# Free boundary cluster with Robin condition on the transmission interface

Serena Guarino Lo Bianco, Domenico Angelo La Manna, Bozhidar Velichkov  Serena
Guarino Lo Bianco
Università degli studi di Napoli “Federico II”
Dipartimento di Agraria
Via Università 100, 80055 Portici (NA)- ITALY.
<EMAIL_ADDRESS>Domenico Angelo La Manna
Università degli studi di Napoli “Federico II”
Dipartimento di Matematica ”Renato Caccioppoli”
Via Cintia, Monte S. Angelo - 80126 Napoli - ITALY<EMAIL_ADDRESS>Bozhidar Velichkov:
Dipartimento di Matematica, Università di Pisa
Largo Bruno Pontecorvo, 5, 56127 Pisa - ITALY<EMAIL_ADDRESS>
###### Abstract.

We formulate and study a variational two-phase free boundary problem with
Robin condition on the interface between the two phases, and we prove
existence and regularity of solutions in dimension two.

###### Key words and phrases:

Free boundary problems, free interface, optimal transmission problems, Robin
boundary conditions, regularity

###### 1991 Mathematics Subject Classification:

35R35, 49Q10

##  1\. Introduction

Free boundary problems with two and more phases are often used to describe
models in different areas of Physics, Engineering and Life Sciences, for
instance in Fluid Dynamics (Bernoulli free boundary problems), Dynamics of
Populations (optimal partitions problems), Mechanics and Phase Transition
(obstacle problems). The different phases are called _segregated_ if they
occupy different space regions; segregation occurs for instance in the two-
phase Bernoulli problem, the two-phase obstacle problem, optimal partitions
problems.

In all these cases the interaction between the different phases is supposed to
be _competitive_ , in particular, the interfaces are not formed because it is
convenient energetically, but due to the lack of space. For instance, if we
have two disjoint one-phase solutions of the variational Bernoulli (or
obstacle) problem, then the couple they form is a minimizer to the
corresponding two-phase problem, and even if the two phases are very close to
each other, an interface is not formed (we briefly discuss this phenomenon in
Section 1.1).

In this paper, we consider a two-phase problem, in which the phases are still
_segregated_ , but the interaction along the free interface is
_collaborative_. In this case, if two or more disjoint one-phase solutions are
sufficiently close, then it is energetically convenient for them to create a
free interface, that is, the formation of clusteres is incentivized.

We introduce the functional in Section 1.2, while in Section 1.3 we state the
variational problem and the main results of the paper.

###  1.1. The classical one-phase and two-phase Bernoulli free boundary
problems

Let $D$ be a smooth bounded open set in $\mathbb{R}^{d}$. Let $g:\partial
D\to\mathbb{R}$ be a given nonnegative function and $\lambda>0$ a given
constant. The classical one-phase Bernoulli problem can be stated as follows.
Find a domain $\Omega\subset D$ and a function $u:D\to\mathbb{R}$ such that
$u=g$ on $\partial D$ and

$\Delta u=0\quad\text{in}\quad\Omega\,,\qquad u=0\quad\text{and}\quad|\nabla
u|=\lambda\quad\text{on}\quad\partial\Omega\cap D\,.$

In the seminal paper [1] Alt and Caffarelli showed that the existence of such
a couple $(u,\Omega)$ can be obtained by minimizing the functional

$J_{\lambda}(u)=\int_{D}|\nabla u|^{2}+\lambda^{2}|\\{u>0\\}\cap D|,$

among all functions in $H^{1}(D)$ such that $u=g$ on $\partial D$, and then
taking $\Omega:=\\{u>0\\}$.

In the two-phase problem, we are given two constants $\lambda_{1}>0$ and
$\lambda_{2}>0$ and two nonnegative functions $g_{1}:\partial D\to\mathbb{R}$
and $g_{2}:\partial D\to\mathbb{R}$ with disjoint supports. Then, the two-
phase free boundary problem is the following. Find two _disjoint_ sets
$\Omega_{1}$ and $\Omega_{2}$ in $D$ and two functions $u_{1}:D\to\mathbb{R}$
and $u_{2}:D\to\mathbb{R}$ such that $u_{1}=g_{1}$ and $u_{2}=g_{2}$ on
$\partial D$ and

$\begin{cases}\begin{array}[]{ll}\Delta
u_{1}=0\quad\text{in}\quad\Omega_{1}\qquad\text{and}\qquad\Delta
u_{2}=0\quad\text{in}\quad\Omega_{2}\,;\\\ u_{1}=0\quad\text{and}\quad|\nabla
u_{1}|=\lambda_{1}\quad\text{on}\quad
D\cap\partial\Omega_{1}\setminus\partial\Omega_{2}\,;\\\
u_{2}=0\quad\text{and}\quad|\nabla u_{2}|=\lambda_{2}\quad\text{on}\quad
D\cap\partial\Omega_{2}\setminus\partial\Omega_{1}\,;\\\
u_{1}=u_{2}=0\quad\text{and}\quad|\nabla u_{1}|^{2}-|\nabla
u_{2}|^{2}=\lambda_{1}^{2}-\lambda_{2}^{2}\quad\text{on}\quad
D\cap\partial\Omega_{1}\cap\partial\Omega_{2}\,.\end{array}\end{cases}$ (1.1)

The existence of a solution can be obtained by minimizing the functional

$J_{\lambda_{1},\lambda_{2}}(u)=\int_{D}|\nabla
u|^{2}+\lambda_{1}^{2}|\\{u>0\\}\cap D|+\lambda_{2}^{2}|\\{u<0\\}\cap D|,$

among all functions in $H^{1}(D)$ such that $u=g_{1}-g_{2}$ on $\partial D$,
and then taking $u_{1}=\max\\{u,0\\}$, $u_{2}=\max\\{-u,0\\}$,
$\Omega_{1}=\\{u_{1}>0\\}$ and $\Omega_{2}=\\{u_{2}>0\\}$ (see [2], [14] and
[6]).

In the two-phase problem, the two-phase interface
$\partial\Omega_{1}\cap\partial\Omega_{2}$ is formed when the two sets
$\Omega_{1}$ and $\Omega_{2}$ act as geometric obstacles to each other; if
$\Omega_{1}$ and $\Omega_{2}$ are disjoint one-phase solutions, then the two-
phase interface is simply not formed. In other words, if $u_{1}$ and $u_{2}$
are minimizers of the one-phase functionals $J_{\lambda_{1}}$ and
$J_{\lambda_{2}}$ such that $u_{1}u_{2}\equiv 0$, then it is immediate to
check that $u=u_{1}-u_{2}$ is a minimizer of the two-phase functional
$J_{\lambda_{1},\lambda_{2}}$. In fact, if $v\in H^{1}(D)$ is such that $v=u$
on $\partial D$, then $v_{+}=u_{1}$ and $v_{-}=u_{2}$ on $\partial D$ and so,
by the optimality of $u_{1}$ and $u_{2}$, we get

$J_{\lambda_{1},\lambda_{2}}(v)=J_{\lambda_{1}}(v_{+})+J_{\lambda_{2}}(v_{-})\geq
J_{\lambda_{1}}(u_{1})+J_{\lambda_{2}}(u_{2})=J_{\lambda_{1},\lambda_{2}}(u).$

###  1.2. A two-phase problem with Robin condition on the free interface

In this paper we study a different type of two-phase problem in which the two
state functions $u_{1}$ and $u_{2}$ might not vanish on the interface
$\partial\Omega_{1}\cap\partial\Omega_{2}$. Precisely, given $\beta>0$,
$\Lambda>0$ and a fixed set $D$, we consider the functional

$J_{\beta,\Lambda}(u,\Omega_{1},\Omega_{2})=\int_{D}|\nabla
u|^{2}\,dx+\beta\int_{\partial\Omega_{1}\cap\partial\Omega_{2}}u^{2}\,d\mathcal{H}^{d-1}+\Lambda|\\{u>0\\}\cap
D|,$

defined for couples of disjoint domains $\Omega_{1},\Omega_{2}$ in $D$ and
functions $u\in H^{1}(D)$ with $u=0$ on
$D\setminus(\Omega_{1}\cup\Omega_{2})$. We will then show that if
$\big{(}\Omega_{1},\Omega_{2},u\big{)}$ locally minimize $J_{\beta,\Lambda}$
in $D$, then the functions

$u_{1}=u\text{1}_{\Omega_{1}}\qquad\text{and}\qquad
u_{2}=u\text{1}_{\Omega_{2}}\ ,$

are solutions to the problem

$\begin{cases}\begin{array}[]{ll}\Delta
u_{1}=0\quad\text{in}\quad\Omega_{1}\qquad\text{and}\qquad\Delta
u_{2}=0\quad\text{in}\quad\Omega_{2}\\\ u_{1}=0\quad\text{and}\quad|\nabla
u_{1}|=\sqrt{\Lambda}\quad\text{on}\quad
D\cap\partial\Omega_{1}\setminus\partial\Omega_{2}\\\
u_{2}=0\quad\text{and}\quad|\nabla u_{2}|=\sqrt{\Lambda}\quad\text{on}\quad
D\cap\partial\Omega_{2}\setminus\partial\Omega_{1}\\\
u_{1}=u_{2}\quad\text{and}\quad|\nabla u_{1}|+|\nabla
u_{2}|=\beta(u_{1}+u_{2})\quad\text{on}\quad
D\cap\partial\Omega_{1}\cap\partial\Omega_{2}\,,\end{array}\end{cases}$ (1.2)

and satisfy an additional condition on
$\partial\Omega_{1}\cap\partial\Omega_{2}$ involving the mean curvature of the
interface (see [9]).

Notice that if $(u_{1},\Omega_{1})$ and $(u_{2},\Omega_{2})$ are two
minimizers of the one-phase Bernoulli functional $J_{\sqrt{\Lambda}}$ with
disjoint supports ($\Omega_{1}\cap\Omega_{2}=\emptyset$), the triple
($\Omega_{1}$, $\Omega_{2}$, $u=u_{1}+u_{2}$) might not be a minimizer of
$J_{\beta,\Lambda}$, even if the Hausdorff distance between $\Omega_{1}$ and
$\Omega_{2}$ is strictly positive. In fact, it might be convenient to enlarge
the domains $\Omega_{1}$ and $\Omega_{2}$ in order to obtain a non-empty
interface $\partial\Omega_{1}\cap\partial\Omega_{2}$ that will allow to have
competitors which are not vanishing identically on the entire free boundaries
$\partial\Omega_{1}$ and $\partial\Omega_{2}$. This is illustrated by the
following one-dimensional example.

######  Example 1.1 (Formation of an interface in 1D).

Let ${\varepsilon}>0$ and $\beta>0$ be fixed. We consider the interval
$D=[-1-{\varepsilon},1+{\varepsilon}]$ and the boundary data
$g_{1},g_{2}:\partial D\to\mathbb{R}$ given by

$g_{1}(-1-{\varepsilon})=1\,,\quad g_{1}(1+{\varepsilon})=0\,,\quad
g_{2}(-1-{\varepsilon})=0\,,\quad g_{2}(1+{\varepsilon})=1.$

The minimizers of the one-phase function

$J_{1}=\int_{D}|u^{\prime}(x)|^{2}\,dx+|\\{u>0\\}\cap D|$

with boundary conditions $g_{1}$ and $g_{2}$ are respectively the functions

$u_{1}(x)=(-x-{\varepsilon})_{+}\qquad\text{and}\qquad
u_{2}(x)=(x-{\varepsilon})_{+}.$

If we consider the sets $\Omega_{1}=(-1-{\varepsilon},-{\varepsilon})$ and
$\Omega_{2}=({\varepsilon},1+{\varepsilon})$, then we have that

$J_{\beta,1}(u_{1}+u_{2},\Omega_{1},\Omega_{2})=J_{1}(u_{1})+J_{1}(u_{2})=4.$

On the other hand, by taking

$\widetilde{\Omega}_{1}=[-1-{\varepsilon},0]\
,\quad\widetilde{\Omega}_{2}=[0,1+{\varepsilon}]\ ,\quad
u(x)=\begin{cases}1\quad\text{if}\quad x=-1-{\varepsilon},\\\
\ell\quad\text{if}\quad x=0,\\\ 1\quad\text{if}\quad
x=1+{\varepsilon},\end{cases}$

and extending $u$ linearly on the intervals $[-1-{\varepsilon},0]$ and
$[0,1+{\varepsilon}]$ we obtain that

$J_{\beta,1}\big{(}u,\widetilde{\Omega}_{1},\widetilde{\Omega}_{2}\big{)}=2\frac{(1-\ell)^{2}}{1+{\varepsilon}}+\beta\ell^{2}+2+2{\varepsilon}.$

Setting the parameter $\ell$ to be the optimal one,
$\displaystyle\ell=\frac{2}{2+\beta+{\varepsilon}\beta},$ we get that

$J_{\beta,1}\big{(}u,\widetilde{\Omega}_{1},\widetilde{\Omega}_{2}\big{)}=2\frac{(1+{\varepsilon})\beta^{2}}{(2+\beta+{\varepsilon}\beta)^{2}}+\beta\left(\frac{2}{2+\beta+{\varepsilon}\beta}\right)^{2}+2+2{\varepsilon}.$

When ${\varepsilon}=0$, we get

$J_{\beta,1}\big{(}u,\widetilde{\Omega}_{1},\widetilde{\Omega}_{2}\big{)}=2\frac{\beta^{2}}{(2+\beta)^{2}}+\beta\left(\frac{2}{2+\beta}\right)^{2}+2=2\frac{\beta^{2}+2\beta}{\beta^{2}+4\beta+4}+2<4.$

In conclusion, if we fix $\beta>0$ we can find ${\varepsilon}_{0}>0$ such that

$J_{\beta,1}\big{(}u,\widetilde{\Omega}_{1},\widetilde{\Omega}_{2}\big{)}<4\qquad\text{for
all}\qquad{\varepsilon}\in(0,{\varepsilon}_{0}),$

which means that for those choices of $\beta$ and ${\varepsilon}$, the
combination of the two one-phase solutions is not optimal.

###  1.3. Setting of the problem and main theorem

We will define the variational problem for the functional $J_{\beta,\Lambda}$
in the class of sets of finite perimeter and Sobolev functions. Then, we will
prove an existence theorem in this class and we will show that the minimizers
are regular. We fix the boundary data for $\Omega_{1}$, $\Omega_{2}$ and $g$.
Precisely, let

  * •

$E_{1}$ and $E_{2}$ be two smooth, bounded and disjoint sets of positive
distance in $\mathbb{R}^{d}$;

  * •

$D:=\mathbb{R}^{d}\setminus\Big{(}\overline{E}_{1}\cup\overline{E}_{2}\Big{)}$;

  * •

$\Omega_{i}=E_{i}$ in $\mathbb{R}^{d}\setminus D$;

  * •

$g\in H^{1}(\mathbb{R}^{d})\cap L^{\infty}(\mathbb{R}^{d})$ be a non-negative
function such that

$g\equiv 1\quad\text{on}\quad E_{1}\cup E_{2}\,.$

We define the following admissible set of functions

$\mathcal{V}=\Big{\\{}u\in H^{1}(\mathbb{R}^{d})\,:\,u\geq 0\,\text{ in
}\,\mathbb{R}^{d}\,\,\,\text{and}\,\,\,u-g\in H^{1}_{0}(D)\Big{\\}}.$

Then, fixed $u\in\mathcal{V}$, we define the admissible set $\mathcal{A}(u)$
as the set of all couples $(\Omega_{1},\Omega_{2})$ of Lebesgue measurable
sets such that:

  * •

$\Omega_{1}\cap\Omega_{2}=\emptyset$, $E_{1}\subset\Omega_{1}$ and
$E_{2}\subset\Omega_{2}$ Lebesgue almost-everywhere;

  * •

$\Omega_{1}$ and $\Omega_{2}$ have finite perimeter (as subsets of
$\mathbb{R}^{d}$);

  * •

$\\{u>0\\}\subset{\Omega_{1}\cup\Omega_{2}}$ Lebesgue almost-everywhere.

