# Regularity Results for an Optimal Design Problem with lower order terms

L. Esposito  Dipartimento di Matematica, Via Giovanni Paolo II, 132, 84084
Fisciano (SA<EMAIL_ADDRESS>and  L. Lamberti<EMAIL_ADDRESS>
###### Abstract.

We study the regularity of the interface for optimal energy configurations of
functionals involving bulk energies with an additional perimeter penalization
of the interface. It is allowed the dependence on $(x,u)$ for the bulk energy.
For a minimal configuration $(E,u)$, the Hölder continuity of $u$ is well
known. We give an estimate for the singular set of the boundary $\partial E$.
Namely we show that the Hausdorff dimension of the singular set is strictly
smaller than $n-1$.

###### 1991 Mathematics Subject Classification:

2010 Mathematics Subject Classification: 49Q10, 49N60, 49Q20

##  1\. Introduction and statements

In this paper we will deal with energy functionals of the type

${\mathcal{F}}(E,u;\Omega)=\int_{\Omega}\ \bigl{[}F(x,u,\nabla
u)+\mathbbm{1}_{E}G(x,u,\nabla u)\bigr{]}\,\,dx+P(E,\Omega)\,,$ (1)

where $u\in H^{1}(\Omega)$ and $\mathbbm{1}_{E}$ denotes the characteristic
function of a set $E\subset\Omega$ with finite perimeter in $\Omega$, denoted
as $P(E,\Omega)$. In mathematical and physical literature, the problem of
finding the minimal energy configuration of a mixture of two materials in a
bounded connected open set $\Omega\subset\mathbb{R}^{n}$ has been widely
investigated (see for instance [1], [2], [10], [13], [16], [17]). The energy
functional employed in such problems involves both bulk and interface energies
in order to describe a large class of phenomenon in many applied sciences,
such as non linear elasticity, material sciences and image segmentations in
the computer vision. A relevant case deeply studied by several authors is the
following model functional,

$\int_{\Omega}\sigma_{E}(x)|\nabla u|^{2}\,dx+P(E,\Omega),$ (2)

where $u=u_{0}$ is prescribed on $\partial\Omega$ and
$\sigma_{E}(x)=\beta\mathbbm{1}_{E}+\alpha\mathbbm{1}_{\Omega\setminus E}$,
with $0<\alpha<\beta$ given constant.

In $1993$ L. Ambrosio and G. Buttazzo in [1] proved that, if $(E,u)$ is a
minimizer of the functional $\eqref{model}$, then $u$ is locally Hölder
continuous in $\Omega$ and $E$ is relatively open in $\Omega$. In the same
year F.H. Lin proved the regularity of the interface (see [17]), that is
$\partial E$ is regular outside a relatively closed set of vanishing
$\mathcal{H}^{n-1}$-measure. To be more precise, we define the set of regular
points of $\partial E$ as follows:

$\mbox{Reg}(E):=\left\\{x\in\partial E\cap\Omega:\ \partial E\mbox{ is a
}C^{1,\gamma}\mbox{ hypersurface in some }I(x)\right\\}$ (3)

where $I(x)$ denotes a neighborhood of $x$. Accordingly, we define the set of
singular points of $\partial E$

$\Sigma(E):=(\partial E\cap\Omega)\setminus\mbox{Reg}(E).$ (4)

In the paper by F.H. Lin (see [17]) it was proved that
$\mathcal{H}^{n-1}(\Sigma(E))=0$ for minimal configurations of the functional
$\eqref{model}$. More recently G. De Philippis and A. Figalli in [7], Fusco
and Julin in [11], independently and by different approaches, were able to
improve this result indeed proving that

$dim_{\mathcal{H}}(\Sigma(E))\leq s,$ (5)

for some $s<n-1$ depending only on $\alpha,\beta$. Regarding this dependence,
it is worth noticing that in [8] it has been proven, assuming $\alpha=\beta$
and $\beta\leq\gamma$ with $\gamma=\gamma(n)$, that $u\in
C^{0,\frac{1}{2}+\varepsilon}$, $\partial^{*}E$ is a
$C^{1,\varepsilon}-$hypersurface and $\mathcal{H}^{s}(\partial
E\setminus\partial^{*}E)=0$ for all $s>n-8$.

