$

is a local minimizer of $J_{Q}$ in (1.2). Namely, there are a boundary value
${\bf g}=(g_{1},g_{2},\cdots g_{m})\in W^{1,2}(\Omega,[0,+\infty)^{m})$ and a
number $\varepsilon_{\bf u}>0$ such that

$J_{Q}({\bf u})\leqslant J_{Q}({\bf v}),\qquad\forall\ {\bf
v}\in\mathscr{A}_{\bf g}\ \ {\rm with}\ \ \operatorname{d}({\bf u},{\bf
v})<\varepsilon_{\bf u},$

where $\mathscr{A}_{\bf g}$ and $\operatorname{d}({\bf u},{\bf v})$ are given
in (1.4) and (1.5) respectively.

To begin, we argue that the components of local minimizers are subharmonic, so
that powerful analytic tools can be applied later on.

######  Lemma 4.1 (Subharmonicity).

Let ${\bf u}=(u_{1},\ldots,u_{m})$ be a local minimizer of $J_{Q}$ on $\Omega$
with $Q\in L^{\infty}(\Omega)$. Then $\boldsymbol{\Delta}u_{i}\geqslant 0$ on
$\Omega$ in the sense of distributions, for all $i=1,\ldots,m$.

###### Proof.

Fixed any ball $B_{R}(x)\subset\Omega$ such that
$\mu(B_{R}(x))\leqslant\frac{\varepsilon_{\bf u}}{2},$ it suffices to show
that $u_{i}$ is subharmonic on $B_{R}(x)$, where $i=1,\cdots,m.$

For each $0\leqslant\phi\in\operatorname{Lip}_{0}(B_{R}(x))$, $\phi\not=0$,
let

(4.1) ${\bf
v}_{i,\delta}:=(u_{1},\ldots,u_{i-1},(u_{i}-\delta\phi)^{+},u_{i+1},\ldots,u_{m})\text{,}$

where $i\in\left\\{1,\ldots,m\right\\}$ and $\delta>0$, then ${\bf
v}_{i,\delta}\in\mathscr{A}_{\mathbf{g}}$ and, by (1.5),

$\begin{split}\operatorname{d}({\bf v}_{i,\delta},{\bf u})&\leqslant\|{\bf
u}-{\bf v}_{i,\delta}\|_{W^{1,2}(\Omega,\mathbb{R}^{m})}+\|\chi_{\\{|{\bf
u}-{\bf v}_{i,\delta}|>0\\}}\|_{L^{1}(\Omega)}\\\
&\leqslant\delta\|\phi\|_{W^{1,2}(B_{R}(x))}+\mu(B_{R}(x)).\end{split}$

This implies $\operatorname{d}({\bf v}_{i,\delta},{\bf u})<\varepsilon_{\bf
u}$ provided the $\delta<\delta_{0}:=\frac{\varepsilon_{\bf
u}}{2\|\phi\|_{W^{1,2}(B_{R}(x))}}$. Noted that $\left\\{\left|{\bf
v}_{i,\delta}\right|>0\right\\}\subseteq\left\\{\left|{\bf
u}\right|>0\right\\}$, the local minimality of ${\bf u}$ gives

(4.2) $\displaystyle\int_{\Omega}(\left|\nabla{\bf
u}\right|^{2}+Q\chi_{\left\\{\left|{\bf u}\right|>0\right\\}})$
$\displaystyle\leqslant\int_{\Omega}(\left|\nabla{\bf
v}_{i,\delta}\right|^{2}+Q\chi_{\left\\{\left|{\bf
v}_{i,\delta}\right|>0\right\\}})$
$\displaystyle\leqslant\int_{\Omega}(\left|\nabla{\bf
v}_{i,\delta}\right|^{2}+Q\chi_{\left\\{\left|{\bf u}\right|>0\right\\}})$

for all $\delta\in(0,\delta_{0})$. Hence, we get
$\int_{\Omega}\left|\nabla{\bf
u}\right|^{2}\leqslant\int_{\Omega}\left|\nabla{\bf v}_{i,\delta}\right|^{2}$.
This implies

(4.3) $\int_{\Omega}\left|\nabla
u_{i}\right|^{2}\leqslant\int_{\Omega}\left|\nabla(u_{i}-\delta\phi)\right|^{2}=\int_{\Omega}(\left|\nabla
u_{i}\right|^{2}-2\delta\left\langle{\nabla
u_{i}},{\nabla\phi}\right\rangle+\delta^{2}\left|\nabla\phi\right|^{2})$

for all sufficiently small $\delta$. Therefore, by letting $\delta\to 0$, we
obtain

(4.4) $\int_{\Omega}\left\langle{\nabla
u_{i}},{\nabla\phi}\right\rangle\leqslant 0\text{.}$

The arbitrariness of $\phi$ implies $\boldsymbol{\Delta}u_{i}\geqslant 0$ on
$B_{R}(x)$ in the sense of distributions. The proof is finished. ∎

An immediate consequence is the local boundedness of ${\bf u}$.

######  Remark 4.2.

Let ${\bf u}=(u_{1},\ldots,u_{m})$ be a local minimizer of $J_{Q}$ on $\Omega$
with $Q\in L^{\infty}(\Omega)$, then for all $R>0$, there exists a constant
$C=C_{N,K,R}>0$, depending only on $N,K,R$, such that

$\sup_{B_{R/2}(x)}|{\bf u}|\leqslant Cm\cdot\fint_{B_{R}(x)}|{\bf u}|$

provided the ball $B_{R}(x)\subset\Omega$. Indeed, from Lemma 4.1 and [KS01,
Theorem 4.2], we conclude $\sup_{B_{R/2}(x)}|{u_{i}}|\leqslant
C\cdot\fint_{B_{R}(x)}|{u_{i}}|$ for each $i=1,2,\cdots,m.$

We will prove the Hölder continuity of ${\bf u}$ by using Campanato theory, so
we need to obtain a decay estimate on $\int_{B_{r}(x_{0})}\left|\nabla{\bf
u}\right|^{2}{\rm d}\mu$ (see, for example, [Gór09]).

######  Lemma 4.3 (Hölder continuity).

Let ${\bf u}=(u_{1},\ldots,u_{m})$ be a local minimizer of $J_{Q}$ in (1.2)
with $Q\in L^{\infty}(\Omega)$. Then ${\bf u}\in
C_{\mathrm{loc}}^{\alpha}(\Omega)$ for some $\alpha\in(0,1)$ (means that it
has a $C_{\rm loc}^{\alpha}$ representative).

###### Proof.

Fixed any ball $B_{\bar{R}}(\bar{x})\subset\subset\Omega$ such that
$\bar{R}<{\rm diam}(\Omega)/3$, it suffices to show ${\bf u}\in
C^{\alpha}(B_{\bar{R}/2}(\bar{x}))$. Since ${\bf u}\in L^{\infty}_{\rm
loc}(\Omega)$. We denote $M_{1}:=\sup_{B_{\bar{R}}(\bar{x})}|{\bf u}|.$

For each $x_{0}\in B_{\bar{R}/2}(\bar{x})$ and $R\in(0,\bar{R}/4)$, there
exists some ${\bf v}\in W^{1,2}(B_{R}(x_{0}),\mathbb{R}^{m})$ that solves the
following (relaxed) Dirichlet problem [Che99, Theorem 7.12]:

(4.5) $\begin{cases}\boldsymbol{\Delta}{\bf v}=0\qquad\text{ on
}B_{R}(x_{0})\text{,}\\\ {\bf u}-{\bf v}\in
W_{0}^{1,2}(B_{R}(x_{0}),\mathbb{R}^{m})\text{.}\end{cases}$

After extending ${\bf v}$ by ${\bf u}$ on $\Omega\backslash B_{R}(x_{0})$,
(that is, ${\bf v}:={\bf u}$ in $\Omega\backslash B_{R}(x_{0})$, see
Proposition 2.2), we have ${\bf v}\in\mathscr{A}_{\mathbf{g}}$ because all
components of ${\bf v}$ are nonnegative on $B_{R}(x_{0})$ by the maximum
principle (see [Che99, Theorem 7.17]).

(i) We first check that $\operatorname{d}({\bf u},{\bf v})<\varepsilon_{\bf
u}$ whenever $R<R_{0}$ for some $R_{0}>0$ depending only on $K,N,\bar{R}$ and
${\bf u}$.

By the Poincaré inequality

$\int_{B_{R}(x_{0})}|{\bf u}-{\bf v}|^{2}\leqslant
C_{1}\int_{B_{R}(x_{0})}|\nabla({\bf u}-{\bf v})|^{2}$

for some constant $C_{1}$ depending only on $K,N,\bar{R}$ (see, for example,
Proposition 2.1(5)), we get

$\|{\bf v}-{\bf
u}\|_{W^{1,2}(\Omega,\mathbb{R}^{m})}\leqslant(\sqrt{C_{1}}+1)(\int_{B_{R}(x_{0})}|\nabla({\bf
u}-{\bf v})|^{2})^{1/2},$

and hence we have

(4.6) $\displaystyle\operatorname{d}({\bf u},{\bf v})$ $\displaystyle\leqslant
C_{2}(\int_{B_{R}(x_{0})}|\nabla({\bf u}-{\bf
v})|^{2})^{1/2}+\int_{B_{R}(x_{0})}\left|\chi_{\left\\{\left|{\bf
u}\right|>0\right\\}}-\chi_{\left\\{\left|{\bf v}\right|>0\right\\}}\right|$
$\displaystyle\leqslant C_{2}(\int_{B_{R}(x_{0})}|\nabla({\bf u}-{\bf
v})|^{2})^{1/2}+\mu(B_{R}(x_{0})),$

where $C_{2}:=\sqrt{C_{1}}+1.$ Note that

(4.7) $\displaystyle\int_{B_{R}(x_{0})}\left|\nabla({\bf u}-{\bf
v})\right|^{2}$ $\displaystyle\leqslant 2\int_{B_{R}(x_{0})}\left|\nabla{\bf
u}\right|^{2}+2\int_{B_{R}(x_{0})}\left|\nabla{\bf v}\right|^{2}$
$\displaystyle\leqslant 4\int_{B_{R}(x_{0})}\left|\nabla{\bf u}\right|^{2},$

by the Dirichlet energy minimizing property of ${\bf v}$. By the combination
of (4.6), (4.7) and the facts that $|\nabla{\bf u}|^{2}\in L^{1}(\Omega)$, we
conclude that there is $R_{0}\in(0,1)$ (depending only on $K,N,\bar{R}$ and
${\bf u}$) such that $\operatorname{d}({\bf u},{\bf v})<\varepsilon_{\bf u}$
for all $R\in(0,R_{0})$.

(ii) Now by the local minimality of ${\bf u}$, we have

(4.8) $\displaystyle\int_{B_{R}(x_{0})}(\left|\nabla{\bf
u}\right|^{2}-\left|\nabla{\bf v}\right|^{2})$
$\displaystyle\leqslant\int_{B_{R}(x_{0})}(\chi_{\left\\{\left|{\bf
v}\right|>0\right\\}}-\chi_{\left\\{\left|{\bf u}\right|>0\right\\}})Q$
$\displaystyle\leqslant\|Q\|_{L^{\infty}}\cdot\mu(B_{R}(x_{0}))$

for all $R\in(0,R_{0})$. On the other hand, from ${\bf u}-{\bf v}\in
W^{1,2}(\Omega,\mathbb{R}^{m})$ and Remark 2.4(1), it can be taken as test
functions for ${\bf\Delta}{\bf u}$ and ${\bf\Delta}{\bf v}$. Hence, we have
(recalling ${\bf\Delta}{\bf v}=0$) that

(4.9) $\displaystyle\int_{B_{R}(x_{0})}\left|\nabla({\bf u}-{\bf
v})\right|^{2}$ $\displaystyle=-\int_{B_{R}(x_{0})}({\bf u}-{\bf v}){\rm
d}\boldsymbol{\Delta}({\bf u}-{\bf v})$
$\displaystyle=-\int_{B_{R}(x_{0})}({\bf u}-{\bf v}){\rm
d}\boldsymbol{\Delta}({\bf u}+{\bf v})$
$\displaystyle=\int_{B_{R}(x_{0})}(\left|\nabla{\bf
u}\right|^{2}-\left|\nabla{\bf v}\right|^{2}).$

Recall that $M_{1}:=\sup_{B_{\bar{R}}(\bar{x})}|{\bf u}|.$ By the maximum
principle and the Cheng-Yau’s estimate (Theorem 2.6), we have

(4.10) $\begin{split}\sup_{B_{R/2}(x_{0})}|\nabla{\bf v}|&\leqslant
C_{3}\sup_{B_{R}(x_{0})}|{\bf v}|\leqslant
C_{3}\sup_{B_{3\bar{R}/4}(\bar{x})}|{\bf u}|\leqslant C_{3}M_{1},\end{split}$

where we have used $B_{R}(x_{0})\subset B_{3\bar{R}/4}(\bar{x})$. By combining
the equations (4.8)-(4.10), we conclude that for all $R\in(0,R_{0})$ and
$r<R/2$,

(4.11) $\displaystyle\int_{B_{r}(x_{0})}\left|\nabla{\bf u}\right|^{2}$
$\displaystyle\leqslant 2\int_{B_{r}(x_{0})}\left|\nabla({\bf u}-{\bf
v})\right|^{2}+2\int_{B_{r}(x_{0})}\left|\nabla{\bf v}\right|^{2}$
$\displaystyle\leqslant 2\int_{B_{R}(x_{0})}\left|\nabla({\bf u}-{\bf
v})\right|^{2}+2C_{3}M_{1}\cdot\mu(B_{r}(x_{0}))$ $\displaystyle\leqslant
2\|Q\|_{L^{\infty}}\cdot\mu(B_{R}(x_{0}))+2C_{3}M_{1}\cdot\mu(B_{r}(x_{0})).$