For every $\beta>0$ and $\Lambda>0$, we consider the functional
$J_{\beta,\Lambda}$, defined for functions $u\in\mathcal{V}$ and couples of
sets $(\Omega_{1},\Omega_{2})\in\mathcal{A}(u)$, as

$J_{\beta,\Lambda}(u,\Omega_{1},\Omega_{2}):=\int_{D}|\nabla
u|^{2}\,dx+\beta\int_{\partial^{\ast}\Omega_{1}\cap\partial^{\ast}\Omega_{2}}u^{2}\,d\mathcal{H}^{d-1}+\Lambda|\\{u>0\\}\cap
D|,$

where $\partial^{\ast}\Omega_{j}$ is the reduced boundary of $\Omega_{j}$; we
recall that since $u$ is a bounded Sobolev function, the second integral is
well-defined (see Section 2).

In this paper we consider the variational problem

$\min\Big{\\{}J_{\beta,\Lambda}(u,\Omega_{1},\Omega_{2})\,:\,u\in\mathcal{V}\,,\
(\Omega_{1},\Omega_{2})\in\mathcal{A}(u)\Big{\\}}.$ (1.3)

The main result is the following.

######  Theorem 1.2.

Let $D$ be a smooth bounded open set in $\mathbb{R}^{d}$, $d\geq 2$. Given
sets $E_{1}$ and $E_{2}$, and a function $g$ as above, there are a function
$u\in\mathcal{V}$ and sets $(\Omega_{1},\Omega_{2})\in\mathcal{A}(u)$ that
solve the variational problem (1.3). Conversely, if
$(u,\Omega_{1},\Omega_{2})$ is a solution to (1.3), then also
$(u,\widetilde{\Omega}_{1},\widetilde{\Omega}_{2})$ is a solution to (1.3),
where

$\widetilde{\Omega}_{1}=\\{u>0\\}\cap\Omega_{1}\qquad\text{and}\qquad\widetilde{\Omega}_{2}=\\{u>0\\}\cap\Omega_{2}.$

Moreover,

  1. (i)

the boundary $\partial\\{u>0\\}$ is $C^{1,\alpha}$-regular in $D$;

  2. (ii)

the interface $\partial\Omega_{1}\cap\partial\Omega_{2}$ is $C^{\infty}$ in
the open set $D\cap\\{u>0\\}$ and is $C^{1}$ regular up to the boundary
$D\cap\partial\\{u>0\\}$. Moreover, $\partial\Omega_{1}\cap\partial\Omega_{2}$
reaches $\partial\\{u>0\\}$ orthogonally.

### Sketch of the proof and plan of the paper

In order to prove 1.2, we first introduce a family of approximating problems
in Section 4. Then, passing to the limit, we obtain a function
$u\in\mathcal{V}$ and a couple of disjoint sets $\Omega_{1}$ and $\Omega_{2}$.
We cannot obtain immediately that $(u,\Omega_{1},\Omega_{2})$ is a solution to
(1.3), since there is not a uniform bound on the perimeter of the
approximating sets, so we do not a priori have that $\Omega_{1}$ and
$\Omega_{2}$ are sets of locally finite perimeter in $D$. Instead, we are able
to prove that $u$ satisfies an almost-minimality condition involving the one-
phase Alt-Caffarelli functional, which allows to prove that the set
$\\{u>0\\}$ is regular (9.1). This solves the problem only in part because we
only have that

${\\{u>0\\}}={\Omega_{1}\cup\Omega_{2}}\quad\text{in}\quad D.$

We then show that the sets $\Omega_{1}$ and $\Omega_{2}$ are almost-minimizers
of the perimeter in $\\{u>0\\}\cap D$, which implies that (in low dimension)
the free interface $\partial\Omega_{1}\cup\partial\Omega_{2}$ is smooth in
$D\cap\\{u>0\\}$. Thus, in order to prove that $\Omega_{1}$ and $\Omega_{2}$
have finite perimeter it is sufficient to study the behavior of the interface
$\partial\Omega_{1}\cup\partial\Omega_{2}$ close to the free boundary
$\partial\\{u>0\\}$ (see 10.1). We show that $\Omega_{1}$ and $\Omega_{2}$ are
minimizers in $\\{u>0\\}$ of a weighted perimeter functional, the weight being
precisely the function $u^{2}$, which is $C^{0,\alpha}$ and positive in
$\\{u>0\\}$, but as it approaches the free boundary $\partial\\{u>0\\}$ we
have that

$u^{2}(x)\sim\text{dist}^{2}\big{(}x,\partial\\{u>0\\}\big{)}.$

In order to deal with this degenerate weight, we perform a 2D conformal change
of coordinates, which flattens $\partial\\{u>0\\}$ to a line; then we rotate
$\Omega_{1}$ around this line in order to obtain an almost-minimizer of the
perimeter in $\mathbb{R}^{4}$. This allows to conclude that
$\partial\Omega_{1}\cup\partial\Omega_{2}$ is the union of $C^{1}$ curves that
meet $\partial\\{u>0\\}$ orthogonally in a (locally) finite number of points.
This concludes the proof of 10.1 and also shows that $\Omega_{1}$ and
$\Omega_{2}$ have locally finite perimeter. In order to show that $\Omega_{1}$
and $\Omega_{2}$ have finite perimeter, in 8.1 we prove that $\\{u>0\\}$
contains strictly both $\overline{E}_{1}$ and $\overline{E}_{2}$. Then, in
Section 11 we show that $\Omega_{1}$ and $\Omega_{2}$ are actually minimizers
of (1.3) and we complete the proof of 1.2.

##  2\. Sets of finite perimeter and Sobolev functions

###  2.1. Caccioppoli sets

For any Lebesgue measurable set $\Omega\subset\mathbb{R}^{d}$, we define

$\text{\rm Per}(\Omega):=\sup\Big{\\{}\int_{\Omega}\text{\rm div}\,\xi(x)\,dx\
:\ \xi\in C^{1}_{c}(\mathbb{R}^{d}),\ \|\xi\|_{L^{\infty}(\mathbb{R}^{d})}\leq
1\Big{\\}},$

and we say that $\Omega$ is of finite perimeter (Caccioppoli set) if

$\text{\rm Per}\,(\Omega)<+\infty\,.$

Given $\alpha\in[0,1]$, we say that the set $\Omega$ has a Lebesgue density
$\alpha$ at $x\in\mathbb{R}^{d}$ if

$\lim_{r\to 0}\frac{|\Omega\cap B_{r}(x)|}{|B_{r}|}=\alpha\,.$

We define the set $\Omega^{(\alpha)}$ as

$\Omega^{(\alpha)}:=\Big{\\{}x\in\mathbb{R}^{d}\ :\ \lim_{r\to
0}\frac{|\Omega\cap B_{r}(x)|}{|B_{r}|}=\alpha\Big{\\}}.$

Given a set of finite perimeter $\Omega\subset\mathbb{R}^{d}$ we will denote
by $\partial^{\ast}\Omega$ its reduced boundary and by $\nu_{\Omega}$ the
generalized exterior normal. We recall that

$\text{Per}(\Omega)=\mathcal{H}^{d-1}(\partial^{\ast}\Omega),$

and that for any $\xi\in C^{1}_{c}(\mathbb{R}^{d})$

$\int_{\Omega}\text{\rm
div}\,\xi(x)\,dx=\int_{\partial^{\ast}\Omega}\xi\cdot\nu_{\Omega}\,d\mathcal{H}^{d-1},$

where $\mathcal{H}^{d-1}$ denotes the $(d-1)$-dimensional Hausdorff measure in
$\mathbb{R}^{d}$. Moreover, we recall that at every point of the reduced
boundary, $\partial^{\ast}\Omega$ has Lebesgue density $\nicefrac{{1}}{{2}}$,
that is,

$\partial^{\ast}\Omega\subset\Omega^{(\nicefrac{{1}}{{2}})}.$

We also recall the following well-known result by Federer

$\mathcal{H}^{d-1}\Big{(}\mathbb{R}^{d}\setminus\big{\\{}\Omega^{(0)}\cup\Omega^{(1)}\cup\partial^{\ast}\Omega\big{\\}}\Big{)}=0,$

which can also be stated as in the lemma below.

######  Lemma 2.1.

If $\Omega$ is a set of finite perimeter in $\mathbb{R}^{d}$, then up to a set
of zero $\mathcal{H}^{d-1}$ measure

$\Omega^{(\nicefrac{{1}}{{2}})}=\partial^{\ast}\Omega\qquad\text{and}\qquad\Omega^{(0)}\cup\Omega^{(\nicefrac{{1}}{{2}})}\cup\Omega^{(1)}=\mathbb{R}^{d}.$

Finally, we conclude this section with the following proposition

######  Proposition 2.2.

Let $A$ and $B$ be two disoint sets of finite perimeter in $\mathbb{R}^{d}$.
Then,

$\displaystyle\partial^{\ast}A=\Big{(}\partial^{\ast}A\cap\partial^{\ast}B\Big{)}\cup\Big{(}\partial^{\ast}A\setminus\partial^{\ast}B\Big{)}\qquad\text{and}\qquad\partial^{\ast}B=\Big{(}\partial^{\ast}A\cap\partial^{\ast}B\Big{)}\cup\Big{(}\partial^{\ast}B\setminus\partial^{\ast}A\Big{)},$

the set $A\cup B$ is a set of finite perimeter and, up to a set of zero
$\mathcal{H}^{d-1}$-measure, and

$\displaystyle\partial^{\ast}(A\cup
B)=\Big{(}\partial^{\ast}A\setminus\partial^{\ast}B\Big{)}\cup\Big{(}\partial^{\ast}B\setminus\partial^{\ast}A\Big{)}.$
(2.1)

In particular,

$\displaystyle\text{\rm Per}(A)+\text{\rm Per}(B)=\text{\rm Per}(A\cup
B)+2\,\mathcal{H}^{d-1}\big{(}\partial^{\ast}A\cap\partial^{\ast}B\big{)}.$
(2.2)

###### Proof.

Up to a set of zero $\mathcal{H}^{d-1}$ measure, we have that

$\partial^{\ast}A\setminus\partial^{\ast}B=A^{(\nicefrac{{1}}{{2}})}\cap\Big{(}B^{(0)}\cup
B^{(1)}\Big{)}=A^{(\nicefrac{{1}}{{2}})}\cap B^{(0)}.$

Analogously,
$\partial^{\ast}B\setminus\partial^{\ast}A=B^{(\nicefrac{{1}}{{2}})}\cap
A^{(0)}.$ On the other hand

$\partial^{\ast}(A\cup B)=(A\cup
B)^{(\nicefrac{{1}}{{2}})}=\Big{(}A^{(\nicefrac{{1}}{{2}})}\cap
B^{(0)}\Big{)}\cup\Big{(}B^{(\nicefrac{{1}}{{2}})}\cap A^{(0)}\Big{)},$

which proves (2.1). Finally, (2.2) follows since the sets

$\partial^{\ast}A\setminus\partial^{\ast}B\
,\quad\partial^{\ast}B\setminus\partial^{\ast}A\quad\text{and}\quad\partial^{\ast}A\cap\partial^{\ast}B\
,$

are disjoint. ∎

As a consequence of Lemma 2.1, one can obtain the following decomposition.

######  Proposition 2.3.

Let $A$ and $B$ be two sets of finite perimeter in $\mathbb{R}^{d}$. Then,
also $A\setminus B$ and $B\setminus A$ have finite perimeter and we have the
following decompositions (up to sets of zero $\mathcal{H}^{d-1}$ measure) of
$\partial^{\ast}A$, $\partial^{\ast}B$, $\partial^{\ast}(A\setminus B)$ and
$\partial^{\ast}(B\setminus A)$ into disjoint sets:

$\displaystyle\partial^{\ast}A$
$\displaystyle=\Big{(}A^{(\nicefrac{{1}}{{2}})}\cap
B^{(0)}\Big{)}\cup\Big{(}A^{(\nicefrac{{1}}{{2}})}\cap
B^{(1)}\Big{)}\cup\Big{(}A^{(\nicefrac{{1}}{{2}})}\cap
B^{(\nicefrac{{1}}{{2}})}\Big{)}$ $\displaystyle\partial^{\ast}B$
$\displaystyle=\Big{(}B^{(\nicefrac{{1}}{{2}})}\cap
A^{(0)}\Big{)}\cup\Big{(}B^{(\nicefrac{{1}}{{2}})}\cap
A^{(1)}\Big{)}\cup\Big{(}B^{(\nicefrac{{1}}{{2}})}\cap
A^{(\nicefrac{{1}}{{2}})}\Big{)}$ $\displaystyle\partial^{\ast}(A\setminus B)$
$\displaystyle=\Big{(}A^{(1)}\cap
B^{(\nicefrac{{1}}{{2}})}\Big{)}\cup\Big{(}A^{(\nicefrac{{1}}{{2}})}\cap
B^{(0)}\Big{)}\cup\Big{(}A^{(\nicefrac{{1}}{{2}})}\cap
B^{(\nicefrac{{1}}{{2}})}\cap(A\cup B)^{(1)}\Big{)}$
$\displaystyle\partial^{\ast}(B\setminus A)$ $\displaystyle=\Big{(}B^{(1)}\cap
A^{(\nicefrac{{1}}{{2}})}\Big{)}\cup\Big{(}B^{(\nicefrac{{1}}{{2}})}\cap
A^{(0)}\Big{)}\cup\Big{(}B^{(\nicefrac{{1}}{{2}})}\cap
A^{(\nicefrac{{1}}{{2}})}\cap(A\cup B)^{(1)}\Big{)}.$

###  2.2. Sobolev functions and capacity

Let $u:\mathbb{R}^{d}\to\mathbb{R}$ be a measurable function. We recall that
$u$ is a Sobolev function ($u\in H^{1}(\mathbb{R}^{d})$), if $u\in
L^{2}(\mathbb{R}^{d})$ and $\nabla u\in L^{2}(\mathbb{R}^{d};\mathbb{R}^{d})$,
where $\nabla u$ is the distributional gradient of $u$. Given a measurable set
$\Omega\subset\mathbb{R}^{d}$ and a Sobolev function $u\in
H^{1}(\mathbb{R}^{d})$, we say that $u\in\widetilde{H}^{1}_{0}(\Omega)$ if

$u=0\quad\text{almost everywhere on}\quad\mathbb{R}^{d}\setminus\Omega.$

If $\Omega$ is an open set, we can also define the space $H^{1}_{0}(\Omega)$
as the closure of $C^{\infty}_{c}(\Omega)$ with respect to the Sobolev norm

$\|u\|_{H^{1}}:=\Big{(}\|u\|_{L^{2}}^{2}+\|\nabla
u\|_{L^{2}}^{2}\Big{)}^{\nicefrac{{1}}{{2}}}.$

It is well-known that both $H^{1}_{0}(\Omega)$ and
$\widetilde{H}^{1}_{0}(\Omega)$ are closed (with respect to both the strong
and the weak $H^{1}$-convergence) linear subspaces of $H^{1}(\mathbb{R}^{d})$
and that, for any open set $\Omega$,
$H^{1}_{0}(\Omega)\subset\widetilde{H}^{1}_{0}(\Omega)$, while the converse
inclusion is in general false.