In 1999 F.H. Lin and R.V. Kohn in [LK] treated a more general quadratic bulk
energy of the type $\eqref{intro0}$ actually proving the same regularity they
proved for the model case $\eqref{model}$. As a matter of fact they proved
that the singular part $\Sigma(E)$ has vanishing $\mathcal{H}^{n-1}$-measure
for minimal configurations $(E,u)$.

In this paper we address the issue of improving the dimensional estimate for
the singular part $\Sigma(E)$ of optimal configurations for the class of
functionals treated by Lin and Kohn. As a matter of fact we prove, for a wide
class of quadratic functional depending also on $x$ and $u$, the same kind of
regularity proved in the model case $\eqref{model}$ by [DF] and [FJ] namely,
$dim_{\mathcal{H}}(\Sigma(E))\leq s$ for some $s<n-1$.

Our path to prove the aforementioned result basically follows the same
strategy used in [FJ]. As we will point out later in more detail, our
technique relies on the linearity of the Euler equation of the functional
$\eqref{intro0}$. For this reason we need a linear structure condition for the
bulk energy. In the rest of the paper we will assume that the density energies
$F$ and $G$ in $\eqref{intro0}$ satisfy the following assumptions:

$\displaystyle
F(x,s,z)=\sum_{i,j=1}^{n}a_{ij}(x,s)z_{i}z_{j}+\sum_{i=1}^{n}a_{i}(x,s)z_{i}+a(x,s),$
(6) $\displaystyle
G(x,s,z)=\sum_{i,j=1}^{n}b_{ij}(x,s)z_{i}z_{j}+\sum_{i=1}^{n}b_{i}(x,s)z_{i}+b(x,s).$
(7)

Concerning the coefficients we assume that

$a_{ij},b_{ij},a_{i},b_{i},a,b\in C^{0,1}(\Omega\times\mathbb{R}).$

We will denote by $L_{D}$ the greatest lipschitz constant of the data
$a_{ij},b_{ij},a_{i},b_{i},a,b$, that is

$|\nabla a_{ij}|\leq L_{D},\;\;|\nabla b_{ij}|\leq L_{D}\;\;\mbox{ in
}\Omega\times\mathbb{R},$ (8)

and the same holds true for $a_{i},b_{i},a,b$.
Moreover to ensure the existence of minimizers we assume the boundedness of
the coefficients and the ellipticity of the matrices $a_{ij}$ and $b_{ij}$,

$\displaystyle\nu|\xi|^{2}\leq a_{ij}(x,s)\xi_{i}\xi_{j}\leq
N|\xi|^{2},\;\;\;\nu|\xi|^{2}\leq b_{ij}(x,s)\xi_{i}\xi_{j}\leq N|\xi|^{2}$
(9)
$\displaystyle\sum_{i=1}^{n}|a_{i}(x,s)|+\sum_{i=1}^{n}|b_{i}(x,s)|+|a(x,s)|+|b(x,s)|\leq
L$ (10)

We are interested in the regularity of minimizers of the following constrained
problem.

######  Definition 1.1.

In the sequel we shall denote by $\eqref{P_c}$ the constrained problem

$\min_{\begin{subarray}{c}E\in\mathcal{A}(\Omega)\\\ v\in
u_{0}+H_{0}^{1}(\Omega)\end{subarray}}\left\\{\mathcal{F}(E,v;\Omega)\,:\,|E|=d\right\\}$
($P_{c}$)

where $u_{0}\in H^{1}(\Omega)$, $0<d<|\Omega|$ are given and
$\mathcal{A}(\Omega)$ is the class of all subsets of $\Omega$ with finite
perimeter.

The problem of handling with the constraint $|E|=d$ is overtaken using an
argument introduced in [8], ensuring that every minimizer of the constrained
problem $\eqref{P_c}$ is also a minimizer of a penalized functional of the
type

${\mathcal{F}}_{\Lambda}(E,v;\Omega)=\mathcal{F}(E,v;\Omega)+\Lambda\big{|}|E|-d\big{|},$
(11)

for some suitable $\Lambda>0$ (see Theorem 2.4 below). Therefore we give in
addition the following definition.