Let $R_{1}:=\min\\{R_{0},2^{-N/2}\\}$. Then for any $R<R_{1}$, by (4.11) and
taking $r=R^{1+2/N}\ (\leqslant\frac{1}{2}R)$, we have for all $R\in(0,R_{1})$
that

(4.12) $\displaystyle r^{2}\fint_{B_{r}(x_{0})}\left|\nabla{\bf u}\right|^{2}$
$\displaystyle\leqslant
2\|Q\|_{L^{\infty}}r^{2}\frac{\mu(B_{R}(x_{0}))}{\mu(B_{r}(x_{0}))}+2C_{3}M_{1}r^{2}$
$\displaystyle\leqslant
2\|Q\|_{L^{\infty}}r^{2}C_{4}(\frac{R}{r})^{N}+2C_{3}M_{1}r^{2}$
$\displaystyle\leqslant(2\|Q\|_{L^{\infty}}C_{4}+2C_{3}M_{1})r^{\frac{4}{N+2}},$

where we used $\frac{\mu(B_{R}(x_{0}))}{\mu(B_{r}(x_{0}))}\leqslant
C_{4}(R/r)^{N}$ for some $C_{4}$ depending only on $K,N$, (by $r<R<1$, see
(2.1)). It follows from the local Poincaré inequality [Raj12] that

(4.13) $\displaystyle\fint_{B_{r}(x_{0})}\left|{\bf
u}-\fint_{B_{r}(x_{0})}{\bf u}\right|$ $\displaystyle\leqslant
C_{N,K,\bar{R}}\Big{(}r^{2}\fint_{B_{2r}(x_{0})}\left|\nabla{\bf
u}\right|^{2}\Big{)}^{\frac{1}{2}}$ $\displaystyle\leqslant
C_{K,N,\bar{R},\|Q\|_{L^{\infty}},M_{1}}\cdot r^{\frac{2}{N+2}}$

for all $r$ such that $r^{\frac{N}{N+2}}\in(0,R_{1})$, which guarantees that
${\bf u}\in C^{1/(N+2)}(B_{\bar{R}/2}(\bar{x}))$, due to the Campanato theorem
on metric measure spaces [Gór09, Theorem 3.2]. ∎

Here the Hölder index $1/(N+2)$ is not optimal. We will show, in the next
section, that ${\bf u}$ is locally Lipschitz continuous provided that the
metric measure space $(X,d,\mu)$ is non-collapsed and $Q$ satisfies (1.3).

In particular, Lemma 4.3 implies that $\left\\{x\in\Omega\mid\left|{\bf
u}(x)\right|>0\right\\}$ is an open set. By combining with the argument in
Lemma 4.1, we get the following consequence.

######  Lemma 4.4 (Harmonicity).

Let ${\bf u}=(u_{1},\ldots,u_{m})$ be a local minimizer of $J_{Q}$ in (1.2)
with $Q\in L^{\infty}(\Omega)$. Then each component $u_{i}$ is harmonic on the
open set $\Omega_{\bf u}:=\left\\{x\in\Omega\mid\left|{\bf
u}(x)\right|>0\right\\}$ for $i=1,\ldots,m$.

###### Proof.

We know from Lemma 4.3 that the set $\Omega_{\bf u}$ is open. It suffices to
show that ${\bf\Delta}u_{i}=0$ in the sense of distributions on each small
ball $B_{R}(x_{0})\subset\Omega_{\bf u}$, for each $i=1,2,\cdots m.$

For each $0\leqslant\phi\in Lip_{0}(B_{R}(x_{0}))$ let

(4.14) ${\bf
v}_{i,\delta}=(u_{1},\ldots,u_{i-1},u_{i}+\delta\phi,u_{i+1},\ldots,u_{m})\text{,}$

where $i\in\left\\{1,\ldots,m\right\\}$ and $\delta>0$, then ${\bf
v}_{i,\delta}\in\mathscr{A}_{\mathbf{g}}$. Noted that $\operatorname{d}({\bf
v}_{i,\delta},{\bf u})\leqslant\delta\|\phi\|_{W^{1,2}}+Q_{\max}\mu({\rm
supp}\phi)$, when both $\delta$ and $\mu(B_{R}(x_{0}))$ are sufficiently
small, the local minimality of ${\bf u}$ yields

$\displaystyle\int_{\Omega}(\left|\nabla{\bf
u}\right|^{2}+Q\chi_{\left\\{\left|{\bf u}\right|>0\right\\}})$
$\displaystyle\leqslant\int_{\Omega}(\left|\nabla{\bf
v}_{i,\delta}\right|^{2}+Q\chi_{\left\\{\left|{\bf
v}_{i,\delta}\right|>0\right\\}})$
$\displaystyle\leqslant\int_{\Omega}(\left|\nabla{\bf
v}_{i,\delta}\right|^{2}+Q\chi_{\left\\{\left|{\bf
u}\right|>0\right\\}})\text{,}$

where we have observed $\\{|{\bf v}_{i,\delta}|>0\\}=\\{|{\bf u}|>0\\}$, as
$\delta$ small enough. Thus, we have

(4.15) $\int_{\Omega}\left|\nabla
u_{i}\right|^{2}\leqslant\int_{\Omega}\left|\nabla(u_{i}+\delta\phi)\right|^{2}=\int_{\Omega}(\left|\nabla
u_{i}\right|^{2}+2\delta\left\langle{\nabla
u_{i}},{\nabla\phi}\right\rangle+\delta^{2}\left|\nabla\phi\right|^{2})$

for all sufficiently small $\delta$. Therefore, the arbitrariness of $\delta$
gives

(4.16) $-\int_{\Omega}\left\langle{\nabla
u_{i}},{\nabla\phi}\right\rangle\leqslant 0\text{.}$

This yields $\boldsymbol{\Delta}u_{i}\leqslant 0$ on $B_{R}(x_{0})$ in the
sense of distributions. Meanwhile, Lemma 4.1 asserts that
$\boldsymbol{\Delta}u_{i}\geqslant 0$ on $\Omega$ in the sense of
distributions. Thus, we conclude that $u_{i}$ is harmonic on
$B_{R}(x_{0})\subset\Omega_{\bf u}$. The proof is finished. ∎

######  Remark 4.5.

Recently, N. Gigli and I. V. Violo [GV21] obtained the locally Hölder
continuity of a solution to an obstructed problem on $RCD(K,N)$-spaces.

##  5\. Lipschitz continuity of local minimizers

In this section, we derive the Lipschitz regularity for local minimizers of
$J_{Q}$ in (1.2) on a non-collapsed $RCD$ metric measure space. For this, we
will begin with a mean value inequality on general $RCD(K,N)$-spaces.

###  5.1. Mean value inequality

Let $(X,d,\mu)$ be an $RCD(K,N)$ metric measure space with $K\in\mathbb{R}$
and $N\in(1,+\infty).$

######  Lemma 5.1 (Stokes formula on balls).

Let $B_{R}(x_{0})\subset X$ and $\rho(\cdot)=d(\cdot,x_{0})$, $\phi\in
C^{2}([0,R])$ and let $\psi=\phi\circ\rho.$ Suppose that $u\in
C(\overline{B_{R}(x_{0})})\cap W^{1,2}(B_{R}(x_{0}))$. If
$\boldsymbol{\Delta}\psi$ is a signed Radon measure, then

(5.1) $\int_{B_{r}(x_{0})}u{\rm
d}\boldsymbol{\Delta}\psi=-\int_{B_{r}(x_{0})}\left\langle{\nabla
u},{\nabla\psi}\right\rangle+\phi^{\prime}(r)\left.\frac{\mathrm{d}}{\mathrm{d}s}\right|_{s=r}\Big{(}\int_{B_{s}(x_{0})}u\Big{)}$

holds for almost all $r\in(0,R)$.

###### Proof.

Since $\boldsymbol{\Delta}\psi$ and $\mu$ are signed Radon measures, we have
for almost all $r\in(0,R)$ that

(5.2)
$\lim_{j\to\infty}\left|\boldsymbol{\Delta}\psi\right|(\overline{B_{r+1/j}(x_{0})}\backslash
B_{r}(x_{0}))=0$

and

(5.3) $\lim_{j\to\infty}\mu(\overline{B_{r+1/j}(x_{0})}\backslash
B_{r}(x_{0}))=0.$

Meanwhile, noted that $s\mapsto\int_{B_{s}(x_{0})}u$ is locally Lipschitz
continuous on $(0,R)$, it is differentiable almost everywhere on $(0,R)$ too.
We fix an $r$ such that both of them hold. For $j$ sufficiently large, let
$u_{j}=\eta_{j}(\rho)u\in W_{0}^{1,2}(B_{R}(x_{0}))$, where

(5.4) $\eta_{j}(t)=\begin{cases}1&\text{if }t\in\left[0,r\right]\text{,}\\\
1-j(t-r)&\text{if }t\in(r,r+\frac{1}{j})\text{,}\\\ 0&\text{if
}t\in\left[r+\frac{1}{j},R\right]\text{.}\end{cases}$

On the one hand,

(5.5) $\displaystyle\int_{B_{R}(x_{0})}u_{j}{\rm d}\boldsymbol{\Delta}\psi$
$\displaystyle=\int_{B_{R}(x_{0})\backslash B_{r}(x_{0})}u_{j}{\rm
d}\boldsymbol{\Delta}\psi+\int_{B_{r}(x_{0})}u_{j}{\rm
d}\boldsymbol{\Delta}\psi$ $\displaystyle=\int_{B_{R}(x_{0})\backslash
B_{r}(x_{0})}\eta_{j}u{\rm d}\boldsymbol{\Delta}\psi+\int_{B_{r}(x_{0})}u{\rm
d}\boldsymbol{\Delta}\psi\text{,}$

where

(5.6) $\left|\int_{B_{R}(x_{0})\backslash B_{r}(x_{0})}\eta_{j}u{\rm
d}\boldsymbol{\Delta}\psi\right|\leqslant\left|\boldsymbol{\Delta}\psi\right|(\overline{B_{r+1/j}(x_{0})}\backslash
B_{r}(x_{0}))\left\|u\right\|_{C^{0}(B_{R}(x_{0}))}.$

By combining this, (5.2) and (5.6), we have

(5.7) $\lim_{j\to\infty}\int_{B_{R}(x_{0})}u_{j}{\rm
d}\boldsymbol{\Delta}\psi=\int_{B_{r}(x_{0})}u{\rm
d}\boldsymbol{\Delta}\psi\text{.}$

On the other hand, from Remark 2.4(1) and $u_{j}\in
W^{1,2}_{0}(B_{R}(x_{0}))$, we have

(5.8) $\displaystyle\int_{B_{R}(x_{0})}u_{j}{\rm d}\boldsymbol{\Delta}\psi=$
$\displaystyle-\int_{B_{R}(x_{0})}\left\langle{\nabla
u_{j}},{\nabla\psi}\right\rangle$ $\displaystyle=$
$\displaystyle-\int_{B_{R}(x_{0})}\left\langle{\nabla
u},{\nabla\psi}\right\rangle\eta_{j}-\int_{B_{R}(x_{0})}\left\langle{\nabla\eta_{j}},{\nabla\psi}\right\rangle
u$ $\displaystyle=$ $\displaystyle-\int_{B_{R}(x_{0})\backslash
B_{r}(x_{0})}\left\langle{\nabla
u},{\nabla\psi}\right\rangle\eta_{j}-\int_{B_{r}(x_{0})}\left\langle{\nabla
u},{\nabla\psi}\right\rangle$
$\displaystyle-\int_{B_{R}(x_{0})}\left\langle{\nabla\eta_{j}},{\nabla\psi}\right\rangle
u\text{.}$

We want to estimate the first and the third terms of right hand side in (5.8).
From (5.3) and

$\displaystyle\left|\int_{B_{R}(x_{0})\backslash
B_{r}(x_{0})}\left\langle{\nabla
u},{\nabla\psi}\right\rangle\eta_{j}\right|\leqslant\int_{B_{r+\frac{1}{j}}(x_{0})\backslash
B_{r}(x_{0})}\left|\nabla u\right|\left|\nabla\psi\right|$
$\displaystyle\leqslant$
$\displaystyle\sup_{\left[0,R\right]}\left|\phi^{\prime}\right|\Big{(}\int_{B_{r+\frac{1}{j}}(x_{0})\backslash
B_{r}(x_{0})}\left|\nabla
u\right|^{2}\Big{)}^{\frac{1}{2}}\mu(B_{r+\frac{1}{j}}(x_{0})\backslash
B_{r}(x_{0}))^{\frac{1}{2}},$

we conclude that the first term of right hand side in (5.8) converges to $0$
as $j\to\infty.$ Noticed that

(5.9)
$\displaystyle\int_{B_{R}(x_{0})}\left\langle{\nabla\eta_{j}},{\nabla\psi}\right\rangle
u=\int_{B_{r+\frac{1}{j}}(x_{0})\backslash
B_{r}(x_{0})}\left\langle{-j\nabla\rho},{\nabla\psi}\right\rangle u$
$\displaystyle=$ $\displaystyle-j\int_{B_{r+\frac{1}{j}}(x_{0})\backslash
B_{r}(x_{0})}\phi^{\prime}(\rho)u$ $\displaystyle=$
$\displaystyle-j\phi^{\prime}(r)\int_{B_{r+\frac{1}{j}}(x_{0})\backslash
B_{r}(x_{0})}u-j\int_{B_{r+\frac{1}{j}}(x_{0})\backslash
B_{r}(x_{0})}\big{(}\phi^{\prime}(\rho)-\phi^{\prime}(r)\big{)}u$
$\displaystyle=$
$\displaystyle-\phi^{\prime}(r)\frac{\int_{B_{r+\frac{1}{j}}(x_{0})}u-\int_{B_{r}(x_{0})}u}{\frac{1}{j}}-j\int_{B_{r+\frac{1}{j}}(x_{0})\backslash
B_{r}(x_{0})}\big{(}\phi^{\prime}(\rho)-\phi^{\prime}(r)\big{)}u\text{,}$

and that

(5.10) $\displaystyle\left|j\int_{B_{r+\frac{1}{j}}(x_{0})\backslash
B_{r}(x_{0})}\big{(}\phi^{\prime}(\rho)-\phi^{\prime}(r)\big{)}u\right|$
$\displaystyle\leqslant$ $\displaystyle
j\int_{B_{r+\frac{1}{j}}(x_{0})\backslash
B_{r}(x_{0})}\frac{1}{j}\sup_{\left[0,R\right]}\left|\phi^{\prime\prime}\right|\left|u\right|$
$\displaystyle\leqslant$
$\displaystyle\sup_{\left[0,R\right]}\left|\phi^{\prime\prime}\right|(\int_{B_{r+\frac{1}{j}}(x_{0})\backslash
B_{r}(x_{0})}\left|u\right|^{2})^{\frac{1}{2}}\mu(B_{r+\frac{1}{j}}(x_{0})\backslash
B_{r}(x_{0}))^{\frac{1}{2}}\text{,}$

we conclude, by (5.3) and the fact that $r\mapsto\int_{B_{r}(x_{0})}u$ is
differentiable at $r$, that the third term of right hand side in (5.8)
converges to $\phi^{\prime}(r)\frac{{\rm d}}{{\rm d}r}(\int_{B_{r}(x_{0})}u),$
as $j\to\infty$. Therefore, letting $j\to\infty$ in (5.8), we obtain for
almost all $r\in(0,R)$ that

(5.11) $\int_{B_{r}(x_{0})}u{\rm
d}\boldsymbol{\Delta}\psi=-\int_{B_{r}(x_{0})}\left\langle{\nabla
u},{\nabla\psi}\right\rangle+\phi^{\prime}(r)\frac{\mathrm{d}}{\mathrm{d}r}(\int_{B_{r}(x_{0})}u).$

The proof is finished. ∎

A similar argument with a different cut-off function yields the following
slight variant.