Given any set $A\subset\mathbb{R}^{d}$ and any ball $B_{2R}(x_{0})$, we define

$\displaystyle\mathop{\rm
cap}\nolimits\Big{(}A;B_{2R}(x_{0})\Big{)}:=\inf\Big{\\{}$
$\displaystyle\int|\nabla\varphi|^{2}\,dx\ :\ \varphi\in
H^{1}_{0}(B_{2R}(x_{0})),$ $\displaystyle\qquad\varphi\geq 1\ \text{in a
neighborhood of}\ B_{R}(x_{0})\cap A\Big{\\}}.$

We say that a set $A$ has zero capacity if

$\mathop{\rm cap}\nolimits\Big{(}A;B_{2R}(x_{0})\Big{)}=0\qquad\text{for every
ball}\qquad B_{2R}(x_{0})\subset\mathbb{R}^{d}.$

We recall the following properties of the capacity.

  * •

If a set $A\subset\mathbb{R}^{d}$ has zero capacity, then $|A|=0$ and
$\mathcal{H}^{d-1}(A)=0$.

  * •

Given $u\in H^{1}(\mathbb{R}^{d})$, there exists a set of zero capacity
$\mathcal{N}_{u}$ such that

$\lim_{r\to
0}\frac{1}{|B_{r}(x_{0})|}\int_{B_{r}(x_{0})}u(x)\,dx\quad\text{exists}\quad\text{for
every}\quad x_{0}\in\mathbb{R}^{d}\setminus\mathcal{N}_{u}\,.$

In particular, to every $u\in H^{1}(\mathbb{R}^{d})$, we can associate a
representative

$\widetilde{u}:\mathbb{R}^{d}\to\mathbb{R}\,$

defined pointwise everywhere as follows:

$\widetilde{u}(x_{0}):=\lim_{r\to
0}\frac{1}{|B_{r}(x_{0})|}\int_{B_{r}(x_{0})}u(x)\,dx\quad\text{if}\quad
x_{0}\in\mathbb{R}^{d}\setminus\mathcal{N}_{u}\ ,$

while $\widetilde{u}(x_{0})=0$ if $x_{0}\in\mathcal{N}_{u}$.

  * •

Suppose that a sequence $u_{n}\in H^{1}(\mathbb{R}^{d})$ converges strongly in
$H^{1}$ to $u\in H^{1}(\mathbb{R}^{d})$. Let $\widetilde{u}_{n}$ and
$\widetilde{u}$ be the representatives defined above and let
$\mathcal{N}_{u_{n}}$ and $\mathcal{N}_{u}$ be the correspondig sets of zero
capacity. Then, there are a subsequence $u_{n_{k}}$ and a set of zero capacity
$\mathcal{N}$ such that

$\mathcal{N}_{u}\subset\mathcal{N}\qquad\text{and}\qquad\bigcup_{n\geq
1}\mathcal{N}_{u_{n}}\subset\mathcal{N},$

and

$\lim_{k\to\infty}\widetilde{u}_{n_{k}}(x)=\widetilde{u}(x)\quad\text{for
every}\quad x\in\mathbb{R}^{d}\setminus\mathcal{N}.$ (2.3)

For simplicity, we will identify any function $u\in H^{1}(\mathbb{R}^{d})$
with its representative $\widetilde{u}$ and if a sequence $u_{n_{k}}\in
H^{1}(\mathbb{R}^{d})$ satisfies (2.3), then we will say that it converges
quasi-everywhere to $u\in H^{1}(\mathbb{R}^{d})$.

###  2.3. Traces of Sobolev functions on the boundary of sets of finite
perimeter

Let $\Omega$ be a set of finite perimeter in $\mathbb{R}^{d}$ and let $u\in
H^{1}(\mathbb{R}^{d})$. Let $\widetilde{u}:\mathbb{R}^{d}\to\mathbb{R}$ be the
representative of $u$ defined for every point $x_{0}$ outside a set of zero
capacity $\mathcal{N}_{u}$ (and defined as zero on $\mathcal{N}_{u}$). Then,
$\widetilde{u}$ is defined at every point of
$\partial^{\ast}\Omega\setminus\mathcal{N}_{u}$. Since $\mathcal{N}_{u}$ has
zero $\mathcal{H}^{d-1}$-measure, we have that $\widetilde{u}$ is defined
$\mathcal{H}^{d-1}$-almost everywhere on $\partial^{\ast}\Omega$. We also
notice that

$\widetilde{u}:\partial^{\ast}\Omega\to\mathbb{R}$

is a $\mathcal{H}^{d-1}$ measurable function. Indeed, since $u\in
H^{1}(\mathbb{R}^{d})$ is a strong $H^{1}$ limit of a sequence of $C^{\infty}$
functions, we have that $\widetilde{u}:\partial^{\ast}\Omega\to\mathbb{R}$ is
a pointwise limit of smooth functions. From now on, we will write $u$ instead
of $\widetilde{u}$.

The next two propositions allow to write the functional $J_{\beta,\Lambda}$ in
an equivalent way.

######  Proposition 2.4.

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded quasi-open set of finite
perimeter and let $\partial^{\ast}\Omega$ be its reduced boundary. Let
$u\in\widetilde{H}^{1}_{0}(\Omega)$ and let
$\widetilde{u}:\mathbb{R}^{d}\to\mathbb{R}$ be a representative of $u$ defined
up to a set of zero capacity. Then,

$\widetilde{u}=0\quad\text{$\mathcal{H}^{d-1}$-almost everywhere
on}\quad\Omega^{(\nicefrac{{1}}{{2}})}.$

In particular,

$\widetilde{u}=0\quad\text{$\mathcal{H}^{d-1}$-almost everywhere
on}\quad\partial^{\ast}\Omega.$

###### Proof.

Without loss of generality, we can suppose that

$0\leq u\leq 1.$

For every $n\geq 1$, we consider the functional

$F_{n}:H^{1}_{0}(\Omega)\to\mathbb{R}\ ,\qquad F_{n}(v)=\int_{\Omega}|\nabla
v|^{2}\,dx+n\int_{\Omega}|v-u|^{2}\,dx.$

The functional $F_{n}$ admits a unique minimizer in $H^{1}_{0}(\Omega)$ that
we denote by $u_{n}$. By construction, testing the optimality of $u_{n}$ with
$v=u$, we get

$\int_{\Omega}|\nabla
u_{n}|^{2}\,dx+n\int_{\Omega}|u_{n}-u|^{2}\,dx\leq\int_{\Omega}|\nabla
u|^{2}\,dx.$

In particular, the sequence $u_{n}$ converges strongly $L^{2}(\Omega)$ and
weakly in $H^{1}_{0}(\Omega)$ to the function $u$. Moreover, $u_{n}$ solves
the PDE

$-\Delta u_{n}=n(u-u_{n})\quad\text{in}\quad\Omega\ ,\qquad u\in
H^{1}_{0}(\Omega).$ (2.4)

We notice that since $u_{n}$ minimizes $F_{n}$ and since $0\leq u\leq 1$, then
also

$0\leq u_{n}\leq 1.$

Thus, the right-hand side $n(u-u_{n})$ of (2.4) is bounded. Let now
$x_{0}\in\Omega^{(\nicefrac{{1}}{{2}})}$, that is,

$\lim_{r\to 0}\frac{|B_{r}(x_{0})\cap\Omega|}{|B_{r}(x_{0})|}=\frac{1}{2}.$

By [7, Proposition 4.6] we have that for $r>0$ small enough

$\|u_{n}\|_{L^{\infty}(B_{r}(x_{0})}\leq r^{\beta}C_{n},$

for some constant $C_{n}$ depending on $u_{n}$ and some dimensional constant
$\beta>0.$ In particular,

$u_{n}=0\quad\text{on}\quad\Omega^{(\nicefrac{{1}}{{2}})}.$

Now, since

$\int_{\Omega}|\nabla u_{n}|^{2}\,dx\leq\int_{\Omega}|\nabla
u|^{2}\,dx\qquad\text{for every}\qquad n\geq 1\ ,$

we get that the convergence of $u_{n}$ to $u$ is strong in
$H^{1}_{0}(\Omega)$. It is well-known that there is a subsequence of $u_{n}$
converging pointwise quasi-everywhere to $u$. In particular, the same
subsequence converges pointwise $\mathcal{H}^{d-1}$-almost everywhere on
$\partial^{\ast}\Omega$. Thus, (the representative of) $u$ vanishes
$\mathcal{H}^{d-1}$-almost everywhere on both $\Omega^{(\nicefrac{{1}}{{2}})}$
and $\partial^{\ast}\Omega$. ∎

######  Proposition 2.5.

Let $\Omega_{1}$ and $\Omega_{2}$ be two sets of finite perimeter in
$\mathbb{R}^{d}$ such that

$|\Omega_{1}\cap\Omega_{2}|=0.$

Then, for any function
$u\in\widetilde{H}^{1}_{0}(\Omega_{1}\cup\Omega_{2})\cap
L^{\infty}(\Omega_{1}\cup\Omega_{2})$, we have that

$\int_{\partial^{\ast}\Omega_{1}\cap\partial^{\ast}\Omega_{2}}u^{2}\,d\mathcal{H}^{d-1}=\frac{1}{2}\left(\int_{\partial^{\ast}\Omega_{1}}u^{2}\,d\mathcal{H}^{d-1}+\int_{\partial^{\ast}\Omega_{2}}u^{2}\,d\mathcal{H}^{d-1}\right).$
(2.5)

###### Proof.

We first notice that the reduced boundaries $\partial^{\ast}\Omega_{1}$ and
$\partial^{\ast}\Omega_{2}$ can be decomposed as

$\partial^{\ast}\Omega_{1}=\Big{(}\partial^{\ast}\Omega_{1}\cap\partial^{\ast}\Omega_{2}\Big{)}\cup\Big{(}\partial^{\ast}\Omega_{1}\setminus\partial^{\ast}\Omega_{2}\Big{)}\quad\text{and}\quad\partial^{\ast}\Omega_{2}=\Big{(}\partial^{\ast}\Omega_{1}\cap\partial^{\ast}\Omega_{2}\Big{)}\cup\Big{(}\partial^{\ast}\Omega_{2}\setminus\partial^{\ast}\Omega_{1}\Big{)}.$

Thus, in order to prove (2.5), it is sufficient to prove that

$u=0\qquad\text{$\mathcal{H}^{d-1}$-almost everywhere
on}\qquad\big{(}\partial^{\ast}\Omega_{1}\setminus\partial^{\ast}\Omega_{2}\big{)}\cup\big{(}\partial^{\ast}\Omega_{2}\setminus\partial^{\ast}\Omega_{1}\big{)}.$
(2.6)

Let $x_{0}\in\partial^{\ast}\Omega_{1}\setminus\partial^{\ast}\Omega_{2}$.
Since

$\mathcal{H}^{d-1}\Big{(}\mathbb{R}^{d}\setminus\big{(}\Omega_{2}^{(0)}\cup\Omega_{2}^{(1)}\cup\partial^{\ast}\Omega_{2}\big{)}\Big{)}=0,$

we can suppose that $x_{0}\in\Omega_{2}^{(0)}\cup\Omega_{2}^{(1)}$, but since
$x_{0}\in\Omega_{1}^{(\nicefrac{{1}}{{2}})}$ and $\Omega_{1}\cap\Omega_{2}=0$,
we get that necessarily $x_{0}\in\Omega_{2}^{(0)}$. But then,

$x_{0}\in(\Omega_{1}\cup\Omega_{2})^{(\nicefrac{{1}}{{2}})}.$

Thus, by Proposition 2.4, we get that $u(x_{0})=0$. This proves (2.6) and
(2.5). ∎

###  2.4. A semicontinuity lemma

In the proof of the main theorem we will repeatedly use the following lemma,
which is a restatement of a lemma from [9].

######  Lemma 2.6 ([9]).

Let $A\subset\mathbb{R}^{d}$ be a bounded open set. Let $u_{n}\in H^{1}(A)$ be
a sequence of functions converging to $u_{\infty}\in H^{1}(A)$ weakly in
$H^{1}(A)$, strongly in $L^{2}(A)$ and pointwise almost-everywhere. Let
$\Omega_{n}\subset A$ be a sequence of sets of locally finite perimeter in $A$
converging almost-everywhere (in $A$) to the set of locally finite perimeter
$\Omega_{\infty}\subset A$. Then,

$\int_{A\cap\partial^{\ast}\Omega_{\infty}}u_{\infty}^{2}\,d\mathcal{H}^{d-1}\leq\liminf_{n\to\infty}\int_{A\cap\partial^{\ast}\Omega_{n}}u_{n}^{2}\,d\mathcal{H}^{d-1}.$
(2.7)

###### Proof.

The proof is precisely the one from [9, Lemma 2.4]. We report it here for the
sake of completeness. The key observation is that given $u\in H^{1}(A)$ and a
set of locally finite perimeter $\Omega\subset A$, we have

$\int_{A\cap\partial^{\ast}\Omega}u^{2}\,d\mathcal{H}^{d-1}=\sup\left\\{\int_{A\cap\Omega}\text{\rm
div}\,\big{(}u^{2}\xi\big{)}\,dx\ :\ \xi\in C^{1}_{c}(A;\mathbb{R}^{d}),\
|\xi|\leq 1\right\\}.$

We now fix a vector field $\xi\in C^{1}_{c}(A;\mathbb{R}^{d}),\ |\xi|\leq 1$
and we compute

$\displaystyle\liminf_{n\to\infty}\int_{A\cap\partial^{\ast}\Omega_{n}}u_{n}^{2}\,d\mathcal{H}^{d-1}$
$\displaystyle\geq\liminf_{n\to\infty}\int_{A\cap\Omega_{n}}\text{div}\big{(}u_{n}^{2}\xi\big{)}\,dx$
$\displaystyle=\liminf_{n\to\infty}\int_{A}\Big{(}2\big{(}u_{n}\xi\big{)}\cdot\big{(}\text{1}_{\Omega_{n}}\nabla
u_{n}\big{)}+\big{(}u_{n}\,\text{1}_{\Omega_{n}}\big{)}\,(u_{n}\,\text{div}\,\xi)\Big{)}\,dx\,$

Now, since $\text{1}_{\Omega_{n}}\nabla u_{n}$ converges weakly in $L^{2}$ to
$\text{1}_{\Omega_{\infty}}\nabla u_{\infty}$, we get

$\displaystyle\liminf_{n\to\infty}\int_{A\cap\partial^{\ast}\Omega_{n}}u_{n}^{2}\,d\mathcal{H}^{d-1}$
$\displaystyle\geq\int_{A}\Big{(}2\big{(}u_{\infty}\xi\big{)}\cdot\big{(}\text{1}_{\Omega_{\infty}}\nabla
u_{\infty}\big{)}+\big{(}u_{\infty}\,\text{1}_{\Omega_{\infty}}\big{)}\,(u_{\infty}\,\text{div}\,\xi)\Big{)}\,dx\,$
$\displaystyle=\int_{A\cap\Omega_{\infty}}\text{div}\big{(}u_{\infty}^{2}\xi\big{)}\,dx.$

Taking the supremum over $\xi$, we get (2.7). ∎

##  3\. Almost-minimality and Hölder estimates

In this section, we prove two general technical results on the continuity of
subharmonic functions which are almost-minimizers of the Dirichlet energy in a
suitable sense. We will use these estimates in Sections 4 and 9.