######  Definition 1.2.

In the sequel we shall denote by $\eqref{P}$ the penalized problem

$\min_{\begin{subarray}{c}E\in\mathcal{A}(\Omega)\\\ v\in
u_{0}+H_{0}^{1}(\Omega)\end{subarray}}\mathcal{F}_{\Lambda}(E,v;\Omega)$ ($P$)

where $u_{0}\in H^{1}(\Omega)$, is given, and $\mathcal{A}(\Omega)$ is the
class of all subsets of $\Omega$ with finite perimeter in $\Omega$.

From the point of view of regularity, the extra term
$\Lambda\big{|}|E|-|F|\big{|}$ is a higher order negligible perturbation. The
main result of the paper is contained in the following theorem.

######  Theorem 1.3.

Let $(E,u)$ be a minimizer of either problem $\eqref{P_c}$ or problem
$\eqref{P}$, under assumptions $\eqref{structure1}-\eqref{ellipticity2}$. Then

  * a)

there exists a relatively open set $\Gamma\subset\partial E$ such that
$\Gamma$ is a $C^{1,\mu}$ hypersurface for all $0<\mu<\frac{1}{2}$,

  * b)

there exists $\varepsilon>0$ depending on $\nu,N,L,n$, such that

$\mathcal{H}^{n-1-\varepsilon}((\partial E\setminus\Gamma)\cap\Omega)=0.$

For reader convenience the paper is structured in sections which reflect the
proof strategy. Section $2$ collects known results and preliminary
definitions. As in the case of minimizers of the Mumford-Shah functional the
proof of regularity is based on the study of interplay between the perimeter
and the bulk energy. We point out that the Hölder exponent $\frac{1}{2}$ is
critical for solutions $u$ of either $\eqref{P}$ or $\eqref{P_c}$, in the
sense that whenever $u\in C^{0,\frac{1}{2}}$, under appropriate scaling the
bulk term locally have the same dimension $n-1$ as the perimeter term. In this
regard, our starting point is to prove suitable energy decay estimates for the
bulk energy. These estimates are contained in section 3. The key point of this
approach is contained in Lemma 3.10, where it is proved that the bulk energy
decay faster than $\rho^{n-1}$, that is, for any $\delta>0$,

$\int_{B_{\rho}(x_{0})}|\nabla u|^{2}\,dx\leq C\rho^{n-\delta},$ (12)

either in the case that

$\min\\{|E\cap B_{\rho}(x_{0})|,|B_{\rho}(x_{0})\setminus
E|\\}<\varepsilon_{0}|B_{\rho}|,$

or in the case that, there exists an half space $H$ such that

$|(E\Delta H)\cap B_{\rho}(x_{0})|\leq\varepsilon_{0}|B_{\rho}|.$

The latter case is the hardest to handle because it relies on the regularity
properties of solutions of a transmission problem which we study in subsection
3.1. Let us notice that, for any given $E\subset\Omega$, local minimizers $u$
of the functional

$\int_{\Omega}\ \bigl{[}F(x,u,\nabla u)+\mathbbm{1}_{E}G(x,u,\nabla
u)\bigr{]}\,dx$ (13)

are Hölder continuous, $u\in C^{0,\alpha}_{loc}(\Omega)$, but the needed bound
$\alpha>\frac{1}{2}$ cannot be expected in the general case without any
information on the set $E$. In section 3.1 we prove, in the case $E$ is an
half space, that minimizers of the functional (13) are in $C^{0,\alpha}$ for
every $\alpha>0$. In this context the linearity of the equation strongly come
in to play ensuring that the derivatives of the Euler equation are again
solutions of the same equation. For the proof in section $3$ we readapt a
technique depicted in the book [3] in the context of the Mumford and Shah
functional and recently used in a paper by E. Mukoseeva and G. Vescovo, [22].
Once the estimates of section $3$ are obtained, we are ready to prove in
section $4$ that, if in a ball $B_{\rho}(x_{0})$ the perimeter of $E$ is
sufficiently small, then the total energy

$\int_{B_{r}(x_{0})}|\nabla u|^{2}\,dx+P(E,B_{r}(x_{0})),\quad\quad 0<r<\rho,$

decays as $r^{n}$ (see Lemma 4.1). Making use of the latter energy density
estimate we are in position to deduce in section $4$ the density lower bound
for the perimeter of $E$ as well. In the subsequent sections the proof
strategy follows the path traced from the regularity theory for perimeter
minimizers. In section 5 it is proved the compactness for sequences of
minimizers which more or less follows in a standard way from the density lower
bound. Section 6 is devoted to prove some additional consequences of the
density lower bound which involve the excess