######  Remark 5.2 (Stokes formula on annuli).

Let $\overline{B_{R_{2}}(x_{0})}\backslash B_{R_{1}}(x_{0})\subset X$,
$\rho(\cdot)=d(\cdot,x_{0})$, $\phi\in C^{2}(\left[R_{1},R_{2}\right])$, and
let $\psi=\phi\circ\rho$. Suppose that

(5.12) $u\in C(\overline{B_{R_{2}}(x_{0})}\backslash B_{R_{1}}(x_{0}))\cap
W^{1,2}(B_{R_{2}}(x_{0})\backslash B_{R_{1}}(x_{0}))\text{,}$

and if $\boldsymbol{\Delta}\psi$ is a signed Radon measure, then

(5.13) $\displaystyle\int_{B_{r_{2}}(x_{0})\backslash B_{r_{1}}(x_{0})}u{\rm
d}\boldsymbol{\Delta}\psi=$ $\displaystyle-\int_{B_{r_{2}}(x_{0})\backslash
B_{r_{1}}(x_{0})}\left\langle{\nabla u},{\nabla\psi}\right\rangle$
$\displaystyle+\phi^{\prime}(r_{2})\left.\frac{\mathrm{d}}{\mathrm{d}s}\right|_{s=r_{2}}(\int_{B_{s}(x_{0})}u)-\phi^{\prime}(r_{1})\left.\frac{\mathrm{d}}{\mathrm{d}s}\right|_{s=r_{1}}(\int_{B_{s}(x_{0})}u)$

holds for almost all $R_{1}<r_{1}<r_{2}<R_{2}$.

######  Lemma 5.3 (Mean value inequality).

Let $(X,d,\mu)$ be an $RCD(K,N)$-space with $K\in\mathbb{R}$ and
$N\in(1,+\infty)$, and let $\Omega\subset X$ be a bounded domain. Suppose that
$B_{R_{0}}(x_{0})\Subset\Omega$ and that

(5.14) $u\in C(\overline{B_{R_{0}}(x_{0})})\cap
W^{1,2}(B_{R_{0}}(x_{0}),\left[0,+\infty\right))\text{,}$

and let $\rho=d(\cdot,x_{0})$. If

(5.15) $\liminf_{r\to 0^{+}}\frac{1}{r^{N}}\int_{B_{r}(x_{0})}u(x){\rm
d}\mu=0,$

then

(5.16) $\fint_{B_{R}(x_{0})}u\leqslant
C_{1}\int_{0}^{R}\frac{e^{-C_{2}s^{2}}}{s^{N+1}}\int_{B_{s}(x_{0})}\langle\nabla
u,\nabla\rho^{2}\rangle\mathrm{d}s$

for all $R\in(0,R_{0})$, where the constants $C_{1},C_{2}$ only depend on
$N,K$ and $\Omega$.

######  Remark 5.4.

On the Euclidean space $\mathbb{R}^{N}$, we have

$\fint_{B_{R}(x_{0})}u-u(x_{0})=\frac{1}{2}\int_{0}^{R}\frac{1}{s}\fint_{B_{s}(x_{0})}\langle\nabla
u,\nabla\rho^{2}\rangle\mathrm{d}s\text{.}$

###### Proof of Lemma 5.3.

Noted that $\boldsymbol{\Delta}\rho^{2}$ is a signed Radon measure [Gig15] and
$u\geqslant 0$, Lemma 5.1 asserts that

$\int_{B_{r}(x_{0})}u{\rm
d}\boldsymbol{\Delta}(\rho^{2})=-\int_{B_{r}(x_{0})}\left\langle{\nabla
u},{\nabla\rho^{2}}\right\rangle+2r\frac{\mathrm{d}}{\mathrm{d}r}\big{(}\int_{B_{r}(x_{0})}u\big{)}$

holds for almost all $r\in(0,R)$. By combining this and the Laplacian
comparison theorem (see (2.8) in Theorem 2.5), we get

(5.17) $\displaystyle\frac{\mathrm{d}}{\mathrm{d}r}(\int_{B_{r}(x_{0})}u)$
$\displaystyle=\frac{1}{2r}\int_{B_{r}(x_{0})}u{\rm
d}\boldsymbol{\Delta}(\rho^{2})+\frac{1}{2r}\int_{B_{r}(x_{0})}\left\langle{\nabla
u},{\nabla\rho^{2}}\right\rangle$
$\displaystyle\leqslant\frac{1}{2r}\int_{B_{r}(x_{0})}2(N+C_{3}\rho^{2})u+\frac{1}{2r}\int_{B_{r}(x_{0})}\left\langle{\nabla
u},{\nabla\rho^{2}}\right\rangle\text{.}$

Here and in the following of this proof, $C_{1},C_{2},C_{3},\cdots,$ will
denote positive constants depending only on $N,K$ and $\Omega$. This gives

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}r}(\frac{1}{r^{N}}\int_{B_{r}(x_{0})}u)\leqslant$
$\displaystyle
C_{3}r\cdot\Big{(}\frac{1}{r^{N}}\int_{B_{r}(x_{0})}u\Big{)}+\frac{1}{2r^{N+1}}\int_{B_{r}(x_{0})}\left\langle{\nabla
u},{\nabla\rho^{2}}\right\rangle$

for almost all $r\in(0,R)$. Multiplying both sides by $\exp(-C_{3}r^{2}/2)$,
we have

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}r}\left(e^{-\frac{1}{2}C_{3}r^{2}}\frac{1}{r^{N}}\int_{B_{r}(x_{0})}u\right)$
$\displaystyle\leqslant\frac{e^{-\frac{1}{2}C_{3}r^{2}}}{2r^{N+1}}\int_{B_{r}(x_{0})}\left\langle{\nabla
u},{\nabla\rho^{2}}\right\rangle\text{.}$

Since $r\mapsto\frac{1}{r^{N}}e^{-\frac{1}{2}C_{3}r^{2}}\int_{B_{r}(x_{0})}u$
is locally Lipschitz on $\left(0,R\right]$, by integrating the above
inequality over $(r,R)$ for any $r<R$, we get

(5.18)
$\begin{split}\frac{e^{-\frac{1}{2}C_{3}R^{2}}}{R^{N}}\int_{B_{R}(x_{0})}u-&\frac{e^{-\frac{1}{2}C_{3}r^{2}}}{r^{N}}\int_{B_{r}(x_{0})}u\\\
&\leqslant\int_{r}^{R}\frac{e^{-\frac{1}{2}C_{3}s^{2}}}{2s^{N+1}}\int_{B_{s}(x_{0})}\left\langle{\nabla
u},{\nabla\rho^{2}}\right\rangle\mathrm{d}s\text{.}\end{split}$

By substituting the assumption (5.15) into (5.18) and let $r\to 0^{+}$, we get

$\displaystyle\frac{e^{-\frac{1}{2}C_{3}R^{2}}}{R^{N}}\int_{B_{R}(x_{0})}u\leqslant\int_{0}^{R}\frac{e^{-\frac{1}{2}C_{3}s^{2}}}{2s^{N+1}}\int_{B_{s}(x_{0})}\left\langle{\nabla
u},{\nabla\rho^{2}}\right\rangle\mathrm{d}s\text{.}$

Therefore, by $\mu(B_{R}(x_{0}))/R^{N}\geqslant C_{5}:=\mu(B_{{\rm
diam}(\Omega)}(x_{0}))/[{\rm diam}(\Omega)]^{N}$ (this follows from (2.1) and
$R<{\rm diam}(\Omega)$), we conclude

$\fint_{B_{R}(x_{0})}u\leqslant
e^{\frac{1}{2}C_{3}R^{2}}\frac{1}{C_{5}}\int_{0}^{R}\frac{e^{-\frac{1}{2}C_{3}s^{2}}}{2s^{N+1}}\int_{B_{s}(x_{0})}\left\langle{\nabla
u},{\nabla\rho^{2}}\right\rangle\mathrm{d}s\text{.}$

This implies (5.16) with $C_{1}:=\frac{e^{\frac{1}{2}C_{3}R^{2}}}{2C_{5}}$ and
$C_{2}:=\frac{C_{3}}{2}$. The proof is finished. ∎

###  5.2. Lipschitz continuity of local minimizers of $J_{Q}$

From now on, we shall suppose that $(X,d,\mu)$ is an $ncRCD(K,N)$ metric
measure space with $K\leqslant 0$ and $N\in(1,+\infty)$. Let $\Omega\subset X$
be a bounded domain and let $Q\in L^{\infty}(\Omega)$ satisfy (1.3) for two
positive numbers $Q_{\min}$ and $Q_{\max}$. Recall that a map

${\bf u}=(u_{1},u_{2},\cdots,u_{m})\in W^{1,2}(\Omega,[0,+\infty)^{m})$

is a local minimizer of $J_{Q}$ in (1.2) if there exist a data ${\bf g}\in
W^{1,2}(\Omega,[0,+\infty)^{m})$ and $\varepsilon_{\bf u}>0$ such that
$J_{Q}({\bf u})\leqslant J_{Q}({\bf v})$ for all ${\bf v}\in\mathscr{A}_{\bf
g}$ with $\operatorname{d}({\bf u},{\bf v})<\varepsilon_{\bf u}$, where the
$\mathscr{A}_{\bf g}$ and $\operatorname{d}({\bf u},{\bf v})$ are given in
(1.4) and (1.5), respectively. From Lemma 4.3, we can assume that ${\bf u}$ is
continuous on $\Omega$. The set $\Omega_{\bf u}:=\\{x\in\Omega|\ |{\bf
u}|(x)>0\\}$ is open.

The following lemma is inspired by the classical Caccioppoli inequality.

######  Lemma 5.5.

Let ${\bf u}=(u_{1},\ldots,u_{m})$ be a local minimizer of $J_{Q}$ in (1.2)
with $Q$ satisfying (1.3), and let $\Omega^{\prime}\Subset\Omega$. Then there
exists a constant $R_{0}\ (\leqslant 1)$ depending only on
$N,K,\Omega,d(\Omega^{\prime},\partial\Omega),Q_{\max}$ and $\varepsilon_{\bf
u}$, such that for all balls $B_{r}(x)$ with $r<R_{0}$ and
$d(x,\Omega^{\prime})<R_{0}$, it holds

(5.19) $-\int_{B_{r}(x)}\langle\nabla
u_{i},\nabla\phi\rangle\leqslant\big{(}Q_{\max}\cdot\mu(B_{r}(x))\big{)}^{1/2}\cdot\|\phi\|_{W^{1,2}(B_{r}(x))}$

for all $i=1,\cdots,m$ and $\phi\in W^{1,2}_{0}(B_{r}(x))$ and $\phi\geqslant
0$.