######  Lemma 3.1 (A growth estimate).

Let $D$ be a bounded open set in $\mathbb{R}^{d}$, $x_{0}\in D$ and let $u\in
H^{1}(D)$ be a function such that

  1. (a)

$u$ is non-negative and subharmonic in $D$;

  2. (b)

there are constants $\alpha\in[0,1]$, $K>0$ and $r_{0}>0$ such that
$B_{r_{0}}(x_{0})\subset D$ and

$\int_{B_{r}(x_{0})}|\nabla
u|^{2}\,dx\leq\int_{B_{r}(x_{0})}|\nabla(u+\varphi)|^{2}\,dx+Kr^{d-1+\alpha},$

for every $r\in(0,r_{0})$ and every $\varphi\in H^{1}_{0}(B_{r}(x_{0}))$ with
$\varphi\geq 0$ in $B_{r}(x_{0})$.

Then, for every $r\leq r_{0}/4$, we have

$\,-\hskip-10.80003pt\int_{B_{r}(x_{0})}{u(x)\,dx}-u(x_{0})\leq\frac{C_{d}\sqrt{K}}{\alpha+1}\,{r^{\frac{1+\alpha}{2}}}.$

In particular, if $u(x_{0})=0$, then

$\|u\|_{L^{\infty}(B_{r}(x_{0}))}\leq\frac{C_{d}\sqrt{K}}{\alpha+1}\,{r^{\frac{1+\alpha}{2}}}\quad\text{for
every}\quad r\leq\frac{r_{0}}{8}.$

###### Proof.

Without loss of generality, we can suppose that $x_{0}=0$ and $D=B_{r_{0}}$.
Then, for every $r\in(0,r_{0})$, we have

$0\leq\,-\hskip-10.80003pt\int_{\partial
B_{r}}{u\,d\mathcal{H}^{d-1}}-u(0)\leq\frac{1}{d\omega_{d}}\int_{0}^{r}{s^{1-d}}\Delta
u(B_{s})\,ds.$

Thus, we only need to estimate $\Delta u(B_{r})$. In order to do so, we test
the optimality of $u$ with $u+t\varphi$, where
$\varphi(x):=\frac{1}{r}(2r-|x|)_{+}\,.$

$-2\int_{B_{r}}\nabla u\cdot\nabla\varphi\,dx\leq
t\int_{B_{r}}|\nabla\varphi|^{2}+\frac{K}{t}r^{d-1+\alpha}\leq
t\omega_{d}r^{d-2}+\frac{K}{t}r^{d-1+\alpha}.$

Taking

$t:=\sqrt{\frac{K}{\omega_{d}}}\,r^{\frac{1+\alpha}{2}},$

we obtain

$-2\int_{B_{r}}\nabla
u\cdot\nabla\varphi\,dx\leq\sqrt{{K}{\omega_{d}}}\,r^{d-\frac{3}{2}+\frac{\alpha}{2}},$

and so, for every $r\leq\frac{r_{0}}{2}$, we get

$\displaystyle\Delta u(B_{r/2})\leq\int_{B_{r}}\varphi(x)\Delta
u(x)\,dx\leq-\int_{B_{r}}\nabla\varphi\cdot\nabla
u\,dx\leq\frac{1}{2}\sqrt{{K}{\omega_{d}}}\,r^{d-\frac{3}{2}+\frac{\alpha}{2}}.$

Then, for every $r\leq\frac{r_{0}}{4}$, we can estimate

$\,-\hskip-10.80003pt\int_{\partial
B_{r}}{u\,d\mathcal{H}^{d-1}}-u(0)\leq\frac{2^{d-1}\sqrt{K\omega_{d}}}{d\omega_{d}}\int_{0}^{r}{s^{1-d}}s^{d-\frac{3}{2}+\frac{\alpha}{2}}\,ds\leq\frac{2^{d}\sqrt{K\omega_{d}}}{d\omega_{d}}\frac{r^{\frac{1+\alpha}{2}}}{\alpha+1}.$

Now, using the non-negativity and the subharmonicity of $u$, we get the claim.
∎

######  Lemma 3.2 (Hölder continuity).

Let $D$ be a bounded open set in $\mathbb{R}^{d}$ let $u\in H^{1}(D)$ be a
function which is subharmonic in $D$ and such that

$0\leq u\leq M\quad\text{in}\quad D,$

for some constant $M>0$. Given $\delta\in(0,1)$, we define the set

$D_{\delta}:=\big{\\{}x\in D\ :\ \text{\rm dist}(x,\partial
D)>\delta\big{\\}},$ (3.1)

and we suppose that there are constants $\alpha\in[0,1]$ and $K>0$ such that

$\int_{B_{r}(x_{0})}|\nabla
u|^{2}\,dx\leq\int_{B_{r}(x_{0})}|\nabla(u+\varphi)|^{2}\,dx+Kr^{d-1+\alpha},$

for every $x_{0}\in D_{\delta}$, every $r\in(0,\delta)$ and every $\varphi\in
H^{1}_{0}(B_{r}(x_{0}))$ with $\varphi\geq 0$ in $B_{r}(x_{0})$. Then,

$|u(x)-u(y)|\leq
C_{d}\left(M+\frac{\sqrt{K}}{\alpha+1}\right)\,{|x-y|^{\frac{1+\alpha}{3+\alpha}}},$

for every $x,y\in D_{\delta}$ such that
$|x-y|<\big{(}\frac{\delta}{16}\big{)}^{2}$

###### Proof.

We apply the previous lemma, to $x_{0}=y$ and $r=|x-y|^{\gamma}$. Then,

$\displaystyle u(x)-u(y)$
$\displaystyle\leq\,-\hskip-10.80003pt\int_{B_{r}(x)}u-u(y)\leq\frac{(r+|x-y|)^{d}}{r^{d}}\,-\hskip-10.80003pt\int_{B_{r+|x-y|}(y)}u-u(y)$
$\displaystyle\leq\frac{(r+|x-y|)^{d}-r^{d}}{r^{d}}M+\,-\hskip-10.80003pt\int_{B_{r+|x-y|}(y)}u-u(y)$
$\displaystyle\leq\frac{|x-y|}{r}Md\frac{(r+|x-y|)^{d-1}}{r^{d-1}}+\frac{C_{d}\sqrt{K}}{\alpha+1}\,{(r+|x-y|)^{\frac{1+\alpha}{2}}}.$

Now, since $|x-y|<1$ and $r=|x-y|^{\gamma}$, with $\gamma\in(0,1)$, we get

$\displaystyle u(x)-u(y)$ $\displaystyle\leq
d2^{d}M|x-y|^{1-\gamma}+\frac{2C_{d}\sqrt{K}}{\alpha+1}\,|x-y|^{\gamma\frac{1+\alpha}{2}}.$

Choosing $\gamma=\frac{2}{3+\alpha}$, we get that

$1-\gamma=\gamma\frac{1+\alpha}{2}=\frac{\alpha+1}{\alpha+3},$

and so we get the claim. Finally, we notice that we should have the inequality
$r+|x-y|<\frac{\delta}{8}$, which is satisfied for instance when
$|x-y|^{\gamma}<\frac{\delta}{16}$. ∎

##  4\. Non-degenerate approximating problems

In this section, we define a sequence of non-degenerate problems,
approximating (1.3), in which the competitors $u$ are a priori bounded from
below by a fixed positive constant.

We consider the family of approximating problems

$\min\Big{\\{}J_{\varepsilon}\big{(}u,\Omega_{1},\Omega_{2}\big{)}\,:\
u\in\mathcal{V},\
\big{(}\Omega_{1},\Omega_{2}\big{)}\in\mathcal{A}(u)\Big{\\}},$ (4.1)

where the functional $J_{\varepsilon}$ is defined as

$\displaystyle J_{\varepsilon}(u,\Omega_{1},\Omega_{2})$
$\displaystyle:=\int_{D}|\nabla u|^{2}\,dx+\Lambda|\\{u>0\\}\cap D|$
$\displaystyle\qquad+\frac{\beta}{2}\left(\int_{\partial^{\ast}\Omega_{1}}u^{2}\,d\mathcal{H}^{d-1}+\int_{\partial^{\ast}\Omega_{2}}u^{2}\,d\mathcal{H}^{d-1}\right)+{\varepsilon}\Big{(}\text{\rm
Per}(\Omega_{1})+\text{\rm Per}(\Omega_{2})\Big{)}.$

######  Proposition 4.1 (Solutions to the approximating problem).

Let $D$ be a bounded open set in $\mathbb{R}^{d}$. Then, for any
${\varepsilon}>0$ there are $u_{\varepsilon}\in\mathcal{V}$ and
$(\Omega^{1}_{\varepsilon},\Omega^{2}_{\varepsilon})\in\mathcal{A}(u_{\varepsilon})$
such that:

  1. (i)

$(u_{\varepsilon},\Omega^{1}_{\varepsilon},\Omega^{2}_{\varepsilon})$ is a
solution to (4.1);

  2. (ii)

the function $u_{\varepsilon}$ is Hölder continuous in $D$ and for every
$\delta>0$ there is a constant $C_{\delta}>0$, depending on $\delta$, $d$,
$\beta$, $\Lambda$ and $\|g\|_{L^{\infty}}$, such that

$|u_{\varepsilon}(x)-u_{\varepsilon}(y)|\leq
C_{\delta}|x-y|^{\nicefrac{{1}}{{3}}}\qquad\text{for every}\qquad x,y\in
D_{\delta},$ (4.2)

where $D_{\delta}$ is the set defined in (3.1).

  3. (iii)

there is a constant $\rho>0$, depending only on $d$, $\Lambda$ and
$\|g\|_{L^{\infty}}$, such that

$\Omega_{\varepsilon}^{1}\cup\Omega_{\varepsilon}^{2}\subset(E_{1}\cup
E_{2})+B_{\rho}.$ (4.3)

###### Proof.

Let ${\varepsilon}>0$ be fixed. We divide the proof into several steps.

Existence. Let
$\big{\\{}\big{(}u_{{\varepsilon},i},\Omega_{{\varepsilon},i}^{1},\Omega_{{\varepsilon},i}^{2}\big{)}\big{\\}}_{i\in\mathbb{N}}$
be a minimizing sequence for (4.1). Since we can use
$\big{(}g,E_{1},E_{2}\big{)}$ as a competitor against
$\big{(}u_{{\varepsilon},i},\Omega_{{\varepsilon},i}^{1},\Omega_{{\varepsilon},i}^{2}\big{)}$,
we have that

$\begin{array}[]{ll}\displaystyle\int_{D}|\nabla
u_{{\varepsilon},i}|^{2}\,dx+{\varepsilon}\Big{(}\text{\rm
Per}(\Omega^{1}_{{\varepsilon},i})+\text{\rm
Per}(\Omega^{2}_{{\varepsilon},i})\Big{)}\leq
J_{{\varepsilon}}(g,E_{1},E_{2})\leq J_{1}(g,E_{1},E_{2}).\end{array}$ (4.4)

Moreover, since $u_{{\varepsilon},i}-g\in H^{1}_{0}(D)$, by the Poincaré
inequality with constant $C_{D}$, we have

$\displaystyle\|u_{{\varepsilon},i}\|_{L^{2}(D)}$
$\displaystyle\leq\|u_{{\varepsilon},i}-g\|_{L^{2}(D)}+\|g\|_{L^{2}(D)}$
$\displaystyle\leq
C_{D}\|\nabla(u_{{\varepsilon},i}-g)\|_{L^{2}(D)}+\|g\|_{L^{2}(D)}.$

Thus, the sequence $\\{u_{{\varepsilon},i}\\}_{i\in\mathbb{N}}$ is bounded in
$H^{1}(D)$. Hence, up to a subsequence, there exists a function
$u_{\varepsilon}\in H^{1}_{0}(\mathbb{R}^{d})$ such that $\nabla
u_{{\varepsilon},i}\to\nabla u_{\varepsilon}$ weakly in
$L^{2}(\mathbb{R}^{d})$ and $u_{{\varepsilon},i}\to u_{\varepsilon}$ strongly
in $L^{2}(\mathbb{R}^{d})$ and pointwise almost-everywhere. In particular, the
almost-everywhere convergence gives that

$\text{1}_{\\{u_{\varepsilon}>0\\}}\leq\liminf_{i\to\infty}\text{1}_{\\{u_{{\varepsilon},i}>0\\}}.$
(4.5)

Using again (4.4), we obtain

$\text{Per}\big{(}\Omega_{{\varepsilon},i}^{1}\big{)}+\text{Per}\big{(}\Omega_{{\varepsilon},i}^{2}\big{)}\leq\frac{1}{{\varepsilon}}J_{1}(g,E_{1},E_{2}).$

Hence, there are sets of finite perimeter $\Omega^{1}_{\varepsilon}$ and
$\Omega_{\varepsilon}^{2}$ such that

$\Omega_{\varepsilon}^{1}\cap\Omega^{2}_{\varepsilon}=\emptyset\ ,\qquad
E_{1}\subset\Omega^{1}_{\varepsilon}\qquad\text{and}\qquad
E_{2}\subset\Omega^{2}_{\varepsilon},$

and such that (up to a subsequence)

$\text{1}_{\Omega_{{\varepsilon},i}^{1}}\to\text{1}_{\Omega_{\varepsilon}^{1}}\qquad\text{and}\qquad\text{1}_{\Omega_{{\varepsilon},i}^{2}}\to\text{1}_{\Omega_{\varepsilon}^{2}},$

strongly in $L^{1}(\mathbb{R}^{d})$ and pointwise almost-everywhere. Together
with (4.5), this implies that

$\\{u_{\varepsilon}>0\\}\subset\Omega_{\varepsilon}^{1}\cup\Omega_{\varepsilon}^{2}\
,$

so
$(\Omega_{\varepsilon}^{1},\Omega_{\varepsilon}^{2})\in\mathcal{A}(u_{\varepsilon})$.
Finally, the semicontinuity of $J_{\varepsilon}$ (see Lemma 2.6) gives that
$(u_{\varepsilon},\Omega^{1}_{\varepsilon},\Omega^{2}_{\varepsilon})$ is a
solution to the variational problem (4.1).

Subharmonicity of $u_{\varepsilon}$. Let $v$ be a function in $H^{1}(D)$ such
that

$v\leq u_{\varepsilon}\quad\text{in}\quad D\qquad\text{and}\qquad
v=u_{\varepsilon}\quad\text{on}\quad\partial D\,.$

Then, testing the optimality of
$(u_{\varepsilon},\Omega^{1}_{\varepsilon},\Omega^{2}_{\varepsilon})$ with
$(v_{+},\Omega^{1}_{\varepsilon},\Omega^{2}_{\varepsilon})$ and using the fact
that

$v_{+}\leq u_{\varepsilon}\quad\text{in}\quad D,$

we get

$\displaystyle\int_{D}|\nabla v|^{2}\,dx$ $\displaystyle\geq\int_{D}|\nabla
v_{+}|^{2}\,dx\geq\int_{D}|\nabla u_{\varepsilon}|^{2}\,dx.$

In particular, if $\varphi$ is a nonnegative function compactly supported in
$D$, then we can apply the above inequality to $v:=u-t\varphi$ for some $t>0$.
Then , by sending $t$ to zero, we get that

$-\int_{D}\nabla\varphi\cdot\nabla u_{\varepsilon}\,dx\geq 0,$

which means that the distributional Laplacian $\Delta u_{\varepsilon}$ is a
positive Radon measure in $D$.