${\mathbf{e}}(x,r)=\inf_{\nu\in\mathbb{S}^{n-1}}{\mathbf{e}}(x,r,\nu):=\inf_{\nu\in\mathbb{S}^{n-1}}\frac{1}{r^{n-1}}\int_{\partial
E\cap B_{r}(x)}\frac{|\nu_{E}(y)-\nu|^{2}}{2}d\mathcal{H}^{n-1}(y).$

Actually we prove the height bound Lemma and the Lipschitz approximation
Theorem. In this section we also compute the Euler equation for
$\mathcal{F}(E,u)$ involving the variation of the set $E$. Section 7 is
devoted to prove the excess improvement which follows from the fact that
whenever the excess ${\mathbf{e}}(x,r)$ goes to zero for $r\rightarrow 0$ the
Dirichlet integral $\int_{B_{\rho}(x_{0})}|\nabla u|^{2}\,dx$ decays as in
(12). In section 8 we give the proof of Theorem 1.3 that is a consequence of
the excess improvement proved before.

##  2\. Preliminary notations and definitions

In the rest of the paper we will write $\langle\xi,\eta\rangle$ for the inner
product of vectors $\xi,\eta\in\mathbb{R}^{n}$, and
$|\xi|:=\langle\xi,\xi\rangle^{\frac{1}{2}}$ will denote the corresponding
Euclidean norm. If $u$ is integrable in $B_{R}(x_{0})$ we set:

$u_{R}=\frac{1}{\omega_{n}R^{n}}\int_{B_{R}(x_{0})}u\,dx=\fint_{B_{R}(x_{0})}u\,dx$

The following definition is standard.

######  Definition 2.1.

Let $v\in H^{1}_{loc}(\Omega)$, and assume that $E\subset\Omega$ is fixed. We
say that $v$ is a local minimizer of the integral functional $\mathcal{F}$
defined in (1) iff

${\mathcal{F}}(E,v;B_{R}(x_{0}))=\min\left\\{{\mathcal{F}}(E,w;B_{R}(x_{0})):\,\,w\in
v+H^{1}_{0}(B_{R}(x_{0}))\right\\}$

for all $B_{R}(x_{0})\subset\subset\Omega$.

It is clear that any minimizer $u$ of problem $\eqref{P_c}$ is a local
minimizer of the functional (1) and therefore satisfies the Euler equation

$\displaystyle\sum_{i=1}^{n}\frac{\partial}{\partial{x_{i}}}\Big{[}F_{z_{i}}(x,u,\nabla
u)+\mathbbm{1}_{E}G_{z_{i}}(x,u,\nabla u)\bigr{)}\Big{]}$ (14)
$\displaystyle=F_{u}(x,u,\nabla u)+G_{u}(x,u,\nabla u)$

######  Lemma 2.2.

Let $Z(t)$ be a bounded non-negative function in the interval $[\rho,R]$ and
assume that for $\rho\leq t<s\leq R$ we have

$Z(t)\leq\theta Z(s)+\frac{A}{(s-t)^{2}}+B,$ (15)

with $A,B\geq 0$, and $0\leq\theta<1$. Then

$Z(\rho)\leq c(\theta)\Bigl{[}\frac{A}{(R-\rho)^{2}}+B\Bigr{]},$ (16)

for some $c(\theta)$ depending only on $\theta$.

###### Proof.

The proof of this lemma is standard and can be found in [12]. Inspecting the
proof in [12] one can also obtain an explicit expression of the constant
$c(\theta)$. An admissible value for $c(\theta)$, but not the best, is
$c(\theta)=1/(1-\theta^{1/3})^{3}$. ∎

The next lemma can be found in [3, Lemma 7.54].

######  Lemma 2.3.

Let $f:(0,a]\rightarrow[0,\infty)$ be an increasing function such that

$f(\rho)\leq
A\Bigl{[}\Bigl{(}\frac{\rho}{R}\Bigr{)}^{p}+R^{s}\Bigr{]}f(R)+BR^{q}\quad\text{whenever
}0<\rho<R\leq a$

for some constants $A,B\geq 0$, $0<q<p$, $s>0$. Then there exist
$R_{0}(p,q,s,A)$ and $c(p,q,A)$ such that

$f(\rho)\leq
c\Bigl{(}\frac{\rho}{R}\Bigr{)}^{q}f(R)+cB\rho^{q}\qquad\mbox{whenever
}0<\rho<R\leq\min\\{R_{0},a\\}.$

###  2.1. From constrained to penalized problem

The next theorem allows to overcome the difficulty of handling with the
constraint $|E|=d$. As a matter of fact it can be proved that every minimizer
of the constrained problem $\eqref{P_c}$ is also a minimizer of a suitable
unconstrained problem with a volume penalization of the type $\eqref{P}$.