###### Proof.

From (2.13), there is a number
$R_{0}\in(0,\frac{1}{2}d(\Omega^{\prime},\partial\Omega))$ with $R_{0}<1$,
(depending only on $N,K,d(\Omega^{\prime},\partial\Omega),\Omega,Q_{\max}$ and
$\varepsilon_{\bf u}$,) such that

(5.20)
$\big{(}Q_{\max}\mu(B_{R_{0}}(x))\big{)}^{1/2}+Q_{\max}\mu(B_{R_{0}}(x))<\varepsilon_{\bf
u},\qquad\forall\ x\in\Omega.$

Fix any ball $B_{r}(x)$ with $r<R_{0}$ and $d(x,\Omega^{\prime})<R_{0}$, where
$R_{0}$ is given in the above (5.20). The inequality (5.19) obviously holds if
$\|\phi\|_{W^{1,2}(B_{r}(x))}=0$, so we are assuming that
$\|\phi\|_{W^{1,2}(B_{r}(x))}>0$ in the following. Put

(5.21)
$\delta:=\big{(}Q_{\max}\cdot\mu(B_{r}(x))\big{)}^{\frac{1}{2}}\Big{/}\|\phi\|_{W^{1,2}(B_{r}(x))}\text{.}$

Let ${\bf
v}=(u_{1},\cdots,u_{i-1},u_{i}+\delta\phi,u_{i+1},\cdots,u_{m})\in\mathscr{A}_{\bf
g}$. Note that

$\displaystyle\operatorname{d}({\bf v},{\bf u})$
$\displaystyle\leqslant\delta\cdot\|\phi\|_{W^{1,2}(B_{r}(x)))}+Q_{\max}\mu({\rm
supp}\phi)$
$\displaystyle\leqslant(Q_{\max}\mu(B_{r}(x)))^{\frac{1}{2}}+Q_{\max}\mu(B_{r}(x))<\varepsilon_{\bf
u}$

provided $r<R_{0}$, by (5.20). The local minimality of ${\bf u}$ implies

$\displaystyle\int_{\Omega}\left|\nabla{\bf u}\right|^{2}=$ $\displaystyle
J_{Q}({\bf u})-\int_{\Omega}Q\chi_{\left\\{\left|{\bf
u}\right|>0\right\\}}\leqslant J_{Q}({\bf
v})-\int_{\Omega}Q\chi_{\left\\{\left|{\bf u}\right|>0\right\\}}$
$\displaystyle\leqslant$ $\displaystyle\int_{\Omega}\left|\nabla{\bf
v}\right|^{2}+\int_{\Omega}(Q\chi_{\left\\{\left|{\bf
v}\right|>0\right\\}}-Q\chi_{\left\\{\left|{\bf u}\right|>0\right\\}})$
$\displaystyle\leqslant$ $\displaystyle\int_{\Omega}\left|\nabla{\bf
v}\right|^{2}+Q_{\max}\mu(B_{r}(x))$ $\displaystyle\leqslant$
$\displaystyle\int_{\Omega}\left|\nabla{\bf
u}\right|^{2}+2\delta\int_{B_{r}(x)}\langle\nabla
u_{i},\nabla\phi\rangle+\delta^{2}\int_{B_{r}(x)}\left|\nabla\phi\right|^{2}+Q_{\max}\mu(B_{r}(x))$

for all $r<R_{0}$. Therefore,

$\displaystyle-2\int_{B_{r}(x)}\langle\nabla u_{i},\nabla\phi\rangle$
$\displaystyle\leqslant\delta\int_{B_{r}(x)}\left|\nabla\phi\right|^{2}+\frac{Q_{\max}}{\delta}\mu(B_{r}(x))$
$\displaystyle\leqslant\delta\cdot\|\phi\|_{W^{1,2}(B_{r}(x))}^{2}+\frac{Q_{\max}}{\delta}\mu(B_{r}(x))\text{,}$

which is equivalent to (5.19) by (5.21), and the proof is finished. ∎

Combining Lemma 5.5 and Lemma 5.3, we are able to control the growth of local
minimizers near the free boundary $\partial\\{|{\bf u}|>0\\}\cap\Omega.$

######  Lemma 5.6 (Optimal linear growth).

Let ${\bf u}=(u_{1},\ldots,u_{m})$ be a local minimizer of $J_{Q}$ in (1.2)
with $Q$ satisfying (1.3), and let $\Omega^{\prime}\Subset\Omega$. Let $R_{0}$
be the constant given in Lemma 5.5. Suppose that $B_{r}(x_{0})$ is a ball with
radius $r<R_{0}$ and $d(x_{0},\Omega^{\prime})<R_{0}$. Then, if
$u_{i}(x_{0})=0$ for some $i=1,\cdots,m$, it holds

(5.22) $\sup_{B_{r/2}(x_{0})}u_{i}(x)\leqslant C\sqrt{Q_{\max}}\cdot r,$

where $C=C_{N,K,\Omega}>0$ depends only on $K,N$ and $\Omega$.

###### Proof.

Since $(X,d,\mu)$ is non-collapsed, from $u_{i}(x_{0})=0$ and (2.13), we know
that the condition (5.15) in Lemma 5.3 holds. For each $s<R_{0}$, by using
Lemma 5.5 to $\phi=s^{2}-\rho^{2}(x)$, we get

$\displaystyle\int_{B_{s}(x_{0})}\langle\nabla u_{i},\nabla\rho^{2}\rangle$
$\displaystyle=-\int_{B_{s}(x_{0})}\langle\nabla
u_{i},\nabla(s^{2}-\rho^{2})\rangle$
$\displaystyle\leqslant\big{(}Q_{\max}\cdot\mu(B_{s}(x_{0}))\big{)}^{1/2}\cdot\|s^{2}-\rho^{2}\|_{W^{1,2}(B_{s}(x_{0}))}$
$\displaystyle\leqslant\big{(}Q_{\max}\cdot\mu(B_{s}(x_{0}))\big{)}^{1/2}\Big{(}\int_{B_{s}(x_{0})}(4\rho^{2}|\nabla\rho|^{2}+(s^{2}-\rho^{2})^{2})\Big{)}^{\frac{1}{2}}$
$\displaystyle\leqslant 3\sqrt{Q_{\max}}\cdot\mu(B_{s}(x_{0}))\cdot
s\quad\qquad({\rm by}\ \ \rho<s,\ \ s<R_{0}\leqslant 1).$
$\displaystyle\leqslant C_{N,K}\sqrt{Q_{\max}}\cdot s^{N+1}\quad\qquad({\rm
by}\ \eqref{equ2.12},\ s\leqslant 1).$

Since ${\bf u}$ is continuous on $\overline{B_{r}(x_{0})}$ by Lemma 4.3, it
follows from Lemma 5.3 and $u_{i}(x_{0})=0$ that

(5.23) $\displaystyle\fint_{B_{r}(x_{0})}u_{i}$ $\displaystyle\leqslant
C_{1}\int_{0}^{r}\frac{e^{-C_{2}s^{2}}}{s^{N+1}}\int_{B_{s}(x_{0})}\langle\nabla
u_{i},\nabla\rho^{2}\rangle\mathrm{d}s\leqslant C_{3}\sqrt{Q_{\max}}\cdot
r\text{,}$

for all $r<R_{0}$. Thus, by using the fact that $u_{i}$ is subharmonic (Lemma
4.1) and $u_{i}\geqslant 0$, we get (see, for example, [KS01, Theorem 4.2])

$\sup_{B_{r/2}(x_{0})}u_{i}\leqslant C_{4}\fint_{B_{r}(x_{0})}u_{i}\leqslant
C_{4}C_{3}\sqrt{Q_{\max}}\cdot r,\qquad\forall\ r<R_{0}.$

The proof is finished. ∎

As a corollary of the combination of the linear growth and the Cheng-Yau’s
gradient estimate for harmonic functions, one can get the following gradient
estimate near the free boundary $\partial\\{|{\bf u}|>0\\}\cap\Omega.$

######  Lemma 5.7.

Let ${\bf u}=(u_{1},\ldots,u_{m})$ be a local minimizer of $J_{Q}$ in (1.2)
with $Q$ satisfying (1.3), and let $\Omega^{\prime}\Subset\Omega$. There
exists a positive constant $C=C_{N,K,\Omega}>0$ (depending only on $K,N$ and
$\Omega$), such that: if $x_{1}\in\Omega^{\prime}$ and if $d(x_{1},\\{|{\bf
u}|=0\\}\cap\Omega)<R_{0}/8$, then it holds

(5.24) $\operatorname{Lip}u_{i}(x_{1})\leqslant C\sqrt{Q_{\max}},\qquad
i=1,2,\cdots,m,$

where $R_{0}$ is given in Lemma 5.5, and $\operatorname{Lip}u_{i}$ is the
pointwise Lipschitz constant defined in (2.4).

###### Proof.

We will finish this proof by considering two cases as follows.
(i) In the case where $d(x_{1},\\{|{\bf u}|=0\\}\cap\Omega)=0$. The continuity
of ${\bf u}$ implies the $\\{|{\bf u}|=0\\}$ is relative closed in $\Omega$.
This implies $x_{1}\in\\{|{\bf u}|=0\\}$ in this case. By Lemma 5.6, we have

$\sup_{y\in B_{r/2}(x_{1})}\frac{|u_{i}(y)-u_{i}(x_{1})|}{r}\leqslant
C\sqrt{Q_{\max}},\quad\ i=1,2,\cdots,m,$

for all $r<R_{0}$. By (2.4) and letting $r\to 0$, this yields
$\operatorname{Lip}u_{i}(x_{1})\leqslant C\sqrt{Q_{\max}}.$

(ii) In the case where $d(x_{1},\\{|{\bf u}|=0\\}\cap\Omega)>0$. We put

$r_{1}:=d(x_{1},\\{|{\bf u}|=0\\}\cap\Omega)\in(0,R_{0}/8).$

Since $B_{r_{1}/2}(x_{1})\subset\\{|{\bf u}|>0\\}$, from Lemma 4.4, we have
known that all $u_{i}$, $i=1,\cdots,m,$ are harmonic on $B_{r_{1}/2}(x_{1})$.
By using Cheng-Yau estimate, Theorem 2.6, we obtain

(5.25) $\operatorname{Lip}u_{i}(x_{1})\leqslant\sup_{y\in
B_{r_{1}/4}(x_{1})}\operatorname{Lip}u_{i}(y)\leqslant\frac{C}{r_{1}}\sup_{y\in
B_{r_{1}/4}(x_{1})}u_{i}(y),$

where the constant $C$ depends only on $N,K$ and $\Omega$. Take $x_{2}\in
B_{2r_{1}}(x_{1})\cap\\{|{\bf u}|=0\\}$. By applying Lemma 5.6 to
$B_{8r_{1}}(x_{2})$, (remark that $d(x_{2},\Omega^{\prime})<2r$ and the
assumption $8r_{1}<R_{0}$,) we have

(5.26) $\sup_{y\in B_{r_{1}/4}(x_{1})}u_{i}(y)\leqslant\sup_{y\in
B_{4r_{1}}(x_{2})}u_{i}(y)\leqslant C\sqrt{Q_{\max}}\cdot r_{1}.$

The combination of (5.25) and (5.26) implies the desired estimate (5.24). Now
the proof is completed. ∎

Now we are in the position to show the local Lipschitz continuity of ${\bf
u}$.

###### Proof of Theorem 1.4.

Let $B_{R}(x)\subset\Omega$. Let $R_{0}$ be the constant given in Lemma 5.5
with respect to $\Omega^{\prime}:=B_{R/2}(x)$.

Take any $x_{1}\in B_{R/2}(x)$. If $d(x_{1},\Omega\cap\\{|{\bf
u}|=0\\})<R_{0}/8$, then Lemma 5.7 asserts

$\operatorname{Lip}u_{i}(x_{1})\leqslant C\sqrt{Q_{\max}},\qquad
i=1,2,\cdots,m,$

If $d(x_{1},\Omega\cap\\{|{\bf u}|=0\\})\geqslant R_{0}/8$, that is,
$B_{R_{0}/10}(x_{1})\subset\\{|{\bf u}|>0\\}$, then Cheng-Yau’s estimate,
Theorem 2.6, asserts

$\operatorname{Lip}u_{i}(x_{1})\leqslant\frac{C}{R_{0}}\sup_{\Omega}|{\bf
u}|,\qquad i=1,2,\cdots,m.$

By summing up the both cases and recalling Remark 4.2, we conclude that there
exists a constant $L$ depending only on
$N,K,\Omega,R,Q_{\max},\varepsilon_{\bf u}$ and $\int_{B_{R}(x)}|{\bf u}|{\rm
d}\mu$, such that

(5.27) $\sup_{B_{R/2}(x)}\operatorname{Lip}u_{i}(x)\leqslant L,\qquad\forall\
i=1,2,\cdots,m.$

Take any $y,z\in B_{R/4}(x).$ Let $\gamma:[0,d(x,y)]\to\Omega$ be a geodesic
from $y$ to $z$. The triangle inequality implies that $\gamma\subset
B_{R/2}(x)$. Noted that $\operatorname{Lip}u_{i}$ is one of the upper gradient
of $u_{i}$ (see [Che99]), the estimate (5.27) implies that

$|u_{i}(y)-u_{i}(z)|\leqslant\int_{0}^{d(y,z)}\operatorname{Lip}u_{i}\circ\gamma(s){\rm
d}s\leqslant L\cdot d(y,z),$

for each $i=1,2,\cdots,m$. The proof is finished. ∎

##  6\. Local finiteness of perimeter for the free boundary

We continue to assume that $(X,d,\mu)$ is an $ncRCD(K,N)$ metric measure space
with $K\leqslant 0$ and $N\in(1,+\infty)$. Let $\Omega\subset X$ be a bounded
domain and let $Q\in L^{\infty}(\Omega)$ satisfy (1.3) for two positive
numbers $Q_{\min}$ and $Q_{\max}$. Let

${\bf u}=(u_{1},u_{2},\cdots,u_{m})$

be a local minimizer of $J_{Q}$ in (1.2) with a boundary data ${\bf g}\in
W^{1,2}(\Omega,[0,+\infty)^{m})$, i.e., there exists $\varepsilon_{\bf u}>0$
such that $J_{Q}({\bf u})\leqslant J_{Q}({\bf v})$ for all ${\bf
v}\in\mathscr{A}_{\bf g}$ with $\operatorname{d}({\bf u},{\bf
v})<\varepsilon_{\bf u}$, where the $\mathscr{A}_{\bf g}$ and
$\operatorname{d}({\bf u},{\bf v})$ are given in (1.4) and (1.5),
respectively. From Theorem 1.4, we know that ${\bf u}$ is locally Lipschitz
continuous in the interior of $\Omega$.

We begin with the nondegeneracy of the local minimizer ${\bf u}$ near the free
boundary.