Hölder continuity of $u_{\varepsilon}$. Let $\delta\in(0,1)$ and let
$D_{\delta}$ be given by (3.1). Let $x_{0}\in D_{\delta}$ and $0<R<\delta$.
Let $\varphi\in H^{1}_{0}(B_{R}(x_{0}))$ be such that $\varphi\geq 0$ in
$B_{R}(x_{0})$. We set

$\widetilde{\Omega}_{\varepsilon}^{1}:=\Omega_{\varepsilon}^{1}\cup
B_{R}(x_{0})\qquad\text{and}\qquad\widetilde{\Omega}_{\varepsilon}^{2}:=\Omega_{\varepsilon}^{2}\setminus
B_{R}(x_{0}),$

so that
$(\widetilde{\Omega}_{\varepsilon}^{1},\widetilde{\Omega}_{\varepsilon}^{2})\in\mathcal{A}(u_{\varepsilon})$.
Using
$(u_{\varepsilon}+\varphi,\widetilde{\Omega}_{\varepsilon}^{1},\widetilde{\Omega}_{\varepsilon}^{2})$
to test the optimality of
$(u_{\varepsilon},\Omega^{1}_{\varepsilon},\Omega_{\varepsilon}^{2})$, we get

$\displaystyle\int_{B_{R}(x_{0})}$ $\displaystyle|\nabla
u_{\varepsilon}|^{2}\,dx+\Lambda|B_{R}(x_{0})\cap\\{u_{\varepsilon}>0\\}|+\frac{\beta}{2}\int_{\partial^{\ast}\Omega_{\varepsilon}^{1}}u_{\varepsilon}^{2}\,d\mathcal{H}^{d-1}+\frac{\beta}{2}\int_{\partial^{\ast}\Omega_{\varepsilon}^{2}}u_{\varepsilon}^{2}\,d\mathcal{H}^{d-1}$
$\displaystyle\leq\int_{B_{R}(x_{0})}|\nabla(u_{\varepsilon}+\varphi)|^{2}\,dx+\Lambda|B_{R}|+\frac{\beta}{2}\int_{\partial^{\ast}\widetilde{\Omega}_{\varepsilon}^{1}}u_{\varepsilon}^{2}\,d\mathcal{H}^{d-1}+\frac{\beta}{2}\int_{\partial^{\ast}\widetilde{\Omega}_{\varepsilon}^{2}}u_{\varepsilon}^{2}\,d\mathcal{H}^{d-1}+{\varepsilon}\mathcal{H}^{d-1}(\partial
B_{R}).$

Moreover, the definition of
$(u+\varphi,\widetilde{\Omega}^{1}_{\varepsilon},\widetilde{\Omega}^{2}_{\varepsilon})$
gives that, for $j=1,2$,

$\begin{split}\int_{\partial^{\ast}\widetilde{\Omega}_{\varepsilon}^{j}}u_{\varepsilon}^{2}\,d\mathcal{H}^{d-1}&\leq\int_{\partial^{\ast}\Omega_{\varepsilon}^{j}}u_{\varepsilon}^{2}\,d\mathcal{H}^{d-1}+\int_{\partial
B_{R}(x_{0})}u_{\varepsilon}^{2}\,d\mathcal{H}^{d-1}\\\
&\leq\int_{\partial^{\ast}\Omega_{\varepsilon}^{j}}u_{\varepsilon}^{2}\,d\mathcal{H}^{d-1}+d\omega_{d}R^{d-1}\|g\|_{L^{\infty}(D)}^{2}.\end{split}$
(4.6)

Combining these two estimates, we obtain

$\displaystyle\int_{B_{R}(x_{0})}|\nabla u_{\varepsilon}|^{2}\,dx$
$\displaystyle\leq\int_{B_{R}(x_{0})}|\nabla(u_{\varepsilon}+\varphi)|^{2}\,dx+\Lambda|B_{R}|+\big{(}{\varepsilon}+\beta\|g\|_{L^{\infty}(D)}^{2}\big{)}\,d\omega_{d}R^{d-1}$
$\displaystyle\leq\int_{B_{R}(x_{0})}|\nabla(u_{\varepsilon}+\varphi)|^{2}\,dx+\Big{(}\omega_{d}\Lambda\delta+\big{(}1+\beta\|g\|_{L^{\infty}(D)}^{2}\big{)}\,d\omega_{d}\Big{)}R^{d-1}.$

By Lemma 3.2 we get that

$|u_{\varepsilon}(x)-u_{\varepsilon}(y)|\leq
C|x-y|^{\nicefrac{{1}}{{3}}}\qquad\text{for every}\qquad x,y\in D_{\delta},$
(4.7)

such that $\displaystyle|x-y|\leq\frac{\delta^{2}}{256}$, where the constant
$C$ depends only on $d$, $\Lambda$, $\beta$ and $\|g\|_{L^{\infty}}$.

Boundedness of $\Omega_{\varepsilon}^{1}$ and $\Omega_{\varepsilon}^{2}$. We
first notice that $u_{\varepsilon}$ is a subsolution of the Alt-Caffarelli
functional, that is,

$\int_{\mathbb{R}^{d}}|\nabla
u_{\varepsilon}|^{2}\,dx+\Lambda|\\{u_{\varepsilon}>0\\}|\leq\int_{\mathbb{R}^{d}}|\nabla
v|^{2}\,dx+\Lambda|\\{v>0\\}|,$

for every $v\in H^{1}(\mathbb{R}^{d})$ such that $u-v\in H^{1}_{0}(D)$ and
$0\leq v\leq u_{\varepsilon}$ on $\mathbb{R}^{d}$. In particular, this implies
(see for instance [15]) that the set $\\{u_{\varepsilon}>0\\}$ lies in a
sufficiently large ball $B_{\rho}$. Now, since outside
$\\{u_{\varepsilon}>0\\}$ the functional $J_{\varepsilon}$ only accounts for
the perimeter of $\Omega_{\varepsilon}^{1}$ and $\Omega_{\varepsilon}^{2}$, we
have that these sets should be contained in the convex envelope of
$\\{u_{\varepsilon}>0\\}$, which in particular gives (4.3). ∎

##  5\. The limit of the non-degenerate solutions

In this section we define the function $u$ (Section 5.1) and the sets
$(\Omega_{1},\Omega_{2})$ (Section 5.2), which we will prove to be solutions
to the initial problem (1.3). Throughout this section, for any
${\varepsilon}>0$, we fix a solution
$\big{(}u_{\varepsilon},\Omega_{\varepsilon}^{1},\Omega_{\varepsilon}^{2}\big{)}$
of the approximating problem (4.1) for $J_{\varepsilon}$.

###  5.1. The limit function

It is immediate to check that, there is a function

$u\in H^{1}(\mathbb{R}^{d}),$

and a sequence ${\varepsilon}_{n}\to 0$ such that

  * •

for every fixed $\delta>0$, $u_{{\varepsilon}_{n}}\to u$ uniformly in
$D_{\delta}$ as $n\to\infty$ ;

  * •

$u_{{\varepsilon}_{n}}\to u$ strongly in $L^{2}(\mathbb{R}^{d})$ and pointwise
almost-everywhere in $\mathbb{R}^{d}$ ;

  * •

$\nabla u_{{\varepsilon}_{n}}\to\nabla u$ weakly in $L^{2}(\mathbb{R}^{d})$.

By construction, we have $u-g\in H^{1}_{0}(D)$, while 4.1 gives that

$u\in H^{1}_{0}\big{(}(E_{1}\cup E_{2})+B_{\rho}\big{)}\qquad\text{and}\qquad
u\in C^{0,\nicefrac{{1}}{{3}}}(\overline{D}_{\delta})\quad\text{for
every}\quad\delta>0.$

Moreover,

$0\leq u\leq\|g\|_{L^{\infty}(\mathbb{R}^{d})}\qquad\text{and}\qquad\Delta
u\geq 0\quad\text{in}\quad D.$

###  5.2. The limit sets

We next construct the sets $\Omega_{1}$ and $\Omega_{2}$. Choose a ball

$B_{R}(x_{0})\subset\subset D\cap\\{u>0\\}.$

Then, there are $t>0$ and $\delta>0$ such that

$\overline{B}_{R}(x_{0})\subset D_{\delta}\cap\\{u\geq t\\},$

where $D_{\delta}$ is given by (3.1). By the uniform convergence of
$u_{{\varepsilon}_{n}}$ to $u$ on $D_{\delta}$, we can find $n_{0}\geq 1$ such
that, the following inequality holds for every $n\geq n_{0}$:

$u_{{\varepsilon}_{n}}\geq\frac{t}{2}\quad\text{in}\quad\overline{B}_{R}(x_{0}).$

Using this inequality and the optimality of
$\big{(}u_{{\varepsilon}_{n}},\Omega^{1}_{{\varepsilon}_{n}},\Omega^{2}_{{\varepsilon}_{n}}\big{)}$,
we can estimate

$\displaystyle\frac{2}{\beta}J_{\varepsilon}\big{(}g,E_{1},E_{2}\big{)}\geq\frac{2}{\beta}J_{\varepsilon}\big{(}u_{{\varepsilon}_{n}},\Omega^{1}_{{\varepsilon}_{n}},\Omega^{2}_{{\varepsilon}_{n}}\big{)}$
$\displaystyle\geq\int_{B_{R}(x_{0})\cap\partial^{\ast}\Omega^{1}_{{\varepsilon}_{n}}}u_{{\varepsilon}_{n}}^{2}\,d\mathcal{H}^{d-1}+\int_{B_{R}(x_{0})\cap\partial^{\ast}\Omega_{{\varepsilon}_{n}}^{2}}u_{{\varepsilon}_{n}}^{2}\,d\mathcal{H}^{d-1}$
$\displaystyle\geq\frac{t^{2}}{4}\Big{(}\text{\rm
Per}\big{(}\Omega_{{\varepsilon}_{n}}^{1};B_{R}(x_{0})\big{)}+\text{\rm
Per}\big{(}\Omega_{{\varepsilon}_{n}}^{2};B_{R}(x_{0})\big{)}\Big{)}.$

Thus, the sets

$\Omega^{1}_{{\varepsilon}_{n}}\cap
B_{R}(x_{0})\qquad\text{and}\qquad\Omega^{2}_{{\varepsilon}_{n}}\cap
B_{R}(x_{0})$

have uniformly bounded perimeter. In particular, up to a subsequence there are
sets

$\Omega_{R,x_{0}}^{1}\cap
B_{R}(x_{0})\qquad\text{and}\qquad\Omega_{R,x_{0}}^{2}\cap B_{R}(x_{0}),$

of finite perimeter and such that, as $n\to\infty$,

$\text{1}_{\Omega^{1}_{{\varepsilon}_{n}}\cap
B_{R}(x_{0})}\to\text{1}_{\Omega_{R,x_{0}}^{1}}\qquad\text{and}\qquad\text{1}_{\Omega^{2}_{{\varepsilon}_{n}}\cap
B_{R}(x_{0})}\to\text{1}_{\Omega_{R,x_{0}}^{2}},$

pointwise almost-everywhere and strongly in $L^{1}(\mathbb{R}^{d})$. Thus, by
a diagonal sequence argument, we can define the sets $\Omega_{1}$ and
$\Omega_{2}$ as the union of $\Omega^{1}_{R,x_{0}}$ and $\Omega^{2}_{R,x_{0}}$
over all balls

$B_{R}(x_{0})\subset\subset D\cap\\{u>0\\},$

of radius $R\in\mathbb{Q}$ and center with rational coordinates
$x_{0}\in\mathbb{Q}^{d}$,

$\Omega_{i}:=E_{i}\cup\bigcup_{R,x_{0}}\Omega^{i}_{R,x_{0}}\qquad\text{for}\quad
i=1,2.$

By construction, $\Omega_{1}$ and $\Omega_{2}$ have locally finite perimeter
in $D\cap\\{u>0\\}$ and satisfy

$E_{i}\subset\Omega_{i}\subset\Big{(}\big{(}E_{1}\cup E_{2}\big{)}\cap
B_{\rho}\Big{)},$

where we recall that
$D:=\mathbb{R}^{d}\setminus\big{(}\overline{E}_{1}\cup\overline{E}_{2}\big{)}$.
Moreover, we still have the pointwise convergence of the corresponding
characteristic functions, that is, for $i=1,2$,

$\text{1}_{\Omega_{i}}=\lim_{n\to\infty}\text{1}_{\Omega^{i}_{{\varepsilon}_{n}}\cap\\{u>0\\}}\quad\text{in}\quad\mathbb{R}^{d},$

which implies that, for almost every $x\in\mathbb{R}^{d}$,

$\text{1}_{\Omega_{1}\cap\,\Omega_{2}}(x)=\text{1}_{\Omega_{1}}(x)\cdot\text{1}_{\Omega_{2}}(x)=\lim_{n\to\infty}\Big{(}\text{1}_{\Omega^{1}_{{\varepsilon}_{n}}}(x)\cdot\text{1}_{\Omega^{2}_{{\varepsilon}_{n}}}(x)\cdot\text{1}_{\\{u>0\\}}\Big{)}=0,$

the sets $\Omega_{1}$ and $\Omega_{2}$ are disjoint,
$|\Omega_{1}\cap\Omega_{2}|=0$.

######  Remark 5.1.

Notice that we do not have a priori that $\Omega_{1}$ and $\Omega_{2}$ have
finite perimeter in $\mathbb{R}^{d}$, so at this stage they might not be in
the admissible class $\mathcal{A}(u)$ defined in Section 1.3.

##  6\. Almost-minimality and Lipschitz estimates of $u$

In this section we will show that $u$ is an almost-minimizer (in some suitable
sense) of the classical one-phase functional of Alt and Caffarelli. From this
we deduce the Lipschitz growth of $u$ on the boundary, which we will use in
Section 9.1 in order to deduce the convergence of the blow-up sequences of
$u$.