######  Theorem 2.4.

Let $0<d<|\Omega|$. There exists $\Lambda_{0}>0$ such that if $(F,v)$ is a
minimizer of the functional

${\mathcal{F}}_{\Lambda}(A,w)=\int_{\Omega}\ \Bigl{(}F(x,u,\nabla
u)+\mathbbm{1}_{A}G(x,u,\nabla
u)\,dx\Bigr{)}\,dx+P(A,\Omega)+\Lambda\big{|}|A|-d\big{|}$ (17)

for some $\Lambda\geq\Lambda_{0}$, among all configurations $(A,w)$ such that
$w=v_{0}$ on $\partial\Omega$, then $|F|=d$ and $(F,v)$ is a minimizer of
problem $\eqref{P_c}$. Conversely, if $(E,u)$ is minimizer of problem
$\eqref{P_c}$, then it is a minimizer of (17), for all
$\Lambda\geq\Lambda_{0}$.

###### Proof.

The proof can be carried out as [8, Theorem 1]. For the convenience of the
reader we give here its sketch, emphasizing main ideas and minor differences
with respect to the case treated in [8].
The first part of the theorem can be proved by contradiction. Assume that
there exists a sequence $(\lambda_{h})_{h\in\mathbb{N}}$ such that
$\lambda_{h}\rightarrow+\infty$ as $h\rightarrow+\infty$ and a sequence of
configurations $(E,u_{h})$ minimizing $\mathcal{F}_{\lambda_{h}}$ and such
that $u_{h}=u_{0}$ on $\partial\Omega$ and $|E_{h}|\neq d$ for all $h$. Le us
choose now an arbitrary fixed $E_{0}\subset\Omega$ with finite perimeter and
such that $|E|=d$. Let us point out that

$\mathcal{F}_{\lambda_{h}}(E_{h},u_{h})\leq\mathcal{F}(E_{0},u_{0}):=\Theta.$
(18)

Without loss of generality we can assume that $|E_{h}|<d$. As a matter of fact
the case $|E_{h}|>d$ can be treated in the same way considering the complement
of $E$ in $\Omega$. Our aim is to show that for $h$ sufficiently large, there
exists a configuration $(\widetilde{E}_{h},\tilde{u}_{h})$ such that
$\mathcal{F}_{\lambda_{h}}(\widetilde{E}_{h},\tilde{u}_{h})<\mathcal{F}_{\lambda_{h}}({E_{h}},{u_{h}})$,
thus proving the result by contradiction.
By condition $\eqref{Theta}$, it follows that the sequence $\\{u_{h}\\}$ is
bounded in $H^{1}(\Omega)$, the perimeters of the sets $E_{h}$ in $\Omega$ are
bounded and $|E_{h}|\rightarrow d$. Therefore, possibly extracting a not
relabelled subsequence, we may assume that there exists a configuration
$(E,u)$ such that $u_{h}\rightarrow u$ weakly in $H^{1}(\Omega)$,
$\mathbbm{1}_{E_{h}}\rightarrow\mathbbm{1}_{E}$ a.e. in $\Omega$, where the
set $E$ is of finite perimeter in $\Omega$ and $|E|=d$. The couple $(E,u)$
will be used as reference configuration for the definition of
$(\widetilde{E}_{h},\tilde{u}_{h})$.