###  6.1. Nondegeneracy

######  Theorem 6.1 (Nondegeneracy).

Let ${\bf u}=(u_{1},\ldots,u_{m})$ be a local minimizer of $J_{Q}$ in (1.2)
with $Q$ satisfying (1.3), and let $\Omega^{\prime}\Subset\Omega.$ Then there
is a constant $R_{1}>0$ (depending only on
$N,K,\Omega^{\prime},Q_{\max},\varepsilon_{\bf u}$ and the Lipschitz constant
of ${\bf u}$ on $\Omega^{\prime}$) such that for any ball
$B_{r}(x_{0})\subset\Omega$ with $x_{0}\in\partial\\{\left|{\bf
u}\right|>0\\}\cap\Omega^{\prime}$ and $r<R_{1}$, it holds

(6.1) $\sup_{B_{r}(x_{0})}\left|{\bf u}\right|\geqslant c\sqrt{Q_{\min}}\cdot
r\text{,}$

where the positive constant $c$ depends only on
$N,K,\Omega^{\prime},\varepsilon_{\bf u}$ and the Lipschitz constant of ${\bf
u}$ on $\Omega^{\prime}$.

###### Proof.

In the Euclidean setting, this assertion was established in [CSY18, Theorem 3]
and [MTV17]. It was extended to smooth Riemannian manifolds in [LS20]. Here we
will extend their arguments to nonsmooth setting. Without loss of the
generality, we can assume that $K<0$.

Fix any $r\in(0,1)$ and let $M=\sup_{B_{r}(x_{0})}\left|{\bf u}\right|$. Since
${\bf u}$ is Lipschitz continuous on $\Omega^{\prime}$ (see Theorem 1.4) and
${\bf u}(x_{0})=0$, we get $M\leqslant L$, the Lipschitz constant of ${\bf u}$
on $\Omega^{\prime}$. Given any $\theta\in(0,1)$, as in [CSY18], we consider
the map ${\bf v}=(v_{1},\ldots,v_{m})$, where

(6.2)
$v_{i}(y)=\begin{cases}\min\left\\{u_{i}(y),M\psi_{\theta}(\frac{\rho(y)}{r})\right\\}&\text{if
}y\in B_{r}(x_{0})\text{,}\\\ u_{i}(y)&\text{if }y\not\in
B_{r}(x_{0})\text{,}\end{cases}$

for all $i=1,\ldots,m$ and $y\in\Omega$, where $\rho(\cdot)=d(\cdot,x_{0})$,

(6.3)
$\psi_{\theta}(t)=\frac{(\phi_{N,K}(t)-\phi_{N,K}(\theta))^{+}}{\phi_{N,K}(1)-\phi_{N,K}(\theta)}$

and

$\phi_{N,K}(s)=-\int^{1}_{s}\left(\frac{\sinh(\sqrt{-K/(N-1)}t)}{\sqrt{-K/(N-1)}}\right)^{1-N}\mathrm{d}t.$

Then it is clear that ${\bf v}\in\mathscr{A}_{\mathbf{g}}$ and $u_{i}-v_{i}\in
W^{1,2}_{0}(B_{r}(x_{0}))$ for all $i=1,2,\cdots,m$.

(i) We first check that $\operatorname{d}({\bf v},{\bf u})<\varepsilon_{\bf
u}$ provided both $r$ and $\theta$ are sufficiently small. The co-area formula
gives

$\begin{split}\int_{B_{r}(x_{0})\setminus B_{\theta
r}(x_{0})}\left|\nabla\psi_{\theta}(\frac{\rho}{r})\right|^{2}{\rm
d}\mu&\leqslant\int_{\theta
r}^{r}\left|\frac{1}{r}\psi^{\prime}_{\theta}(\frac{s}{r})\right|^{2}\cdot\frac{d^{+}}{dr}\mu(B_{s}(x_{0})){\rm
d}s.\end{split}$

Since $(X,d,\mu)$ is non-collapsed, substituting (2.14) into the above
inequality, we obtain

(6.4) $\begin{split}\int_{B_{r}(x_{0})\setminus B_{\theta
r}(x_{0})}\left|\nabla\psi_{\theta}(\frac{\rho}{r})\right|^{2}{\rm
d}\mu&\leqslant C_{1}r^{N-2}\cdot\left(\int_{\theta}^{1}s^{1-N}{\rm
d}s\right)^{-1},\end{split}$

where we have used $|\psi_{\theta}^{\prime}(t)|\leqslant
C_{N,K}\frac{t^{1-N}}{|\phi_{N,K}(1)-\phi_{N,K}(\theta)|}\leqslant\frac{C^{\prime}_{N,K}\cdot
t^{1-N}}{\int_{\theta}^{1}s^{1-N}{\rm d}s}$ for all $t\in(\theta,1)$. Here and
in the following of this proof, all constants $C_{1},C_{2},\cdots$ depend only
on $N,K,\Omega^{\prime}$.

Noticed that $|\psi_{\theta}|\leqslant 1$ and
$u_{i}-v_{i}=(u_{i}-M\psi_{\theta}(\frac{\rho}{r}))^{+}\in
W^{1,2}_{0}(B_{r}(x_{0}))$, we get

(6.5)
$\int_{\Omega}\left|u_{i}-v_{i}\right|^{2}\leqslant\int_{B_{r}(x_{0})}((u_{i}-M\psi_{\theta}(\frac{\rho}{r}))^{+})^{2}\leqslant
M^{2}\mu(B_{r}(x_{0}))$

and (by the fact that [Che99], for any $w\in W^{1,2}(\Omega)$, $|\nabla
w^{+}|\leqslant|\nabla w|$ holds almost everywhere in $\Omega$,)

(6.6) $\displaystyle\int_{\Omega}\left|\nabla(u_{i}-v_{i})\right|^{2}$
$\displaystyle\leqslant\int_{B_{r}(x_{0})}\left|\nabla(u_{i}-M\psi_{\theta}(\frac{\rho}{r}))\right|^{2}$
$\displaystyle\leqslant 2\int_{B_{r}(x_{0})\setminus B_{\theta
r}(x_{0})}\left|\nabla(M\psi_{\theta}(\frac{\rho}{r}))\right|^{2}+2\int_{B_{r}(x_{0})}\left|\nabla
u_{i}\right|^{2}$ $\displaystyle\leqslant 2M^{2}\int_{B_{r}(x_{0})\setminus
B_{\theta
r}(x_{0})}\left|\nabla\psi_{\theta}(\frac{\rho}{r})\right|^{2}+2L^{2}\mu(B_{r}(x_{0})),$

where for the second inequality we have used $\psi_{\theta}=0$ on $B_{\theta
r}(x_{0})$, and for the last inequality we have used $|\nabla u_{i}|\leqslant
L$. Recall $M\leqslant L$.

From the combination of (6.4)–(6.6), the fact $M\leqslant L$, ${\bf u}-{\bf
v}\in W^{1,2}_{0}(B_{r}(x_{0}),\mathbb{R}^{m})$, and taking $\theta$ such that

(6.7) $C_{1}\Big{(}\int_{\theta}^{1}s^{1-N}{\rm
d}s\Big{)}^{-1}=\frac{\varepsilon_{\bf u}}{8L^{2}},$

we conclude that $\operatorname{d}({\bf v},{\bf u})<\varepsilon_{\bf u}$
provided $r<R_{1}$ for some small number $R_{1}>0$ depending only on
$N,K,L,Q_{\max}$ and $\varepsilon_{\bf u}$.

(ii) Fixed any $r\in(0,R_{1})$ and taken $\theta$ in (6.7), the local
minimality of ${\bf u}$ gives

$\displaystyle\int_{B_{\theta r}(x_{0})}(\left|\nabla{\bf
u}\right|^{2}+Q\chi_{\left\\{\left|{\bf u}\right|>0\right\\}})$
$\displaystyle=$ $\displaystyle\int_{B_{r}(x_{0})}(\left|\nabla{\bf
u}\right|^{2}+Q\chi_{\left\\{\left|{\bf
u}\right|>0\right\\}})-\int_{B_{r}(x_{0})\setminus B_{\theta
r}(x_{0})}(\left|\nabla{\bf u}\right|^{2}+Q\chi_{\left\\{\left|{\bf
u}\right|>0\right\\}})$ $\displaystyle\leqslant$
$\displaystyle\int_{B_{r}(x_{0})}(\left|\nabla{\bf
v}\right|^{2}+Q\chi_{\left\\{\left|{\bf
v}\right|>0\right\\}})-\int_{B_{r}(x_{0})\setminus B_{\theta
r}(x)}(\left|\nabla{\bf u}\right|^{2}+Q\chi_{\left\\{\left|{\bf
u}\right|>0\right\\}})$ $\displaystyle=$
$\displaystyle\int_{B_{r}(x_{0})\setminus B_{\theta
r}(x_{0})}(\left|\nabla{\bf v}\right|^{2}+Q\chi_{\left\\{\left|{\bf
v}\right|>0\right\\}})-\int_{B_{r}(x_{0})\setminus B_{\theta
r}(x_{0})}(\left|\nabla{\bf u}\right|^{2}+Q\chi_{\left\\{\left|{\bf
u}\right|>0\right\\}}),$

where we used that ${\bf v}\llcorner B_{\theta r}(x_{0})\equiv 0$. Noticed
that $(B_{r}(x_{0})\setminus B_{\theta r}(x_{0}))\cap\\{|{\bf
u}|>0\\}=(B_{r}(x_{0})\setminus B_{\theta r}(x_{0}))\cap\\{|{\bf v}|>0\\}$,
this yields

(6.8) $\displaystyle\int_{B_{\theta r}(x_{0})}(\left|\nabla{\bf
u}\right|^{2}+Q\chi_{\left\\{\left|{\bf u}\right|>0\right\\}})\leqslant$
$\displaystyle\int_{B_{r}(x_{0})\setminus B_{\theta
r}(x_{0})}(\left|\nabla{\bf v}\right|^{2}-\left|\nabla{\bf u}\right|^{2})$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{m}\int_{B_{r}(x_{0})\setminus
B_{\theta r}(x_{0})}(\left|\nabla v_{i}\right|^{2}-\left|\nabla
u_{i}\right|^{2})\text{.}$

Let $w_{i}=(u_{i}-M\psi_{\theta}(\frac{\rho}{r}))^{+}$ for each
$i=1,2,\cdots,m$. Then $u_{i}=v_{i}+w_{i}$ and

(6.9) $\displaystyle\int_{B_{r}(x_{0})\setminus B_{\theta
r}(x_{0})}(\left|\nabla v_{i}\right|^{2}-\left|\nabla u_{i}\right|^{2})$
$\displaystyle=$ $\displaystyle\int_{B_{r}(x_{0})\setminus B_{\theta
r}(x_{0})}\Big{(}-2\left\langle{\nabla v_{i}},{\nabla
w_{i}}\right\rangle-\left|\nabla w_{i}\right|^{2}\Big{)}$
$\displaystyle\leqslant$ $\displaystyle-2\int_{B_{r}(x_{0})\setminus B_{\theta
r}(x_{0})}\left\langle{\nabla v_{i}},{\nabla w_{i}}\right\rangle$
$\displaystyle=$ $\displaystyle-2\int_{B_{r}(x_{0})\setminus B_{\theta
r}(x_{0})}\left\langle{\nabla(M\psi_{\theta}(\frac{\rho}{r}))},{\nabla
w_{i}}\right\rangle,$

for each $i=1,\ldots,m$, where for the last equality, we have used that
$\left|\nabla w_{i}\right|=0$ $\mu$-a.e. on $\left\\{w_{i}=0\right\\}\cap
B_{r}(x_{0})$ (see [Che99] or Proposition 2.1(3)) and that
$v_{i}=M\psi_{\theta}(\frac{\rho}{r})$ on $\\{w_{i}\not=0\\}\cap
B_{r}(x_{0})$. Remark that $\\{w_{i}\not=0\\}$ is open by the continuity of
$w_{i}$.

(iii) Next we want to estimate $\sum_{i=1,}^{m}I_{\theta r,r}(w_{i}),$ where
we denote

(6.10) $I_{r_{1},r_{2}}(w_{i}):=-\int_{B_{r_{2}}(x_{0})\setminus
B_{r_{1}}(x_{0})}\left\langle{\nabla(M\psi_{\theta}(\frac{\rho}{r}))},{\nabla
w_{i}}\right\rangle$

for any $r_{1},r_{2}\in\left[\theta r,r\right]$ with $r_{1}<r_{2}$.

By the Laplacian comparison theorem (see (2.7) in Theorem 2.5) and that the
space $(M,\frac{d}{r})$ satisfies $RCD(K\cdot r^{2},N)$, we conclude that

$\boldsymbol{\Delta}\psi_{\theta}(\frac{\rho}{r})\leqslant 0\quad\mathrm{on}\
B_{r}(x_{0})\setminus\overline{B_{\theta r}(x_{0})}$

in the sense of distributions. From Remark 5.2, we have for almost all
$r_{1},r_{2}\in(\theta r,r)$ with $r_{1}<r_{2}$, that

(6.11) $\displaystyle I_{r_{1},r_{2}}(w_{i})=$
$\displaystyle\int_{B_{r_{2}}(x_{0})\setminus B_{r_{1}}(x_{0})}w_{i}{\rm
d}\boldsymbol{\Delta}(M\psi_{\rho}(\frac{\rho}{r}))$
$\displaystyle+M\frac{\psi_{\theta}^{\prime}(\frac{r_{1}}{r})}{r}\left.\frac{\mathrm{d}}{\mathrm{d}s}\right|_{s=r_{1}}\int_{B_{s}(x_{0})}w_{i}-M\frac{\psi_{\theta}^{\prime}(\frac{r_{2}}{r})}{r}\left.\frac{\mathrm{d}}{\mathrm{d}s}\right|_{s=r_{2}}\int_{B_{s}(x_{0})}w_{i}$
$\displaystyle\leqslant$ $\displaystyle
C_{N,K,\theta}\frac{M}{r}\left.\frac{\mathrm{d}}{\mathrm{d}s}\right|_{s=r_{1}}\int_{B_{s}(x_{0})}w_{i},$

where we have used that $0\leqslant\phi^{\prime}_{N,K}(t)\leqslant
C_{N,K,\theta}$ for all $t\in(\theta,1)$ and that
$\frac{\mathrm{d}}{\mathrm{d}s}\int_{B_{s}(x_{0})}w_{i}\geqslant 0$ for almost
all $s\in(\theta r,r).$

For almost every $r_{1}\in(\theta r,r)$ such that both
$s\mapsto\int_{B_{s}(x_{0})}w_{i}$ and $s\mapsto\int_{B_{s}(x_{0})}u_{i}$ are
differentiable at $r_{1}$, we have

(6.12)
$\begin{split}\left.\frac{\mathrm{d}}{\mathrm{d}s}\right|_{s=r_{1}}\int_{B_{s}(x_{0})}w_{i}-\left.\frac{\mathrm{d}}{\mathrm{d}s}\right|_{s=r_{1}}\int_{B_{s}(x_{0})}u_{i}=&\lim_{\delta\to
0^{+}}\frac{1}{\delta}\int_{A_{r_{1},r_{1}+\delta}}\big{(}u_{i}-w_{i}\big{)}\\\
\leqslant&\lim_{\delta\to
0^{+}}\frac{1}{\delta}\int_{A_{r_{1},r_{1}+\delta}}\big{|}u_{i}-w_{i}\big{|},\end{split}$

where $A_{r_{1},r_{1}+\delta}:=B_{r_{1}+\delta}(x_{0})\backslash
B_{r_{1}}(x_{0})$.