######  Lemma 6.1.

Let $u\in H^{1}(\mathbb{R}^{d})$ be the function from Section 5.1 and let
$\overline{B}_{r}(x_{0})\subset\mathbb{R}^{d}\setminus\overline{E}_{2}$.
Suppose that $v\in H^{1}(\mathbb{R}^{d})$ is a function such that $v-u\in
H^{1}_{0}(B_{r}(x_{0}))$. Then,

$\displaystyle\displaystyle\int_{B_{r}(x_{0})}|\nabla u|^{2}\,dx$
$\displaystyle+\Lambda|B_{r}(x_{0})\cap\\{u>0\\}|$
$\displaystyle\leq\int_{B_{r}(x_{0})}|\nabla
v|^{2}\,dx+\Lambda|B_{r}(x_{0})\cap\\{v>0\\}|+\beta\int_{\partial
B_{r}(x_{0})}u^{2}d\mathcal{H}^{d-1}.$

###### Proof.

We can suppose that $v\geq 0$. Then, testing the minimality of
$\big{(}u_{{\varepsilon}_{n}},\Omega_{{\varepsilon}_{n}}^{1},\Omega_{{\varepsilon}_{n}}^{2}\big{)}$,
with the function $v$ (which we can do since $v-g\in H^{1}_{0}(D)$) and the
sets

$\Omega_{{\varepsilon}_{n}}^{1}\cup
B_{r}(x_{0})\qquad\text{and}\qquad\Omega_{{\varepsilon}_{n}}^{2}\setminus
B_{r}(x_{0}),$

we obtain

$\displaystyle\int_{D}|\nabla u_{{\varepsilon}_{n}}|^{2}\,dx$
$\displaystyle+\Lambda|D\cap\\{u_{{\varepsilon}_{n}}>0\\}|$
$\displaystyle\leq\int_{D}|\nabla
v|^{2}\,dx+\Lambda|D\cap\\{v>0\\}|+\beta\int_{\partial
B_{r}(x_{0})}u_{{\varepsilon}_{n}}^{2}\,d\mathcal{H}^{d-1}.$

Passing to the limit as $n\to\infty$, we get the claim. ∎

######  Lemma 6.2.

Let $u\in H^{1}(\mathbb{R}^{d})$ be the function defined in Section 5.1. For
every $\delta\in(0,1)$ and every $x_{0}\in D_{\delta}\cap\\{u=0\\}$, we have

$\|u\|_{L^{\infty}(B_{r}(x_{0}))}\leq Cr\quad\text{for every}\qquad
r<\frac{\delta}{256},$

where $C$ is a constant depending only on $d$, $\beta$, $\Lambda$ and
$\|g\|_{L^{\infty}}$.

###### Proof.

We set for simplicity $M:=\|g\|_{L^{\infty}}$. By Lemma 6.1, we have that for
every $r\in(0,\delta)$

$\displaystyle\displaystyle\int_{B_{r}(x_{0})}|\nabla u|^{2}\,dx$
$\displaystyle\leq\int_{B_{r}(x_{0})}|\nabla(u+\varphi)|^{2}\,dx+\Lambda|B_{r}|+\beta\int_{\partial
B_{r}(x_{0})}u^{2}\,d\mathcal{H}^{d-1}$
$\displaystyle\leq\int_{B_{r}(x_{0})}|\nabla(u+\varphi)|^{2}\,dx+\Big{(}\Lambda\omega_{d}\delta+\beta
d\omega_{d}M^{2}\Big{)}r^{d-1},$

for every $\varphi\in H^{1}_{0}(B_{r}(x_{0}))$ such that $\varphi\geq 0$ in
$B_{r}(x_{0})$. Applying Lemma 3.1, we obtain

$\|u\|_{L^{\infty}(B_{r}(x_{0}))}\leq C_{d}\big{(}\Lambda+\beta
M^{2}\big{)}^{\nicefrac{{1}}{{2}}}\,{r^{\nicefrac{{1}}{{2}}}}\quad\text{for
every}\quad r\leq\frac{\delta}{8}.$

Using this estimate in Lemma 6.1, we get that, for every
$r\in(0,\nicefrac{{\delta}}{{8}})$,

$\displaystyle\displaystyle\int_{B_{r}(x_{0})}|\nabla u|^{2}\,dx$
$\displaystyle\leq\int_{B_{r}(x_{0})}|\nabla(u+\varphi)|^{2}\,dx+\Lambda|B_{r}|+\beta\int_{\partial
B_{r}(x_{0})}u^{2}\,d\mathcal{H}^{d-1}$
$\displaystyle\leq\int_{B_{r}(x_{0})}|\nabla(u+\varphi)|^{2}\,dx+C_{d}\Big{(}\Lambda+\beta\big{(}\Lambda+\beta
M^{2}\big{)}\Big{)}r^{d},$

for every nonnegative $\varphi\in H^{1}_{0}(B_{r}(x_{0}))$. Applying again
Lemma 3.1, we get the claim. ∎

##  7\. Non-degeneracy of $u$

In this section we show that $u$ is a subsolution of the Alt-Caffarelli
functional. From this information, we can immediately deduce that $\\{u>0\\}$
has finite perimeter in $D_{\delta}$, for every $\delta>0$. Moreover, the
suboptimality of $u$ implies that it is non-degenerate, which assures that the
blow-up limits of $u$ are not identically zero.

######  Lemma 7.1.

Let $u\in H^{1}(\mathbb{R}^{d})$ be the function from Section 5.1. Then,

$\displaystyle\int_{D}|\nabla u|^{2}\,dx+\Lambda\big{|}\\{u>0\\}\cap
D\big{|}\leq\int_{D}|\nabla v|^{2}\,dx+\Lambda\big{|}\\{v>0\\}\cap D\big{|},$

for any $v\in H^{1}(\mathbb{R}^{d})$ such that

$v-u\in H^{1}_{0}(D)\qquad\text{and}\qquad 0\leq v\leq u\quad\text{in}\quad
D.$

###### Proof.

Testing the optimality of
$\big{(}u_{{\varepsilon}_{n}},\Omega_{{\varepsilon}_{n}}^{1},\Omega_{{\varepsilon}_{n}}^{2}\big{)}$
with
$\big{(}v,\Omega_{{\varepsilon}_{n}}^{1},\Omega_{{\varepsilon}_{n}}^{2}\big{)}$,
we obtain

$\displaystyle\int_{D}|\nabla
u_{{\varepsilon}_{n}}|^{2}\,dx+\Lambda|\\{u_{{\varepsilon}_{n}}>0\\}\cap
D|\leq\int_{D}|\nabla v|^{2}\,dx+\Lambda|\\{v>0\\}\cap D|.$

Passing to the limit as $n\to\infty$, we get the claim. ∎

As an immediate consequence, we have

######  Corollary 7.2.

Let $u\in H^{1}(\mathbb{R}^{d})$ be the function defined in Section 5.1. Then:

  1. (i)

the set $\\{u>0\\}$ has locally finite perimeter in $D$;

  2. (ii)

there is a constant $\eta>0$ such that for every $x_{0}\in
D\cap\overline{\\{u>0\\}}$

$\|u\|_{L^{\infty}(B_{r}(x_{0}))}\geq\eta\,r\qquad\text{for every}\qquad
B_{2r}(x_{0})\subset D.$

###### Proof.

See [1] or [15]. ∎

##  8\. Density estimate and its consequences

In this section, we will show that the free boundary $\partial\\{u>0\\}$ is
not touching $\partial E_{1}$ and $\partial E_{2}$. This is a crucial step in
proving that $\Omega_{1}$ and $\Omega_{2}$ have finite perimeter in
$\mathbb{R}^{d}$. The main result is the following.

######  Proposition 8.1 (Non-collapsing).

Let $u\in H^{1}(\mathbb{R}^{d})$ be the function from Section 5.1. Then, there
is a positive constant $t>0$ such that $u\geq t$ in a neighborhood of
$\overline{E}_{1}\cup\overline{E}_{2}$.

The proof of 8.1 is based on the following lemma.

######  Lemma 8.2 (Density estimate).

Let $u\in H^{1}(\mathbb{R}^{d})$ be the function from Section 5.1. There is a
constant $c>0$ and $R_{0}>0$, depending only on $d$, $\beta$, $\Lambda$ and
$\|g\|_{L^{\infty}}$, such that

$\big{|}B_{R}(x_{0})\cap\\{u=0\\}\big{|}\geq c|B_{R}|\quad\text{for
every}\quad B_{R}(x_{0})\subset D\quad\text{with}\quad u(x_{0})=0.$

###### Proof.

We notice that by Lemma 6.2, we have that

$\|u\|_{L^{\infty}(B_{\rho}(x_{0}))}\leq C\rho\quad\text{for
every}\qquad\rho<\frac{R}{256}.$ (8.1)

Now, we consider the competitor

$\widetilde{\Omega}_{1}:=\Omega_{1}\cup B_{\nicefrac{{R}}{{256}}}(x_{0})\
,\quad\widetilde{\Omega}_{2}:=\Omega_{2}\setminus
B_{\nicefrac{{R}}{{256}}}(x_{0})\
,\quad\widetilde{u}(x):=\begin{cases}u(x)\quad\text{if}\quad
x\in\mathbb{R}^{d}\setminus B_{\nicefrac{{R}}{{256}}}(x_{0})\\\
h(x)\quad\text{if}\quad x\in B_{\nicefrac{{R}}{{256}}}(x_{0})\end{cases},$

$h$ being the harmonic extension of $u$ in $B_{\nicefrac{{R}}{{256}}}(x_{0})$.
Setting for simplicity $r=\frac{R}{256}$ and $x_{0}=0$, by Lemma 6.1, we have
that

$\displaystyle\displaystyle\int_{B_{r}}|\nabla
u|^{2}\,dx+\Lambda|B_{r}\cap\\{u>0\\}|\leq\int_{B_{r}}|\nabla
h|^{2}\,dx+\Lambda|B_{r}\cap\\{h>0\\}|+\beta d\omega_{d}r^{d-1}C^{2}r^{2}.$

The rest of the proof follows the analogous lemma from [1]. Using the fact
that $h$ is harmonic and strictly positive in $B_{r}$, we have

$\displaystyle\displaystyle\int_{B_{r}}|\nabla(u-h)|^{2}\,dx\leq\Lambda|B_{r}\cap\\{u=0\\}|+\beta
C^{2}d\omega_{d}r^{d+1}.$

By the Poincaré inequality, there is a dimensional constant $C_{d}$ such that

$\displaystyle\left(\frac{1}{|B_{r}|}\int_{B_{r}}(h-u)\,dx\right)^{2}\leq\frac{1}{|B_{r}|}\int_{B_{r}}(h-u)^{2}\,dx\leq\frac{C_{d}r^{2}}{|B_{r}|}\Big{(}\Lambda|B_{r}\cap\\{u=0\\}|+\beta
C^{2}r^{d+1}\Big{)}.$

On the other hand, the combination of 7.2 and the subharmonicity of $u$ in
$B_{R}$, gives that for any $\kappa\in(0,1)$

$\frac{1}{|B_{\kappa r}|}\int_{B_{\kappa
r}}h(x)\,dx=\,-\hskip-10.80003pt\int_{\partial B_{r}}u\geq 2^{-d}cr.$

On the other hand, (8.1) implies that

$\frac{1}{|B_{\kappa r}|}\int_{B_{\kappa r}}u(x)\,dx\leq C\kappa r.$

Thus,

$\displaystyle(c-C2^{d}\kappa)^{2}\leq\frac{C_{d}}{|B_{r}|}\Big{(}\Lambda|B_{r}\cap\\{u=0\\}|+\beta
C^{2}r^{d+1}\Big{)}.$

Choosing $\kappa$ and $R>0$ small enough, we get the claim. ∎

###### Proof of 8.1.

Suppose that $x_{0}\in D$ is such that

$u(x_{0})=0\qquad\text{and}\qquad\delta:=\text{\rm dist}(x_{0},\partial
E_{1})<\delta_{0},$

where the constant $\delta_{0}>0$ will be chosen later. Let $y_{0}$ be the
projection of $x_{0}$ on $E_{1}$ and let $R:=2|x_{0}-y_{0}|$. We consider the
competitor

$\widetilde{\Omega}_{1}:=\Omega_{1}\cup B_{R}(y_{0})\
,\quad\widetilde{\Omega}_{2}:=\Omega_{2}\setminus B_{R}(y_{0})\
,\quad\widetilde{u}(x):=\begin{cases}u(x)\quad\text{if}\quad
x\in\mathbb{R}^{d}\setminus\big{(}B_{R}(y_{0})\cap D\big{)}\\\
h(x)\quad\text{if}\quad x\in B_{R}(y_{0})\cap D\end{cases},$

$h$ being the harmonic extension of $u$ in $B_{R}(y_{0})\cap D$. We notice
that $\widetilde{u}$ is the solution to

$\min\Big{\\{}\int_{B_{R}(y_{0})}|\nabla v|^{2}\,dx\ :\ u-v\in
H^{1}_{0}\big{(}B_{R}(y_{0})\big{)}\Big{\\}}.$

Thus, by [15, Lemma 3.7] and the fact that $u=g$ on $E_{1}$, we have that

$\frac{1}{R^{2}}\big{|}\\{u=0\\}\cap
B_{R}(y_{0})\big{|}\left(\,-\hskip-10.80003pt\int_{\partial
B_{R}(y_{0})}{u\,d\mathcal{H}^{d-1}}\right)^{2}\leq
C_{d}\int_{B_{R}(y_{0})}|\nabla(u-h)|^{2}\,dx.$ (8.2)

On the other hand, using Lemma 6.1, we get

$\displaystyle\int_{B_{R}(y_{0})}|\nabla(u-h)|^{2}\,dx\leq\Lambda|B_{R}(y_{0})\cap\\{u=0\\}|+\beta\int_{\partial
B_{R}(y_{0})}u^{2}d\mathcal{H}^{d-1}.$ (8.3)

Now, since $u\equiv 1$ on $E_{1}$ and $u\leq 1$ in $\mathbb{R}^{d}$, we get
that

$\frac{1}{C_{1}}\leq\,-\hskip-10.80003pt\int_{\partial
B_{R}(y_{0})}{u\,d\mathcal{H}^{d-1}}\qquad\text{and}\qquad\int_{\partial
B_{R}(y_{0})}u^{2}d\mathcal{H}^{d-1}\leq d\omega_{d}R^{d-1},$

where $C_{1}>0$ is a constant depending only on $E_{1}$ (notice that since
$E_{1}$ is regular, for $\delta>0$ small, we can choose $C_{1}\simeq 2$).
Thus, combining (8.2) and (8.3), we get

$\frac{1}{R^{2}}\big{|}\\{u=0\\}\cap B_{R}(y_{0})\big{|}\leq
C_{1}^{2}C_{d}\Big{(}\Lambda+\frac{\beta}{R}\Big{)}|B_{R}|\,,$

which by the density estimate Lemma 8.2, gives a contradiction when $R$ is
small enough. ∎

##  9\. Regularity of the free boundary $\partial\\{u>0\\}$

In this section we prove the following.

######  Theorem 9.1.

Let $u\in H^{1}(\mathbb{R}^{d})$ be the function defined in Section 5.1 and
let $d=2$. Then $\partial\\{u>0\\}\cap D$ is a $C^{1,\alpha}$ regular
$(d-1)$-dimensional manifold.

The proof is based on the fact that $u$ satisfies an almost-minimality
condition in $D$. Precisely, by Lemma 3.1 and Lemma 6.1, we have that, given a
compact set $K\subset D$, there are constants $C>0$ and $r_{0}>0$ for which

$\displaystyle\displaystyle\int_{B_{r}(x_{0})}|\nabla u|^{2}\,dx$
$\displaystyle+\Lambda|B_{r}(x_{0})\cap\\{u>0\\}|$
$\displaystyle\leq\int_{B_{r}(x_{0})}|\nabla
v|^{2}\,dx+\Lambda|B_{r}(x_{0})\cap\\{v>0\\}|+Cr^{d+1},$ (9.1)

for every $r<r_{0}$, $x_{0}\in\partial\\{u>0\\}\cap D$ and $v\in
H^{1}(B_{r}(x_{0}))$ such that $u-v\in H^{1}_{0}(B_{r}(x_{0}))$. In Section
9.1, we use the almost-minimality to show that when $d\leq 4$ every blow-up
$u$ is a half-plane solution (that is, solution of the form (9.2)) of the
classical Alt-Caffarelli functional. Then, in Section 9.2, we show that in
dimension $d=2$, we can use the epiperimetric inequality from [14] to conclude
the proof of 9.1.