Step 1. Construction of $(\widetilde{E}_{h},\tilde{u}_{h})$. Proceeding
exactly as in [8], we take a point $x\in\partial^{*}E\cap\Omega$ and observe
that the sets $E_{r}=(E-x)/r$ converge locally in measure to the half space
$H=\\{z\cdot\nu_{E}(x)>0\\}$, i.e., $\chi_{E_{r}}\to\chi_{H}$ in $L^{1}_{\rm
loc}(\mathbb{R}^{n})$, where $\nu_{E}(x)$ is the generalized inner normal to
$E$ at $x$ (see [3, Definition 3.54]). Let $y\in B_{1}(0)\setminus H$ be the
point $y=-\nu_{E}(x)/2$. Given $\varepsilon$ (to be chosen at the end of the
proof), since $\chi_{E_{r}}\to\chi_{H}$ in $L^{1}(B_{1}(0))$ there exists
$0<r<1$ such that

$|E_{r}\cap B_{1/2}(y)|<\varepsilon,\qquad|E_{r}\cap B_{1}(y)|\geq|E_{r}\cap
B_{1/2}(0)|>\frac{\omega_{n}}{2^{n+2}}\,,$

where $\omega_{n}$ denotes the measure of the unit ball of $\mathbb{R}^{n}$
and $x_{r}=x+ry\in\Omega$. Therefore we have

$|E\cap B_{r/2}(x_{r})|<\varepsilon r^{n},\qquad|E\cap
B_{r}(x_{r})|>\frac{\omega_{n}r^{n}}{2^{n+2}}\,.$

Let us assume, without loss of generality, that $x_{r}=0$ and from now on let
us denote the balls centered at the origin by $B_{r}$. From the convergence of
$E_{h}$ to $E$ we have that for all $h$ sufficiently large

$|E_{h}\cap B_{r/2}|<\varepsilon r^{n},\qquad|E_{h}\cap
B_{r}|>\frac{\omega_{n}r^{n}}{2^{n+2}}\,.$ (19)

Let us now define the following bi-Lipschitz map used in [8] which maps
$B_{r}$ into itself

$\Phi(x)=\begin{cases}\bigl{(}1-\sigma(2^{n}-1)\bigr{)}x&\text{if}\,\,\,|x|<\displaystyle\frac{r}{2},\\\
x+\sigma\Bigl{(}\displaystyle
1-\frac{r^{n}}{|x|^{n}}\Bigr{)}x&\text{if}\,\,\,\displaystyle\frac{r}{2}\leq|x|<r,\\\
x&\text{if}\,\,\,|x|\geq r\,,\end{cases}$ (20)

for some fixed $0<\sigma<1/2^{n}$ such that, setting

${\widetilde{E}}_{h}=\Phi(E_{h}),\qquad{\tilde{u}}_{h}=u_{h}\circ\Phi^{-1}\,,$

we have $|{\widetilde{E}}_{h}|<d$. We obtain

$\displaystyle{\mathcal{F}}_{\lambda_{h}}(u_{h},E_{h})-{\mathcal{F}}_{\lambda_{h}}({\tilde{u}}_{h},{\widetilde{E}}_{h})=\biggl{[}\int_{B_{r}}\big{[}F(x,u_{h},\nabla
u_{h})+\mathbbm{1}_{E_{h}}G(x,u_{h},\nabla u_{h})\big{]}\,dx$
$\displaystyle-\int_{B_{r}}\big{[}F(x,\tilde{u}_{h},\nabla\tilde{u}_{h})+\mathbbm{1}_{\widetilde{E}_{h}}G(x,\tilde{u}_{h},\nabla\tilde{u}_{h})\big{]}\,dx\biggr{]}$
(21)
$\displaystyle\quad+\bigl{[}P(E_{h},{\overline{B}}_{r})-P({\widetilde{E}}_{h},{\overline{B}}_{r})\bigr{]}+\lambda_{h}\bigl{(}|{\widetilde{E}}_{h}|-|E_{h}|\bigr{)}$
$\displaystyle=I_{1,h}+I_{2,h}+I_{3,h}\,.$

Step 2. Estimate of $I_{1,h}$. In order to estimate the bulk energy $I_{1,h}$
we write down two preliminary estimates for the map $\Phi$ that can be easily
obtained by direct computation (see [8] for the explicit calculation). There
exists a constant $C_{1}=C_{1}(n)$ depending only on $n$ such that, for
$\sigma<1/4^{n}(n-1)^{2}-1$,

$\displaystyle\bigl{\|}\nabla\Phi^{-1}\bigl{(}\Phi(x)\bigr{)}\bigr{\|}\leq\bigl{(}1-(2^{n}-1)\sigma\bigr{)}^{-1}\leq
1+2^{n}n\sigma\qquad\text{for all}\,\,\,$ $\displaystyle x\in B_{r},$ (22)
$\displaystyle 1+C_{1}(n)\sigma\leq J\Phi(x)\leq 1+2^{n}n\sigma\qquad\text{for
all}\,\,\,$ $\displaystyle x\in B_{r}.$ (23)