From the definition $w_{i}=(u_{i}-M\psi_{\theta}(\frac{\rho}{r}))^{+}$,
$M\leqslant L$ and $|\psi^{\prime}_{\theta}(t)|\leqslant C_{N,K,\theta}$ in
$(\theta,1)$, it follows that $(u_{i}-w_{i})$ is Lipschitz continuous on
$B_{r}(x_{0})$ with a Lipschitz constant $C_{L,N,K,\theta}>0$. By using
$(u_{i}-w_{i})(y)=0$ for any $y\in\partial B_{\theta r}(x_{0})$, we conclude
that

(6.13) $\sup_{A_{r_{1},r_{1}+\delta}}\big{|}u_{i}-w_{i}\big{|}\leqslant
C_{L,N,K,\theta}\cdot(r_{1}+\delta-\theta r).$

Substituting this into (6.12), we have

(6.14)
$\begin{split}&\left.\frac{\mathrm{d}}{\mathrm{d}s}\right|_{s=r_{1}}\int_{B_{s}(x_{0})}w_{i}-\left.\frac{\mathrm{d}}{\mathrm{d}s}\right|_{s=r_{1}}\int_{B_{s}(x_{0})}u_{i}\\\
\leqslant&\lim_{\delta\to 0^{+}}C_{L,N,K,\theta}\cdot(r_{1}+\delta-\theta
r)\cdot\frac{\mu(B_{r_{1}+\delta}(x_{0})\backslash
B_{r_{1}}(x_{0}))}{\delta}\\\
\leqslant&C^{\prime}_{L,N,K,\theta,\Omega^{\prime}}\cdot(r_{1}-\theta
r),\end{split}$

where for the last inequality we have used (2.2).

On the other hand, by using Lemma 5.1 and Theorem 2.5 (the Laplacian
comparison theorem), we have for almost all $r_{1}\in(\theta r,3\theta r/2)$
that

$\displaystyle\left.\frac{\mathrm{d}}{\mathrm{d}s}\right|_{s=r_{1}}\int_{B_{s}(x_{0})}u_{i}$
$\displaystyle=\frac{1}{2r_{1}}\left(\int_{B_{r_{1}}(x_{0})}u_{i}{\rm
d}\boldsymbol{\Delta}(\rho^{2})+\int_{B_{r_{1}}(x_{0})}\left\langle\nabla
u_{i},\nabla\rho^{2}\right\rangle\right)$
$\displaystyle\leqslant\frac{C_{N,K}}{2r_{1}}\int_{B_{r_{1}}(x_{0})}u_{i}+\int_{B_{r_{1}}(x_{0})}\left|\nabla
u_{i}\right|$
$\displaystyle\leqslant\frac{C_{N,K,\theta}}{r}\int_{B_{r_{1}}(x_{0})}\left|{\bf
u}\right|+\int_{B_{r_{1}}(x_{0})}\left|\nabla{\bf u}\right|,$

where we have used $u_{i}\geqslant 0$ and $r_{1}\geqslant\theta r$. This
implies for almost all $r_{1}\in(\theta r,3\theta r/2)$ that

(6.15)
$\displaystyle\sum_{i=1}^{m}\left.\frac{\mathrm{d}}{\mathrm{d}s}\right|_{s=r_{1}}\int_{B_{s}(x_{0})}u_{i}$
$\displaystyle\leqslant$ $\displaystyle
C_{N,K,m,\theta}\int_{B_{r_{1}}(x_{0})}\Big{(}\frac{1}{r}\left|{\bf
u}\right|+\left|\nabla{\bf u}\right|\Big{)}$ $\displaystyle=$ $\displaystyle
C_{N,K,m,\theta}\int_{B_{r_{1}}(x_{0})}\Big{(}\frac{1}{r}\left|{\bf
u}\right|\chi_{\left\\{\left|{\bf u}\right|>0\right\\}}+\left|\nabla{\bf
u}\right|\chi_{\left\\{\left|{\bf u}\right|>0\right\\}}\Big{)}$
$\displaystyle\leqslant$ $\displaystyle
C_{N,K,m,\theta}\int_{B_{r_{1}}(x_{0})}\Big{(}\frac{M}{r}\chi_{\left\\{\left|{\bf
u}\right|>0\right\\}}+\frac{1}{2\sqrt{Q_{\min}}}(Q_{\min}\chi^{2}_{\left\\{\left|{\bf
u}\right|>0\right\\}}+\left|\nabla{\bf u}\right|^{2})\Big{)}$
$\displaystyle\leqslant$ $\displaystyle
C_{N,K,m,\theta}\int_{B_{r_{1}}(x_{0})}\Big{(}\frac{M}{Q_{\min}r}Q\chi_{\left\\{\left|{\bf
u}\right|>0\right\\}}+\frac{1}{2\sqrt{Q_{\min}}}(Q\chi_{\left\\{\left|{\bf
u}\right|>0\right\\}}+\left|\nabla{\bf u}\right|^{2})\Big{)}$
$\displaystyle\leqslant$ $\displaystyle
C_{N,K,m,\theta}\Big{(}\frac{M}{Q_{\min}r}+\frac{1}{2\sqrt{Q_{\min}}}\Big{)}\int_{B_{r_{1}}(x_{0})}\Big{(}Q\chi_{\left\\{\left|{\bf
u}\right|>0\right\\}}+\left|\nabla{\bf u}\right|^{2}\Big{)}\text{,}$

where for the second inequality we have used $|\nabla{\bf u}|=0$ $\mu$-a.e. in
$\\{|{\bf u}|=0\\}$ ([Che99]), and for the third inequality we have used
$\sup_{B_{r_{1}}(x_{0})}\left|{\bf u}\right|\leqslant M$. From the combination
of (6.11), (6.14), (6.15) and the fact

$\lim_{r_{1}\to\theta r}I_{\theta r,r_{1}}(w_{i})=0,\qquad\lim_{r_{2}\to
r}I_{r_{2},r}(w_{i})=0,$

by letting $r_{1}\to\theta r$ and $r_{2}\to r$, we conclude

(6.16) $\displaystyle\sum_{i=1}^{m}I_{\theta
r,r}(w_{i})=\sum_{i=1}^{m}\Big{(}I_{\theta
r,r_{1}}(w_{i})+I_{r_{1},r_{2}}(w_{i})+I_{r_{2},r}(w_{i})\Big{)}$
$\displaystyle\leqslant$ $\displaystyle\lim_{r_{1}\to\theta
r}\sum_{i=1}^{m}C_{N,K,\theta}\frac{M}{r}\left.\frac{\mathrm{d}}{\mathrm{d}s}\right|_{s=r_{1}}\int_{B_{s}(x_{0})}u_{i}$
$\displaystyle\leqslant$ $\displaystyle
C^{\prime}_{N,K,m,\theta}\frac{M}{r}\Big{(}\frac{M}{Q_{\min}r}+\frac{1}{2\sqrt{Q_{\min}}}\Big{)}\int_{B_{\theta
r}(x_{0})}\Big{(}Q\chi_{\left\\{\left|{\bf
u}\right|>0\right\\}}+\left|\nabla{\bf u}\right|^{2}\Big{)}.$

(iv) At last, the assumption $x_{0}\in\partial\\{\left|{\bf u}\right|>0\\}$
and the continuity of ${\bf u}$ implies

$\int_{B_{\theta r}(x_{0})}\Big{(}\left|\nabla{\bf
u}\right|^{2}+Q\chi_{\left\\{\left|{\bf u}\right|>0\right\\}}\Big{)}\geqslant
Q_{\min}\cdot\mu\big{(}B_{\theta r}(x_{0})\cap\\{|{\bf u}|>0\\}\big{)}>0.$

Thus, by combining this with (6.8), (6.9), (6.10) and (6.16), we obtain

$1\leqslant C^{\prime}_{N,K,m,\theta}\frac{2M}{r}\Big{(}\frac{M}{Q_{\min}\cdot
r}+\frac{1}{2\sqrt{Q_{\min}}}\Big{)}.$

This implies

$\frac{M}{\sqrt{Q_{\min}}\cdot
r}\geqslant\min\left\\{\frac{1}{2},\frac{1}{2C^{\prime}_{N,K,m,\theta}}\right\\}\text{.}$

This is the desired estimate (6.1), since $M=\sup_{B_{r}(x)}\left|{\bf
u}\right|$ and $\theta$ is given in (6.7). The proof is finished. ∎

######  Remark 6.2.

If ${\bf u}$ is an absolute minimizer of $J_{Q}$, the previous proof (step
(ii)–(iv)) still works for general $RCD(K,N)$-spaces. That is:

_Let $(X,d,\mu)$ be an $RCD(K,N)$-space with some $K<0$ and $N\in(1,+\infty)$,
and let ${\bf u}$ be an absolute minimizer of $J_{Q}$ in (1.2). Let
$\Omega^{\prime}\Subset\Omega.$ If ${\bf u}$ is Lipschitz continuous on
$\Omega^{\prime}$, then for any ball $B_{r}(x_{0})\subset\Omega$ with
$x_{0}\in\partial\\{\left|{\bf u}\right|>0\\}\cap\Omega^{\prime}$, it holds
(6.1) for a positive constant $c$ depending only on $m,N$ and $K$._

###  6.2. Density estimates near the free boundary

In the subsection, we will show that both $\\{|{\bf u}|>0\\}$ and $\\{|{\bf
u}|=0\\}$ have positive density along the free boundary.

######  Lemma 6.3.

Let ${\bf u}=(u_{1},\ldots,u_{m})$ be a local minimizer of $J_{Q}$ in (1.2)
with $Q$ satisfying (1.3), and let $\Omega^{\prime}\Subset\Omega.$ Then for
any ball $B_{r}(x_{0})\subset\Omega^{\prime}$ with
$x_{0}\in\Omega^{\prime}\cap\partial\\{|{\bf u}|>0\\}$ and $r<2R_{1}$, where
$R_{1}$ is given in Theorem 6.1, we have

(6.17) $\mu\big{(}B_{r}(x_{0})\cap\\{|{\bf u}|>0\\}\big{)}\geqslant
c_{1}\cdot\mu(B_{r}(x_{0}))$

for some constant $c_{1}$ depending on
$m,N,K,\Omega^{\prime},Q_{\min},\varepsilon_{\bf u}$ and $L$, the Lipschitz
constant of ${\bf u}$ on $\Omega^{\prime}$.

###### Proof.

From Theorem 6.1, there exists some $i_{0}\in\\{1,2,\cdots,m\\}$ such that

$\sup_{B_{r/2}(x_{0})}u_{i_{0}}\geqslant\frac{c\sqrt{Q_{\min}}}{m}\cdot r/2$

for all $r/2<R_{1}$, where $R_{1}$ and $c$ are given in Theorem 6.1. Choose
$y_{0}\in B_{r/2}(x_{0})$ such that $u_{i_{0}}(y_{0})\geqslant c_{2}r/2$,
where $c_{2}:=\frac{c\sqrt{Q_{\min}}}{2m}$. Since $u_{i_{0}}$ is Lipschitz
continuous on $B_{r/2}(y_{0})\subset\Omega^{\prime}$ with a Lipschitz constant
$L$, we have

$\inf_{B_{c_{3}r}(y_{0})}u_{i_{0}}\geqslant u_{i_{0}}(y_{0})-c_{3}Lr\geqslant
c_{2}r/4,\qquad c_{3}:=\min\big{\\{}\frac{1}{2},\frac{c_{2}}{8L}\big{\\}}.$

In particular, this yields $B_{c_{3}r}(y_{0})\subset\\{|{\bf u}|>0\\}\cap
B_{r}(x_{0})$. It follows

$\mu\big{(}B_{r}(x_{0})\cap\\{|{\bf
u}|>0\\}\big{)}\geqslant\mu(B_{c_{3}r}(y_{0})).$

By combining this and the Bishop-Gromov inequality

$\mu(B_{c_{3}r}(y_{0}))\geqslant
c_{N,K}{(c_{3}/2)^{N}}\mu(B_{2r}(y_{0}))\geqslant
c_{N,K}{(c_{3}/2)^{N}}\mu(B_{r}(x_{0})),$

we get the desired estimate (6.17). ∎

######  Lemma 6.4.

Let ${\bf u}=(u_{1},\ldots,u_{m})$ be a local minimizer of $J_{Q}$ in (1.2)
with $Q$ satisfying (1.3), and let $\Omega^{\prime}\Subset\Omega.$ Then there
exists a number $R_{0}\in(0,1)$ depending only on
$N,K,\Omega^{\prime},Q_{\max}$ and ${\bf u}$, such that for any ball
$B_{r}(x_{0})\subset\Omega^{\prime}$ with
$x_{0}\in\Omega^{\prime}\cap\partial\\{|{\bf u}|>0\\}$ and $r<R_{0}$, we have

(6.18) $\mu\big{(}B_{r}(x_{0})\cap\\{|{\bf u}|=0\\}\big{)}\geqslant
c_{4}\cdot\mu(B_{r}(x_{0}))$

for some constant $c_{4}$ depending on
$m,N,K,\Omega^{\prime},Q_{\min},Q_{\max},\varepsilon_{\bf u}$ and the
Lipschitz constant of ${\bf u}$ on $\Omega^{\prime}$.