######  Remark 9.2.

We notice that the function $u$ might not be smooth in the open set
$\\{u>0\\}$. In fact, $u$ is not even $C^{1}$ as the gradient is not
continuous across $\partial\Omega_{1}\cap\partial\Omega_{2}$. We stress that
we can still use the 2D epiperimetric inequality from [14] together with the
almost-minimality condition (9.1) to prove the $C^{1,\alpha}$ regularity of
the free boundary, but we cannot improve this regularity to $C^{\infty}$.

######  Remark 9.3.

We expect 9.1 to hold in every dimension $2\leq d\leq 4$, as there are several
epsilon-regularity results for functions $u$ satisfying almost-minimality
conditions similar to (9.1) (see for instance [5, 4, 8, 13]), but we stress
that non of these results directly apply to (9.1). In fact, the almost-
minimality of $u$ only holds around points at the boundary $\partial\\{u>0\\}$
(and in our case $u$ is not even $C^{1,\alpha}$ in $\\{u>0\\}$), which
essentially requires [5, 4, 8, 13] to be revisited in order to be used in our
context. We choose the approach from [13] which limits 9.1 to the case $d=2$,
but on the other hand is based on the epiperimetric inequality from [14],
which works without any modifications in our case.

###  9.1. Blow-up sequences and blow-up limits

Let $x_{0}\in\partial\\{u>0\\}\cap D$. We define

$u_{r}(x):=\frac{1}{r}u(x_{0}+rx).$

Let $r_{n}$ be an infinitesimal sequence. Then, for $n$ large enough, the
sequence $u_{r_{n}}$ is uniformly bounded in $L^{\infty}$ in every ball
$B_{2R}\subset\mathbb{R}^{d}$. Moreover, by Lemma 3.2, $u_{r_{n}}$ is
uniformly bounded also in $C^{0,\nicefrac{{1}}{{3}}}(B_{R})$. Thus, up to a
(non-relabelled) subsequence $u_{r_{n}}$ it converges uniformly in $B_{R}$).
By a diagonal sequence argument, there is a continuous function

$u_{0}:\mathbb{R}^{d}\to\mathbb{R}$

and a subsequence $u_{r_{n}}$ such that $u_{r_{n}}$ converges to $u_{0}$
uniformly on every ball $B_{R}\subset\mathbb{R}^{d}$. We will say that $u_{0}$
is a blow-up limit of $u$ at $x_{0}$.

######  Proposition 9.4.

Let $2\leq d\leq 4$ and let $u$ be the function from Section 5.1. Then every
blow-up limit $u_{0}:\mathbb{R}^{d}\to\mathbb{R}$ of $u$ at a point
$x_{0}\in\partial\\{u>0\\}\cap D$ is of the form

$u_{0}(x)=\sqrt{\Lambda}\,(x\cdot\nu)_{+}\ ,$ (9.2)

for some unit vector $\nu\in\mathbb{R}^{d}$.

###### Proof.

Let $u_{r_{n}}$ be a blow-up sequence converging to $u_{0}$. We notice that,
by 7.2, $u_{0}$ is non-trivial. Moreover, using the almost-minimality
condition (9.1) we get that $u_{0}$ is a local minimizer of the Alt-Caffarelli
functional (see for instance [1] of [15]). Precisely,

$\int_{B_{R}}|\nabla
u|^{2}\,dx+\Lambda|B_{R}\cap\\{u>0\\}|\leq\int_{B_{R}}|\nabla
v|^{2}\,dx+\Lambda|B_{R}\cap\\{v>0\\}|,$

for every $B_{R}\subset\mathbb{R}^{d}$ and every $v\in H^{1}(B_{R})$ such that
$u-v\in H^{1}_{0}(B_{R})$. Moreover, the almost-minimality condition (9.1)
implies that every blow-up limit $u_{0}$ is $1$-homogeneous (see [13]). When
$d\leq 4$, using [3] and [10], this gives that every blow-up limit $u_{0}$ is
of the form (9.2). ∎

###  9.2. Epiperimetric inequality and regularity of $\partial\\{u>0\\}$

For any $\varphi\in H^{1}(B_{1})$ we consider the Weiss’ boundary adjusted
energy introduced in [16]

$W(\varphi):=\int_{B_{1}}|\nabla\varphi|^{2}\,dx+\Lambda|\\{\varphi>0\\}\cap
B_{1}|-\int_{\partial B_{1}}\varphi^{2}\,d\mathcal{H}^{d-1}.$

Let $K$ be a compact set contained in $D$ and let $C>0$ and $r_{0}$ be the
constants from the almost-minimality condition (9.1). Let
$x_{0}\in\partial\\{u>0\\}\cap K$ be fixed and let
$u_{r}(x):=\frac{1}{r}u(x_{0}+rx)$. Then, the derivative of $W(u_{r})$ is
given by

$\frac{\partial}{\partial
r}W(u_{r})=\frac{d}{r}\big{(}W(z_{r})-W(u_{r})\big{)}+\frac{1}{r}\int_{\partial
B_{1}}|x\cdot\nabla u-u|^{2}\,d\mathcal{H}^{d-1}(x),$

where $z_{r}:B_{1}\to\mathbb{R}$,
$z_{r}(x):=|x|u_{r}\big{(}\nicefrac{{x}}{{|x|}}\big{)}$, is the
$1$-homogeneous extension of $u_{r}$ in $B_{1}$. Now, by the 2D epiperimetric
inequality of [14], we have that there is a constant ${\varepsilon}\in(0,1)$
such that for every $r>0$, there exists a function $h_{r}:B_{1}\to\mathbb{R}$
with $h_{r}=u_{r}=z_{r}$ on $\partial B_{1}$ and

$W(h_{r})-\Theta\leq(1-{\varepsilon})\big{(}W(z_{r})-\Theta\big{)}\,,\quad\text{where}\quad\Theta:=\Lambda\frac{|B_{1}|}{2}.$

Now, using the almost minimality (9.1) of $u$, we have that for every $r\leq
r_{0}$

$\displaystyle\frac{\partial}{\partial r}W(u_{r})$
$\displaystyle=\frac{d}{r}\big{(}W(z_{r})-W(u_{r})\big{)}+\frac{1}{r}\int_{\partial
B_{1}}|x\cdot\nabla u-u|^{2}\,d\mathcal{H}^{d-1}(x)$
$\displaystyle=\frac{d}{r}\Big{(}\big{(}W(z_{r})-\Theta\big{)}-\big{(}W(u_{r})-\Theta\big{)}\Big{)}+\frac{1}{r}\int_{\partial
B_{1}}|x\cdot\nabla u-u|^{2}\,d\mathcal{H}^{d-1}(x)$
$\displaystyle\geq\frac{d}{r}\Big{(}\frac{1}{1-{\varepsilon}}\big{(}W(h_{r})-\Theta\big{)}-\big{(}W(u_{r})-\Theta\big{)}\Big{)}+\frac{1}{r}\int_{\partial
B_{1}}|x\cdot\nabla u-u|^{2}\,d\mathcal{H}^{d-1}(x)$
$\displaystyle\geq\frac{d}{r}\Big{(}\frac{1}{1-{\varepsilon}}\big{(}W(u_{r})-\Theta\big{)}-\frac{Cr}{1-{\varepsilon}}-\big{(}W(u_{r})-\Theta\big{)}\Big{)}+\frac{1}{r}\int_{\partial
B_{1}}|x\cdot\nabla u-u|^{2}\,d\mathcal{H}^{d-1}(x)$
$\displaystyle\geq\frac{d}{r}\frac{{\varepsilon}}{1-{\varepsilon}}\big{(}W(u_{r})-\Theta\big{)}-\frac{dC}{1-{\varepsilon}}+\frac{1}{r}\int_{\partial
B_{1}}|x\cdot\nabla u-u|^{2}\,d\mathcal{H}^{d-1}(x).$

Taking $\displaystyle\gamma=\frac{d\,{\varepsilon}}{1-{\varepsilon}}$, we get
that

$\frac{\partial}{\partial
r}\left(\frac{W(u_{r})}{r^{\gamma}}+\frac{dCr^{1-\gamma}}{1-(d+1){\varepsilon}}\right)\geq\frac{1}{r^{1+\gamma}}\int_{\partial
B_{1}}|x\cdot\nabla u-u|^{2}\,d\mathcal{H}^{d-1}(x),$

which implies that

$W(u_{r})\leq C_{0}r^{\gamma}\ ,\qquad\text{where}\qquad
C_{0}:=\frac{W(u_{r_{0}})}{r_{0}^{\gamma}}+\frac{dCr_{0}^{1-\gamma}}{1-(d+1){\varepsilon}}.$

By a standard argument (see for instance [14], [13] or [15]), this implies the
uniqueness of the blow-up limit at $x_{0}$ and the $C^{1,\alpha}$-regularity
of $\partial\\{u>0\\}$ in $K$, which concludes the proof of 9.1.∎

##  10\. Regularity of the free interface

In this section we prove the following theorem.

######  Theorem 10.1.

Let $u$ be the function from Section 5.1 and let $\Omega_{1}$ and $\Omega_{2}$
be the sets constructed in Section 5.2. Then,

  1. (i)

in any dimension $d\geq 2$, the free interface
$\partial^{\ast}\Omega_{1}\cap\big{(}D\cap\\{u>0\\}\big{)}$ is a $C^{\infty}$
manifold up to a closed singular set of Hausdorff dimension at most $d-8$;

  2. (ii)

in dimension $d=2$, the contact set
$\partial\Omega_{1}\cap\partial\Omega_{2}\cap(D\cap\partial\\{u>0\\})$ is
discrete in $D$;

  3. (iii)

in dimension $d=2$, in a neighborhood of every point
$x\in\partial\Omega_{1}\cap\partial\Omega_{2}\cap(D\cap\partial\\{u>0\\})$ the
boundary $\partial\Omega_{1}\cap\partial\Omega_{2}\cap\\{u>0\\}$ is a
$C^{\infty}$ curve, $C^{1}$ regular up to the endpoint $x$, and attaches
orthogonally to $\partial\\{u>0\\}$ at $x$.

###  10.1. Minimality and regularity of the free interface in $D\cap\\{u>0\\}$

######  Proposition 10.2 (Minimality of the limit sets).

In any $d\geq 2$, let $\Omega_{1}$ and $\Omega_{2}$ be the sets from Section
5.2 and $u$ be the function from Section 5.1. Then, for every open set
$A\subset\subset D\cap\\{u>0\\}$,

$\int_{A\cap\partial^{\ast}\Omega_{1}}u^{2}\,d\mathcal{H}^{d-1}\leq\int_{A\cap\partial^{\ast}\widetilde{\Omega}_{1}}u^{2}\,d\mathcal{H}^{d-1}\,,$
(10.1)

for every set $\widetilde{\Omega}_{1}$ of locally finite perimeter in
$D\cap\\{u>0\\}$ such that
$\widetilde{\Omega}_{1}\Delta\Omega_{1}\subset\subset A$.

In particular, $\partial^{\ast}\Omega_{1}\cap\big{(}D\cap\\{u>0\\}\big{)}$ is
a $C^{1,\alpha}$-regular manifold up to a closed singular set of Hausdorff
dimension at most $d-8$.

###### Proof.

Without loss of generality we can suppose that $A$ is a finite union of balls

$A=\bigcup_{j=1}^{N}B_{r_{j}}(x_{j}).$

Let
$(u_{n},\Omega_{n}^{1},\Omega_{n}^{2}):=(u_{{\varepsilon}_{n}},\Omega_{{\varepsilon}_{n}}^{1},\Omega_{{\varepsilon}_{n}}^{2})$
be the sequence of minimizers from Section 5.1 and Section 5.2 converging to
$(u,\Omega_{1},\Omega_{2})$. Then, by construction

$\Omega_{n}^{1}\cap\Omega_{n}^{2}=\emptyset\qquad\text{and}\qquad\Big{(}\Omega_{n}^{1}\cap
A\Big{)}\cup\Big{(}\Omega_{n}^{2}\cap A\Big{)}=A,$

and the same holds for the limit sets $\Omega_{1}$ and $\Omega_{2}$. Let now
$\widetilde{\Omega}_{1}$ be such that
$\widetilde{\Omega}_{1}\Delta\Omega_{1}\subset\subset A$. Notice that we can
find a family of balls $B_{\rho_{j}}(x_{j})$, $j=1,\dots,N$, such that

$B_{\rho_{j}}(x_{j})\subset B_{r_{j}}(x_{j})\qquad\text{for every}\qquad
j=1,\dots,N\,;$
$\widetilde{\Omega}_{1}\Delta\Omega_{1}\subset\subset\bigcup_{j=1}^{N}B_{\rho_{j}}(x_{j})\subset\subset\bigcup_{j=1}^{N}B_{r_{j}}(x_{j})\,,$

and such that for every $j=1,\dots,N$, we have

$\mathcal{H}^{d-1}\Big{(}\partial^{\ast}\Omega_{1}\cap\partial
B_{\rho_{j}}(x_{j})\Big{)}=0\qquad\text{and}\qquad\mathcal{H}^{d-1}\Big{(}\partial^{\ast}\Omega_{n}^{1}\cap\partial
B_{\rho_{j}}(x_{j})\Big{)}=0\quad\text{for every}\quad n\geq 1.$

We define

$B:=\bigcup_{j=1}^{N}B_{\rho_{j}}(x_{j}),$

and consider the sets

$\widetilde{\Omega}_{n}^{1}:=\Big{(}\widetilde{\Omega}_{1}\cap
B\Big{)}\cup\Big{(}\Omega_{n}^{1}\setminus
B\Big{)}\qquad\text{and}\qquad\widetilde{\Omega}_{n}^{2}:=\Big{(}B\setminus\widetilde{\Omega}_{1}\Big{)}\cup\Big{(}\Omega_{n}^{2}\setminus
B\Big{)}.$

Testing the optimality of $(u_{n},\Omega_{n}^{1},\Omega_{n}^{2})$ with
$(u_{n},\widetilde{\Omega}_{n}^{1},\widetilde{\Omega}_{n}^{2})$, we obtain

$\int_{B\cap\partial^{\ast}\widetilde{\Omega}_{1}}u_{n}^{2}+\int_{\partial^{\ast}\Omega_{n}^{1}\setminus
B}u_{k}^{2}=\int_{\partial^{\ast}\widetilde{\Omega}_{n}^{1}}u_{n}^{2}\geq\int_{\partial^{\ast}\Omega_{n}^{1}}u_{n}^{2}=\int_{B\cap\partial^{\ast}\Omega_{n}^{1}}u_{n}^{2}+\int_{\partial^{\ast}\Omega_{n}^{1}\setminus
B}u_{n}^{2}\,$

which we can write simply as

$\int_{B\cap\partial^{\ast}\widetilde{\Omega}_{1}}u_{n}^{2}\geq\int_{B\cap\partial^{\ast}\Omega_{n}^{1}}u_{n}^{2}\,.$

Using Lemma 2.6 and the uniform convergence of $u_{n}$ to $u$, we get

$\int_{B\cap\partial^{\ast}\widetilde{\Omega}_{1}}u^{2}=\lim_{n\to\infty}\int_{B\cap\partial^{\ast}\widetilde{\Omega}_{1}}u_{n}^{2}\qquad\text{and}\qquad\int_{B\cap\partial^{\ast}\Omega_{1}}u^{2}\leq\liminf_{n\to\infty}\int_{B\cap\partial^{\ast}\Omega_{n}^{1}}u_{n}^{2}\,,$

which concludes the proof of (10.1). The regularity of $\partial\Omega_{1}$ in
$A$ then follows (as in [9]) from the fact that $u\in C^{0,\alpha}(A)$ and
$u\geq t>0$ in $A$, for some constant $t>0$. ∎

###  10.2. Minimality of the free interface up to the boundary of $\\{u>0\\}$

######  Proposition 10.3 (Minimality of the limit sets).

In any $d\geq 2$, let $\Omega_{1}$ and $\Omega_{2}$ be the sets from Section
5.2 and $u$ be the function from Section 5.1. Then, for every ball
$B_{r}(x_{0})\subset\subset D$,

$\int_{B_{r}(x_{0})\cap\partial^{\ast}\Omega_{1}}u^{2}\,d\mathcal{H}^{d-1}\leq\int_{B_{r}(x_{0})\cap\partial^{\ast}\widetilde{\Omega}_{1}}u^{2}\,d\mathcal{H}^{d-1}\,,$
(10.2)

for every set $\widetilde{\Omega}_{1}$ such that
$\widetilde{\Omega}_{1}\Delta\Omega_{1}\subset\subset B_{r}(x_{0})$ and which
is of finite perimeter in $B_{r}(x_{0})$.