###### Proof.

Let ${\bf v}=(v_{1},v_{2},\cdots,v_{m})\in
W^{1,2}(B_{r}(x_{0}),\mathbb{R}^{m})$ be the map such that each component
$v_{i}$ is the (unique) solution of the following (relaxed) Dirichlet problem
[Che99, Theorem 7.12]:

(6.19) $\begin{cases}\boldsymbol{\Delta}v_{i}=0\quad\text{ on }\
B_{r}(x_{0}),\\\ v_{i}-u_{i}\in W_{0}^{1,2}(B_{r}(x_{0})).\end{cases}$

After extending ${\bf v}$ by ${\bf u}$ on $\Omega\backslash B_{r}(x_{0})$, we
have ${\bf v}\in\mathscr{A}_{\mathbf{g}}$ because all $v_{i}\geqslant 0$ on
$B_{R}(x_{0})$ by the maximum principle (see [Che99, Theorem 7.17]). In the
proof of Lemma 4.3, it was showed that there is $R_{0}\in(0,1)$ (depending
only on $N,K,\Omega^{\prime}$ and ${\bf u}$) such that $\operatorname{d}({\bf
u},{\bf v})<\varepsilon_{\bf u}$ for all $r\in(0,R_{0})$.

The local minimality of ${\bf u}$ implies

(6.20) $\int_{B_{r}(x_{0})}(|\nabla{\bf u}|^{2}-|\nabla{\bf
v}|^{2})\leqslant\int_{B_{r}(x_{0})}Q(\chi_{\left\\{\left|{\bf
v}\right|>0\right\\}}-\chi_{\left\\{\left|{\bf u}\right|>0\right\\}})$

for all $r\in(0,R_{0})$. From the harmonicity of ${\bf v}$, the same argument
in (4.9) gives

(6.21) $\int_{B_{r}(x_{0})}(|\nabla{\bf u}|^{2}-|\nabla{\bf
v}|^{2})=\int_{B_{r}(x_{0})}\left|\nabla({\bf u}-{\bf v})\right|^{2}.$

Note that $|{\bf v}|$ is not identical zero on $B_{r}(x_{0})$ by Theorem 6.1.
We get from the strong maximum principle (see, for example, [KS01, Corollary
6.4]) that $|{\bf v}|>0$ on $B_{r}(x_{0})$. Hence,

(6.22) $\begin{split}\int_{B_{r}(x_{0})}Q(\chi_{\left\\{\left|{\bf
v}\right|>0\right\\}}-\chi_{\left\\{\left|{\bf
u}\right|>0\right\\}})&=\int_{B_{r}(x_{0})}Q\chi_{\left\\{\left|{\bf
u}\right|=0\right\\}}\\\ &\leqslant
Q_{\max}\cdot\mu\big{(}B_{r}(x_{0})\cap\\{|{\bf u}|=0\\}\big{)}.\end{split}$

By combining (6.20)–(6.22) and the Poincaré inequality (see Proposition
2.1(5)), we get

(6.23) $\int_{B_{r}(x_{0})}|{\bf u}-{\bf v}|^{2}\leqslant C_{P}\cdot
r^{2}\cdot Q_{\max}\cdot\mu\big{(}B_{r}(x_{0})\cap\\{|{\bf u}|=0\\}\big{)},$

where the Poincaré constant $C_{P}$ depends only on $N,K$ and
$\Omega^{\prime}$.

From the nondegeneracy Theorem 6.1, there exists some
$i_{0}\in\\{1,2,\cdots,m\\}$ such that
$\sup_{B_{r/2}(x_{0})}u_{i_{0}}\geqslant c\sqrt{Q_{\min}}r/2.$ Recalling
${\bf\Delta}(u_{i_{0}}-v_{i_{0}})\geqslant 0$ on $B_{r}(x_{0})$ in the sense
of distributions, the maximum principle implies that $v_{i_{0}}\geqslant
u_{i_{0}}$ on $B_{r}(x_{0})$. Hence, from this and the Harnack inequality, we
have

(6.24) $\inf_{B_{r/2}(x_{0})}v_{i_{0}}\geqslant
C_{1}\sup_{B_{r/2}(x_{0})}v_{i_{0}}\geqslant
C_{1}\sup_{B_{r/2}(x_{0})}u_{i_{0}}\geqslant C_{1}c\sqrt{Q_{\min}}r/2,$

where the constant $C_{1}$ depends only on $N,K,\Omega^{\prime}.$ Since
$u_{i_{0}}(x_{0})=0$ and that $u_{i_{0}}$ is Lipschitz continuous on
$B_{r}(x_{0})\subset\Omega^{\prime}$ with a Lipschitz constant $L$, we have

$\sup_{B_{c_{1}r}(x_{0})}u_{i_{0}}\leqslant Lc_{1}r,\quad{\rm with}\ \
c_{1}:=\min\Big{\\{}\frac{C_{1}c\sqrt{Q_{\min}}}{4L},\frac{1}{4}\Big{\\}}.$

Combining this with (6.24), we conclude that

$\inf_{B_{c_{1}r}(x_{0})}(v_{i_{0}}-u_{i_{0}})\geqslant\frac{C_{1}c\sqrt{Q_{\min}}}{4}r:=C_{2}\sqrt{Q_{\min}}\cdot
r,$

where the constant $C_{2}$ depends on $m,N,K,\Omega^{\prime},\varepsilon_{\bf
u}$ and $L$. This yields

(6.25) $\int_{B_{c_{1}r}(x_{0})}|{\bf u}-{\bf
v}|^{2}\geqslant\int_{B_{c_{1}r}(x_{0})}(v_{i_{0}}-u_{i_{0}})^{2}\geqslant
C_{2}^{2}Q_{\min}\cdot r^{2}\cdot\mu(B_{c_{1}r}(x_{0})).$

From (6.23), (6.25) and that $\mu(B_{c_{1}r}(x_{0}))\geqslant
C_{N,K}c_{1}^{N}\cdot\mu(B_{r}(x_{0}))$, it follows

$C_{P}Q_{\max}\cdot\mu\big{(}B_{r}(x_{0})\cap\\{|{\bf u}|=0\\}\big{)}\geqslant
C_{2}^{2}Q_{\min}C_{N,K}c_{1}^{N}\cdot\mu(B_{r}(x_{0})).$

This is the desired (6.18) with the constant
$c_{4}:=C_{P}^{-1}C^{2}_{2}C_{N,K}c_{1}^{N}\cdot Q_{\min}/Q_{\max}$. The proof
is finished. ∎

###  6.3. Local finiteness of perimeter

In this subsection, we will derive the local finiteness of perimeter of the
free boundary in the sense of [Amb01, Mir03, ABS19], via the estimates of
density, Lemma 6.3 and Lemma 6.4, and a similar argument in [MTV20]. We need
an estimate on perimeter which is similar to the one in [MTV20, Lemma 2.4].

######  Lemma 6.5.

Let $(X,d,\mu)$ be an $RCD(K,N)$ space with $K\in\mathbb{R}$ and
$N\in(1,+\infty)$, and let $D\subset X$ be an open subset. Suppose that
$0\leqslant\psi\in\operatorname{Lip}_{\mathrm{loc}}(D)\cap W^{1,1}(D)$. If
there are positive constants $\bar{\varepsilon}$ and $C$ such that

(6.26) $\int_{\left\\{0<\psi<\varepsilon\right\\}\cap
D}\left|\nabla\psi\right|\leqslant C\varepsilon$

for all $\varepsilon\in(0,\bar{\varepsilon})$, then

(6.27) $\mathcal{P}(\left\\{\psi>0\right\\},D)\leqslant 2C\text{.}$

###### Proof.

Given $\varepsilon\in(0,\bar{\varepsilon}/2)$, then, by the coarea formula
[Amb01, Theorem 3.3] (see also Proposition 2.15(2)) to $\chi_{D}$, we have

(6.28)
$\displaystyle\int_{\varepsilon}^{2\varepsilon}\mathcal{P}(\left\\{\psi>t\right\\},D)\mathrm{d}t$
$\displaystyle=\int_{\varepsilon\leqslant\psi<2\varepsilon}\chi_{D}{\rm
d}|D\psi|=\int_{\left\\{\varepsilon\leqslant\psi<2\varepsilon\right\\}\cap
D}\left|\nabla\psi\right|$
$\displaystyle\leqslant\int_{\left\\{0<\psi<2\varepsilon\right\\}\cap
D}\left|\nabla\psi\right|\leqslant 2C\varepsilon.$

For each integer $j$ so large that $2^{-j}\leqslant\bar{\varepsilon}/2$, by
setting $\varepsilon_{j}=2^{-j}$, there exists some
$\varepsilon^{\prime}_{j}\in(\varepsilon_{j},2\varepsilon_{j})$ such that

(6.29)
$\mathcal{P}(\left\\{\psi>\varepsilon^{\prime}_{j}\right\\},D)\leqslant\varepsilon_{j}^{-1}\int_{\varepsilon_{j}}^{2\varepsilon_{j}}\mathcal{P}(\left\\{\psi>t\right\\},D)\mathrm{d}t\leqslant\varepsilon_{j}^{-1}{2C\varepsilon_{j}}\leqslant{2C}\text{.}$

It is clear that
$\chi_{\left\\{\psi>\varepsilon^{\prime}_{j}\right\\}}\xrightarrow{L_{\mathrm{loc}}^{1}(D)}\chi_{\left\\{\psi>0\right\\}},$
by Lebesgue’s dominated converge theorem. From (6.29) and the lower
semicontinuity of perimeter [Amb01], it follows that

(6.30)
$\mathcal{P}(\left\\{\psi>0\right\\},D)\leqslant\liminf_{j\to\infty}\mathcal{P}(\left\\{\psi>\varepsilon^{\prime}_{j}\right\\},D)\leqslant{2C}\text{.}$

This finishes the proof. ∎

Now let us check the condition (6.26) of the above lemma for the local
minimizers of $J_{Q}$.

######  Lemma 6.6.

Let ${\bf u}=(u_{1},\ldots,u_{m})$ be a local minimizer of $J_{Q}$ in (1.2)
with $Q$ satisfying (1.3), and let $\Omega^{\prime}\Subset\Omega$. Then there
exist $\bar{r}>0,\bar{\varepsilon}>0$ and $C>0$ such that

(6.31) $\int_{\left\\{\left|{\bf u}\right|\leqslant\varepsilon\right\\}\cap
B_{r/2}(x_{0})}\left|\nabla{\bf u}\right|^{2}+\mu\big{(}\\{0<|{\bf
u}|\leqslant\varepsilon\\}\cap B_{r/2}(x_{0})\big{)}\leqslant
C\mu\big{(}B_{r}(x_{0})\big{)}\cdot\frac{\varepsilon}{r^{2}}$

for all $\varepsilon\in(0,\bar{\varepsilon})$, $r\in(0,\bar{r})$ and
$B_{r}(x_{0})\subset\Omega^{\prime}$. Here the constants
$\bar{r},\bar{\varepsilon},C$ depend only on $N,K,\Omega^{\prime},{\rm
diam}(\Omega),Q_{\min},\varepsilon_{\bf u}$ and $L$, the Lipschitz constant of
${\bf u}$ on $\Omega^{\prime}$.

###### Proof.

Fix any ball $B_{r}(x_{0})\subset\Omega^{\prime}$ such that $r\in(0,\bar{r})$,
where $\bar{r}<1$ will be determined later. Let $\phi:M\to\left[0,1\right]$ be
a cutoff function with $\phi\equiv 1$ on $B_{r/2}(x_{0})$,
$\operatorname{supp}(\phi)\subset B_{r}(x_{0})$, and
$r\left|\nabla\phi\right|\leqslant C_{N}$. For each $\varepsilon>0$, we set

(6.32) ${\bf v}:=\begin{cases}(1-\phi){\bf u}&\text{if }\left|{\bf
u}\right|\leqslant\varepsilon\text{,}\\\ (1-\varepsilon\frac{\phi}{\left|{\bf
u}\right|}){\bf u}&\text{if }\left|{\bf
u}\right|>\varepsilon\text{.}\end{cases}$

It is easy to check ${\bf v}\in\mathscr{A}_{\bf g}$ and ${\bf v}-{\bf u}\in
W^{1,2}_{0}(B_{r}(x_{0}),\mathbb{R}^{m})$.

(i) We will first show that $d({\bf u},{\bf v})<\varepsilon_{\bf u}$ provided
both $r$ and $\varepsilon$ are sufficiently small. By the Poincaré inequality
(Proposition 2.1(5)) and ${\bf v}-{\bf u}\in
W^{1,2}_{0}(B_{r}(x_{0}),\mathbb{R}^{m})$, we have

(6.33) $\operatorname{d}({\bf u},{\bf v})\leqslant
C_{P}\Big{(}\int_{B_{r}(x_{0})}|\nabla({\bf u}-{\bf
v})|^{2}\Big{)}^{1/2}+\mu\big{(}B_{r}(x_{0})\big{)},$

where $C_{P}$ depends on $N,K$ and ${\rm diam}(\Omega).$ From ${\bf u}-{\bf
v}=\phi{\bf u}$ on $\\{|{\bf u}|\leqslant\varepsilon\\}\cap B_{r}(x_{0})$ and
[Che99, Corollary 2.25], we have

$|\nabla({\bf u}-{\bf v})|=|\nabla(\phi{\bf u})|\leqslant
L+\varepsilon\cdot\frac{C_{N}}{r},\quad\mu{\rm-a.e.\ \ on}\ \ \\{|{\bf
u}|\leqslant\varepsilon\\}\cap B_{r}(x_{0}),$

where $L$ is a Lipschitz constant of ${\bf u}$ on $\Omega^{\prime}$, and we
have used $|\nabla\phi|\leqslant\frac{C_{N}}{r}.$ Thus,

(6.34) $\int_{B_{r}(x_{0})\cap\\{|{\bf u}|\leqslant\varepsilon\\}}|\nabla({\bf
u}-{\bf
v})|^{2}\leqslant\big{(}L+C_{N}\cdot\frac{\varepsilon}{r}\big{)}^{2}\cdot\mu(B_{r}(x_{0})).$

On $\\{|{\bf u}|>\varepsilon\\}\cap B_{r}(x_{0})$, we denote by
$h:=\frac{\varepsilon\phi}{|{\bf u}|}.$ Then, from the Chain rule (see
[Gig15]) it follows

$|\nabla h|\leqslant\frac{\varepsilon}{|\bf
u|}\cdot\frac{C_{N}}{r}+\frac{\varepsilon\phi|\nabla|{\bf u}||}{|{\bf
u}|^{2}}\leqslant\frac{\varepsilon}{|\bf
u|}\cdot\frac{C_{N}}{r}+\frac{L}{|{\bf u}|}$

$\mu{\rm-a.e.}$ on $\\{|{\bf u}|>\varepsilon\\}\cap B_{r}(x_{0})$, where we
have used $|\nabla|{\bf u}||\leqslant|\nabla{\bf u}|\leqslant L$,
$\varepsilon\phi\leqslant|{\bf u}|$ and
$|\nabla\phi|\leqslant\frac{C_{N}}{r}.$ From this and ${\bf u}-{\bf v}=h{\bf
u}$ on $\\{|{\bf u}|>\varepsilon\\}$, we have by [Che99, Corollary 2.25], that

$|\nabla({\bf u}-{\bf v})|=|\nabla(h{\bf u})|\leqslant
L+\varepsilon\cdot\frac{C_{N}}{r}+h|\nabla{\bf u}|\leqslant
2L+\varepsilon\cdot\frac{C_{N}}{r},$

$\mu{\rm-a.e.}$ on $\\{|{\bf u}|>\varepsilon\\}\cap B_{r}(x_{0})$. Thus, we
obtain

(6.35) $\int_{B_{r}(x_{0})\cap\\{|{\bf u}|>\varepsilon\\}}|\nabla({\bf u}-{\bf
v})|^{2}\leqslant\big{(}2L+C_{N}\cdot\frac{\varepsilon}{r}\big{)}^{2}\cdot\mu(B_{r}(x_{0})).$

From (6.33), (6.34) and (6.35), it follows that

(6.36) $\begin{split}\operatorname{d}({\bf u},{\bf v})&\leqslant
C_{1}\Big{(}L+\frac{\varepsilon}{r}\Big{)}\cdot\sqrt{\mu(B_{r}(x_{0}))}+\mu(B_{r}(x_{0})),\end{split}$

where $C_{1}$ depending only on $N,K$ and ${\rm diam}(\Omega)$. By the non-
collapsing property of $X$, we have $N\geqslant 2$ (since $N$ is an integer
and $N>1$), and then $\mu(B_{r}(x_{0}))\leqslant C_{2}r^{N}\leqslant
C_{2}r^{2}$ for $r<1$, where $C_{2}$ depends only on $N,K$ (see (2.13)).
Therefore, we conclude $\operatorname{d}({\bf u},{\bf v})<\varepsilon_{\bf u}$
provided

(6.37) $\varepsilon\leqslant\bar{\varepsilon}:=\frac{\varepsilon_{\bf
u}}{4C_{1}C_{2}^{1/2}}\qquad
r\leqslant\bar{r}:=\min\Big{\\{}1,\frac{\varepsilon_{\bf
u}}{4(C_{1}LC^{1/2}_{2}+C_{2})}\Big{\\}}.$

(ii) Fix any $r\in(0,\bar{r})$ and $\varepsilon\in(0,\bar{\varepsilon})$ given
in (6.37). From the local minimality of ${\bf u}$, we obtain

(6.38) $\displaystyle\int_{B_{r}(x_{0})}\big{(}|\nabla{\bf u}|^{2}-|\nabla{\bf
v}|^{2}\big{)}$
$\displaystyle\leqslant\int_{B_{r}(x_{0})}Q\big{(}\chi_{\\{|{\bf
v}|>0\\}}-\chi_{\\{|\bf u|>0\\}}\big{)}$
$\displaystyle\leqslant\int_{B_{r/2}(x_{0})\cap\\{|{\bf
u}|\leqslant\varepsilon\\}}Q\big{(}\chi_{\\{|{\bf v}|>0\\}}-\chi_{\\{|\bf
u|>0\\}}\big{)}$ $\displaystyle\leqslant-\int_{B_{r/2}(x_{0})}Q\chi_{\\{0<|\bf
u|\leqslant\varepsilon\\}}$ $\displaystyle\leqslant-
Q_{\min}\cdot\mu\big{(}B_{r/2}(x_{0})\cap\\{0<|{\bf
u}|\leqslant\varepsilon\\}\big{)},$

where for the second inequality we have used $\chi_{\\{|{\bf
v}|>0\\}}-\chi_{\\{|\bf u|>0\\}}\leqslant 0$ on $B_{r}(x_{0})$, and for the
third inequality we have used ${\bf v}=0$ on $B_{r/2}(x_{0})\cap\\{|{\bf
u}|\leqslant\varepsilon\\}$.

On the one hand, from ${\bf v}=(1-\phi){\bf u}$ on $\\{|{\bf
u}|\leqslant\varepsilon\\}\cap B_{r}(x_{0})$ and [Che99, Corollary 2.25], it
follows $|\nabla{\bf v}|=|\nabla((1-\phi){\bf u})|$ $\mu$-a.e. in $\\{|{\bf
u}|\leqslant\varepsilon\\}\cap B_{r}(x_{0})$. Thus, we have

(6.39) $\displaystyle\int_{\left\\{|{\bf u}|\leqslant\varepsilon\right\\}\cap
B_{r}(x_{0})}(\left|\nabla{\bf u}\right|^{2}-\left|\nabla{\bf v}\right|^{2})$
$\displaystyle=$ $\displaystyle\int_{\left\\{\left|{\bf
u}\right|\leqslant\varepsilon\right\\}\cap B_{r}(x_{0})}(\left|\nabla{\bf
u}\right|^{2}-\left|\nabla((1-\phi){\bf u})\right|^{2})$ $\displaystyle=$
$\displaystyle\int_{\\{|{\bf u}|\leqslant\varepsilon\\}\cap
B_{r}(x_{0})}((2\phi-\phi^{2})|\nabla{\bf
u}|^{2}+2(1-\phi)\sum_{i=1}^{m}u_{i}\left\langle{\nabla\phi},{\nabla
u_{i}}\right\rangle-\left|\nabla\phi\right|^{2}\left|{\bf u}\right|^{2})$
$\displaystyle\geqslant$ $\displaystyle\int_{\left\\{\left|{\bf
u}\right|\leqslant\varepsilon\right\\}\cap
B_{r}(x_{0})}\big{(}\chi_{B_{r/2}(x_{0})}\left|\nabla{\bf
u}\right|^{2}\big{)}-\int_{\left\\{\left|{\bf
u}\right|\leqslant\varepsilon\right\\}\cap
B_{r}(x_{0})}\big{(}2\frac{C_{N}L}{r}|{\bf u}|+\frac{C^{2}_{N}}{r^{2}}|{\bf
u}|^{2}\big{)}$ $\displaystyle\geqslant$
$\displaystyle\int_{\left\\{\left|{\bf
u}\right|\leqslant\varepsilon\right\\}\cap B_{r/2}(x_{0})}\left|\nabla{\bf
u}\right|^{2}-C_{N,L}\frac{\varepsilon}{r^{2}}\mu\big{(}B_{r}(x_{0})\big{)},$

where for the first inequality we have used $(2\phi-\phi^{2})=1$ on
$B_{r/2}(x_{0})$, $|\nabla{\bf u}|\leqslant L$ and $|\nabla\phi|\leqslant
C_{N}/r$ on $B_{r}(x_{0})$.

On the other hand, we denote $h=\frac{\varepsilon\phi}{|{\bf u}|}$. Then ${\bf
v}=(1-h){\bf u}$ on $\\{|{\bf u}|>\varepsilon\\}\cap B_{r}(x_{0})$, and by
[Che99, Corollary 2.25], we have that, for $\mu$-a.e. in $\\{|{\bf
u}|>\varepsilon\\}\cap B_{r}(x_{0})$,

$\begin{split}&|\nabla{\bf u}|^{2}-|\nabla{\bf v}|^{2}=|\nabla{\bf
u}|^{2}-|\nabla((1-h){\bf u})|^{2}\\\ =&(2h-h^{2})|\nabla{\bf u}|^{2}-|\nabla
h|^{2}|{\bf u}|^{2}+2(1-h)\sum_{i=1}^{m}u_{i}\left\langle{\nabla h},{\nabla
u_{i}}\right\rangle\\\ =&(2h-h^{2})|\nabla{\bf u}|^{2}-|\nabla h|^{2}|{\bf
u}|^{2}+(1-h)\left\langle{\nabla h},{\nabla(|{\bf u}|^{2})}\right\rangle\\\
=&(2h-h^{2})|\nabla{\bf u}|^{2}-|\nabla h|^{2}|{\bf u}|^{2}+2(1-h)|{\bf
u}|\left\langle{\nabla h},{\nabla|{\bf u}|}\right\rangle\\\
=&(2h-h^{2})|\nabla{\bf u}|^{2}-|\nabla(h|{\bf
u}|)|^{2}+2\left\langle{\nabla|{\bf u}|},{\nabla(h|{\bf
u}|)}\right\rangle+(h^{2}-2h)|\nabla|{\bf u}||^{2}\\\
=&(2h-h^{2})\big{(}|\nabla{\bf u}|^{2}-|\nabla|{\bf
u}||^{2}\big{)}-|\nabla(\varepsilon\phi)|^{2}+2\left\langle{\nabla|{\bf
u}|},{\nabla(\varepsilon\phi)}\right\rangle.\end{split}$

Thus, by $|\nabla{\bf u}|\geqslant|\nabla|{\bf u}||$ $\mu$-almost everywhere
and $h\leqslant 1$, we have

(6.40) $\displaystyle\int_{\left\\{\left|{\bf
u}\right|>\varepsilon\right\\}\cap B_{r}(x_{0})}(\left|\nabla{\bf
u}\right|^{2}-\left|\nabla{\bf v}\right|^{2})$ $\displaystyle\geqslant$
$\displaystyle\int_{\left\\{\left|{\bf u}\right|>\varepsilon\right\\}\cap
B_{r}(x_{0})}\big{(}-\varepsilon^{2}\left|\nabla\phi\right|^{2}+2\varepsilon\left\langle{\nabla|{\bf
u}|},{\nabla\phi}\right\rangle\big{)}$ $\displaystyle\geqslant$
$\displaystyle\big{(}-\varepsilon^{2}\frac{C^{2}_{N}}{r^{2}}-2\varepsilon\frac{C_{N}L}{r}\big{)}\mu\big{(}B_{r}(x_{0})\big{)}\geqslant-
C_{N,L}\frac{\varepsilon}{r^{2}}\mu\big{(}B_{r}(x_{0})\big{)},$

where we have used $|\nabla{\bf u}|\leqslant L$ and $|\nabla\phi|\leqslant
C_{N}/r$. The inequality (6.31) follows from the combination of (6.38)–(6.40).
The proof is finished. ∎

Now we are in the position to show the free boundary of ${\bf u}$ is a set of
locally finite perimeter in the sense of [Amb01, Mir03, ABS19].

######  Proposition 6.7 (Local finiteness of perimeter).

Let ${\bf u}=(u_{1},\ldots,u_{m})$ be a local minimizer of $J_{Q}$ in (1.2)
and let $Q$ satisfy(1.3). Then $\Omega_{\bf u}=\Omega\cap\\{|{\bf u}|>0\\}$ is
of locally finite perimeter. Moreover, the followings hold:

  1. (1)

For all $\Omega^{\prime}\Subset\Omega$,
$\mathscr{H}^{N-1}\big{(}\partial\\{|{\bf
u}|>0\\}\cap\Omega^{\prime}\big{)}<+\infty$;

  2. (2)

There exist nonnegative Borel functions $q_{i},i=1,2,\cdots,m,$ such that

${\bf\Delta}u_{i}=q_{i}\cdot\mathscr{H}^{N-1}\llcorner(\partial\\{|{\bf
u}|>0\\}\cap\Omega).$

###### Proof.

Fix any $\Omega^{\prime}\Subset\Omega$. By using Lemma 6.6, we get

$\begin{split}\int_{\left\\{0<|{\bf u}|\leqslant\varepsilon\right\\}\cap
B_{r/2}(x_{0})}|\nabla{\bf u}|&\leqslant\int_{\left\\{0<|{\bf
u}|\leqslant\varepsilon\right\\}\cap B_{r/2}(x_{0})}\frac{|\nabla{\bf
u}|^{2}+1}{2}\\\ &\leqslant
C_{N,K,L,r,\mu(B_{r}(x_{0}))}\cdot\varepsilon\end{split}$

for all $\varepsilon\in(0,\bar{\varepsilon}),$ which holds for any ball
$B_{r}(x_{0})\subset\Omega^{\prime}$ with radius $r<\bar{r}$. Then, by using
Lemma 6.5, we conclude that $\\{|{\bf u}|>0\\}\cap B_{r/2}(x_{0})$ has finite
perimeter. Hence, $\\{|{\bf u}|>0\\}\cap\Omega^{\prime}$ has finite perimeter.