###### Proof.

Let $\widetilde{\Omega}_{1}$ be such that
$\widetilde{\Omega}_{1}\Delta\Omega_{1}\subset\subset B_{r}(x_{0})$. Since $u$
satisfies the minimality condition from Lemma 7.1, we have that there are a
sequence $\delta_{n}\to 0$ and a constant $C>0$ such that

$\mathcal{H}^{d-1}\Big{(}B_{r}(x_{0})\cap\partial^{\ast}\\{u>\delta_{n}\\}\Big{)}\leq
C\qquad\text{for every}\qquad n\geq 1\,.$

Moreover, we can suppose that

$\mathcal{H}^{d-1}\Big{(}\partial^{\ast}\\{u>\delta_{n}\\}\cap\partial^{\ast}\Omega_{1}\Big{)}=\mathcal{H}^{d-1}\Big{(}\partial^{\ast}\\{u>\delta_{n}\\}\cap\partial^{\ast}\widetilde{\Omega}_{1}\Big{)}=0.$

Now, consider the sets

$A_{n}:=B_{r}(x_{0})\cap\\{u>\delta_{n}\\}\qquad\text{and}\qquad\Omega_{n}:=\Big{(}A_{n}\cap\widetilde{\Omega}_{1}\Big{)}\cup\Big{(}\Omega_{1}\setminus
A_{n}\Big{)}.$

Since $\Omega_{n}\Delta\Omega_{1}\subset\subset B_{r}(x_{0})\cap\\{u>0\\}$, by
10.2, we have that

$\int_{B_{r}(x_{0})\cap\partial^{\ast}\Omega_{1}}u^{2}\leq\int_{B_{r}(x_{0})\cap\partial^{\ast}\Omega_{n}}u^{2},$

which we write as

$\displaystyle\int_{\big{(}B_{r}(x_{0})\cap\partial^{\ast}\Omega_{1}\big{)}\cap
A_{n}}$ $\displaystyle
u^{2}+\int_{\big{(}B_{r}(x_{0})\cap\partial^{\ast}\Omega_{1}\big{)}\setminus
A_{n}}u^{2}$
$\displaystyle\leq\int_{\big{(}B_{r}(x_{0})\cap\partial^{\ast}\widetilde{\Omega}_{1}\big{)}\cap
A_{n}}u^{2}+\int_{\big{(}B_{r}(x_{0})\cap\partial^{\ast}\Omega_{1}\big{)}\setminus
A_{n}}u^{2}+2\int_{B_{r}(x_{0})\cap\partial^{\ast}A_{n}}u^{2}$
$\displaystyle\leq\int_{\big{(}B_{r}(x_{0})\cap\partial^{\ast}\widetilde{\Omega}_{1}\big{)}\cap
A_{n}}u^{2}+\int_{\big{(}B_{r}(x_{0})\cap\partial^{\ast}\Omega_{1}\big{)}\setminus
A_{n}}u^{2}+2\delta_{n}^{2}C,$

which gives

$\displaystyle\int_{\big{(}B_{r}(x_{0})\cap\partial^{\ast}\Omega_{1}\big{)}\cap
A_{n}}u^{2}\leq\int_{\big{(}B_{r}(x_{0})\cap\partial^{\ast}\widetilde{\Omega}_{1}\big{)}\cap
A_{n}}u^{2}+2\delta_{n}^{2}C.$

Passing to the limit as $n\to\infty$, we get (10.2). ∎

###  10.3. Regularity of the free interface up to the boundary
$\partial\\{u>0\\}$

In this section we will need the $C^{1,\alpha}$ regularity of the free
boundary $\partial\\{u>0\\}$ in $D$, so in order to have 9.1 we assume that
$d=2$.

Let $x_{0}\in D\cap\\{u>0\\}$ and $B_{r}(x_{0})$ be a (small) ball contained
in $D$. Without loss of generality, we suppose that $x_{0}=(0,0)$. We define
the function $h:B_{r}\to\mathbb{R}$ as

$\Delta h=0\,\text{ in }\,B_{r}\cap\\{u>0\\}\,,\quad h=u\,\text{ in
}\,\partial B_{r}\,,\quad h=0\,\text{ in }\,B_{r}\cap\partial\\{u>0\\}\,.$

Then, $h$ is $C^{1,\alpha}$ regular in $B_{r}\cap\\{u>0\\}$ up to the boundary
$B_{r}\cap\partial\\{u>0\\}$ and moreover, there is a $C^{0,\alpha}$-regular
strictly positive function

$a:B_{r}\cap\overline{\\{u>0\\}}\to\mathbb{R}$

such that

$a(x)=\frac{h(x)}{u(x)}\quad\text{for}\quad x\in B_{r}\cap\\{u>0\\}\,;\quad
a(x)=\frac{|\nabla h|(x)}{\sqrt{\Lambda}}\quad\text{for}\quad x\in
B_{r}\cap\partial\\{u>0\\}.$

Moreover, choosing $r>0$ small enough the set $B_{r}\cap\\{u>0\\}$ is simply
connected, so we can find a function $w:B_{r}\cap\\{u>0\\}\to\mathbb{R}$ such
that

$\partial_{x}w=-\partial_{y}h\quad\text{and}\quad\partial_{y}w=\partial_{x}h\quad\text{
in }\quad B_{r}\cap\\{u>0\\}\,,$

$w$ being defined as

$w(x,y)=\int_{\gamma_{x,y}}\Big{(}\partial_{y}h\,dx-\partial_{x}h\,dy\Big{)},$

where $\gamma_{x,y}:[0,1]\to\mathbb{R}$ is any $C^{1}$ curve connecting
$(0,0)$ to $(x,y)$ in $B_{r}\cap\\{u>0\\}$. Then, the map

$\Phi:B_{r}\cap\overline{\\{u>0\\}}\
,\qquad\Phi(x,y):=\Big{(}w(x,y),h(x,y)\Big{)},$

is $C^{1,\alpha}$ smooth in $B_{r}\cap\overline{\\{u>0\\}}$ (up to the
boundary $B_{r}\cap\partial{\\{u>0\\}}$) and the set

$A:=\Phi\Big{(}B_{r}\cap\overline{\\{u>0\\}}\Big{)}$

is a relatively open subset of the upper half-plane
$\\{(w,h)\in\mathbb{R}^{2}\ :\ h\geq 0\\}$. We notice that for $r$ small
enough the function $\Phi$ is invertible. Then, we define

$\varphi:A\cap\\{(w,h)\in\mathbb{R}^{2}\ :\ h\geq
0\\}\to\mathbb{R},\qquad\varphi:=\frac{1}{|\nabla h|a^{2}}\circ\Phi^{-1}.$

Then, $\varphi$ is $C^{0,\alpha}$ and is bounded from below by a positive
constant. We will show that in the new coordinates the set
$\Omega:=\Phi(\Omega_{1})$ locally minimizes the functional

$\mathcal{F}_{2}(\Omega):=\int_{\\{h\geq
0\\}\cap\partial^{\ast}\Omega}h^{2}\varphi(w,h)\,d\mathcal{H}^{1}(w,h).$

In fact, since $\partial^{\ast}\Omega_{1}$ is a $C^{1}$ curve, it is
sufficient to check that for any

$\gamma:[0,1]\to
B_{r}\cap\overline{\\{u>0\\}},\quad\gamma(t)=\big{(}x(t),y(t)\big{)},$

we have

$\displaystyle\int_{\Phi(\gamma)}h^{2}\varphi(w,h)$
$\displaystyle:=\int_{0}^{1}h^{2}\big{(}\gamma(t)\big{)}\varphi\big{(}\Phi(\gamma(t))\big{)}\sqrt{\big{(}x^{\prime}\partial_{x}w+y^{\prime}\partial_{y}w\big{)}^{2}+\big{(}x^{\prime}\partial_{x}h+y^{\prime}\partial_{y}h\big{)}^{2}}\,dt$
$\displaystyle=\int_{0}^{1}h^{2}\big{(}\gamma(t)\big{)}\varphi\big{(}\Phi(\gamma(t))\big{)}\sqrt{\big{(}x^{\prime}\partial_{y}h-y^{\prime}\partial_{x}h\big{)}^{2}+\big{(}x^{\prime}\partial_{x}h+y^{\prime}\partial_{y}h\big{)}^{2}}\,dt$
$\displaystyle=\int_{0}^{1}h^{2}\big{(}\gamma(t)\big{)}\varphi\big{(}\Phi(\gamma(t))\big{)}\sqrt{(\partial_{x}h)^{2}+(\partial_{y}h)^{2}}\sqrt{(x^{\prime}(t))^{2}+(y^{\prime}(t))^{2}}\,dt$
$\displaystyle=\int_{0}^{1}u^{2}\big{(}\gamma(t)\big{)}\sqrt{(x^{\prime}(t))^{2}+(y^{\prime}(t))^{2}}\,dt,$

which concludes the proof. In order to conclude the proof of the $C^{1}$
regularity of $\partial\Omega_{1}$, it is sufficient to prove that at the
point $(w_{0},h_{0})=(0,0)$ the set $\Omega:=\Phi(\Omega_{1})$ has a unique
blow-up limit given by

$\Omega_{0}:=\\{(w,h)\in\mathbb{R}^{2}\ :\ h\geq 0,\ w>0\\}.$

In order to do so, we consider the set

$\mathcal{R}:=\Big{\\{}(w,X)\in\mathbb{R}\times\mathbb{R}^{3}\ :\
(w,|X|)\in\Omega\Big{\\}}.$

Then, $\mathcal{R}$ is a local minimizer of the functional

$\mathcal{F}_{4}(\mathcal{R}):=\int_{\partial^{\ast}R}\varphi(w,|X|)\,d\mathcal{H}^{3}(w,X),$

among all sets with the same simmetries as $\mathcal{R}$, that is, all sets of
the form

$\widetilde{\mathcal{R}}:=\Big{\\{}(w,X)\in\mathbb{R}\times\mathbb{R}^{3}\ :\
(w,|X|)\in\widetilde{\Omega}\Big{\\}},$

for some $\widetilde{\Omega}\subset\\{(w,h)\in\mathbb{R}^{2}\ :\ h\geq 0\\}$.
Now, by the monotonicity formula for the local minimizers of the area (see for
instance [11]), we have that any blow-up limit $\mathcal{R}_{0}$ of
$\mathcal{R}$ is a cone in $\mathbb{R}^{4}$, which is area-minimizing with
respect to perturbations that preserve the simmetries of $\mathcal{R}$. But
then, since the dimension of $\partial\mathcal{R}_{0}$ is less than $7$, we
have that $\partial\mathcal{R}_{0}$ is necessary a plane (with the same
simmetries as $\mathcal{R}$). Thus, $\partial\mathcal{R}_{0}$ is ortogonal to
the line $\\{0,0,0\\}\times\mathbb{R}$, which concludes the proof of the
uniqueness of the blow-up, which implies points (ii) and (iii) of 10.1.

##  11\. Proof of Theorem 1.2

In order to prove the existence of a solution to (1.3), we observe that as a
consequence of an almost-minimality condition involving the one-phase Alt-
Caffarelli functional and 10.1, we have that the sets $\Omega_{1}$ and
$\Omega_{2}$ constructed in Section 5.2 have locally finite perimeter in $D$.
It remains to prove that

  1. (i)

$\Omega_{1}$ and $\Omega_{2}$ are sets of finite perimeter in
$\mathbb{R}^{d}$;

  2. (ii)

$(u,\Omega_{1},\Omega_{2})$ is a solution to (1.3).

First, for $j=1,2$ and $\delta>0$, we define

$E_{j}^{\delta}:=\Big{\\{}x\in\mathbb{R}^{d}\setminus E_{j}\ :\ \text{\rm
dist}(x,E_{j})>\delta\Big{\\}}.$

Then, $E_{1}^{\delta}\cap\Omega_{1}$ and $E_{2}^{\delta}\cap\Omega_{2}$ are
sets of finite perimeter and, by 8.1,

$\text{\rm Per}(E_{1}^{\delta}\cap\Omega_{1})+\text{\rm
Per}(E_{2}^{\delta}\cap\Omega_{2})<C,$

where $C$ is a constant that does noit depend on $\delta$. Thus, passing to
the limit as $\delta\to 0$, we get that $\Omega_{1}$ and $\Omega_{2}$ have
finite perimeter. Next, in order to prove (ii), we consider
$\widetilde{u}\in\mathcal{V}$ and
$(\widetilde{\Omega}_{1},\widetilde{\Omega}_{2})\in\mathcal{A}(u)$. Testing
the optimality of
$(u_{\varepsilon},\Omega_{\varepsilon}^{1},\Omega_{\varepsilon}^{2})$ we get
that

$J_{\varepsilon}(u_{\varepsilon},\Omega_{\varepsilon}^{1},\Omega_{\varepsilon}^{2})\leq
J_{\varepsilon}(\widetilde{u},\widetilde{\Omega}_{1},\widetilde{\Omega}_{2}).$

Now, the semicontinuity lemma (Lemma 2.6) gives that

$J_{\beta,\Lambda}(u,\Omega_{1},\Omega_{2})=\liminf_{{\varepsilon}\to
0}J_{\varepsilon}(u_{\varepsilon},\Omega_{\varepsilon}^{1},\Omega_{\varepsilon}^{2})\leq\lim_{{\varepsilon}\to
0}J_{\varepsilon}(\widetilde{u},\widetilde{\Omega}_{1},\widetilde{\Omega}_{2})=J_{\beta,\Lambda}(\widetilde{u},\widetilde{\Omega}_{1},\widetilde{\Omega}_{2}),$

which concludes the proof of the existence.

Moreover, if $(u,\Omega_{1},\Omega_{2})$ is any solution to (1.3), then it
satisfies the minimality conditions from Lemma 6.1, Lemma 7.1, 10.2 and 10.3,
so by 9.1 and 10.1, the claims (i) and (ii) of 1.2 follow.

## Acknowledgments

The first author was partially supported by PRIN 2017 Nonlinear Differential
Problems via Variational, Topological and Set-valued Methods (Grant
2017AYM8XW). The third author has been supported by the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement VAREG, No. 853404).

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