We can now perform the change of variables $y=\Phi(x)$, and observing that
$\mathbbm{1}_{\widetilde{E}_{h}}(\Phi(x))=\mathbbm{1}_{E_{h}}(x)$, we get

$\displaystyle I_{1,h}=\int_{B_{r}}\bigl{[}F(x,u_{h},\nabla
u_{h})-J\Phi(x)F(\Phi(x),u_{h}(x),\nabla
u_{h}(x)\circ\nabla\Phi^{-1}(\Phi(x)))\bigr{]}\,dx$
$\displaystyle+\int_{B_{r}\cap E_{h}}\bigl{[}G(x,u_{h},\nabla
u_{h})-J\Phi(x)G(\Phi(x),u_{h}(x),\nabla
u_{h}(x)\circ\nabla\Phi^{-1}(\Phi(x)))\bigr{]}\,dx$ (24)
$\displaystyle:=J_{1,h}+J_{2,h}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$

The two terms $J_{1,h}$ and $J_{2,h}$, involving $F$ and $G$ in $B_{r}$ and
$B_{r}\cap E_{h}$ respectively, can be treated in the same way. Therefore we
just perform the calculation for $J_{1,h}$.
To make the exposition more clear when using the structure conditions
$\eqref{structure1}$ and $\eqref{structure2}$ we introduce the following
notations. $A_{2}(x,s)$ denotes the quadratic form and $A_{1}(x,s)$ denotes
the linear form defined as follows

$\displaystyle A_{2}(x,s)[\xi]:=a_{ij}(x,s)\xi_{i}\xi_{j}\qquad\text{for
all}\,\,\,\xi\in\mathbb{R}^{n},$ $\displaystyle
A_{1}(x,s)[\xi]:=a_{i}(x,s)\xi_{i}\qquad\text{for
all}\,\,\,\xi\in\mathbb{R}^{n},$

analogously $A_{0}(x,s)=a(x,s)$. Accordingly we can write down

$\displaystyle
J_{1,h}=\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad$
$\displaystyle\int_{B_{r}}\\!\Bigl{\\{}A_{2}(x,u_{h}(x))[\nabla
u_{h}(x)]\\!-\\!A_{2}(\Phi(x),u_{h}(x))[\nabla
u_{h}(x)\\!\circ\\!\nabla\Phi^{-1}(\Phi(x))]J\Phi(x)\Big{\\}}\,dx$
$\displaystyle+\\!\int_{B_{r}}\\!\Bigl{\\{}A_{1}(x,u_{h}(x))[\nabla
u_{h}(x)]\\!-\\!A_{1}(\Phi(x),u_{h}(x))[\nabla
u_{h}(x)\\!\circ\\!\nabla\Phi^{-1}(\Phi(x))]J\Phi(x)\Big{\\}}\,dx$
$\displaystyle+\\!\int_{B_{r}}\\!\Bigl{\\{}A_{0}(x,u_{h}(x))\\!-\\!A_{0}(\Phi(x),u_{h}(x))J\Phi(x)\Big{\\}}\,dx\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
(25)

We proceed estimating the first difference in the previous inequality, being
the other similar and indeed easier to handle.

$\displaystyle\int_{B_{r}}\\!\Bigl{\\{}A_{2}(x,u_{h}(x))[\nabla
u_{h}(x)]\\!-\\!A_{2}(\Phi(x),u_{h}(x))[\nabla
u_{h}(x)\\!\circ\\!\nabla\Phi^{-1}(\Phi(x))]J\Phi(x)\Big{\\}}\,dx$
$\displaystyle=\int_{B_{r}}\\!\Bigl{\\{}A_{2}(\Phi(x),u_{h}(x))[\nabla
u_{h}(x)]\\!-\\!A_{2}(\Phi(x),u_{h}(x))[\nabla
u_{h}(x)\\!\circ\\!\nabla\Phi^{-1}(\Phi(x))]J\Phi(x)\Big{\\}}\,dx$
$\displaystyle+\int_{B_{r}}\\!\Bigl{\\{}A_{2}(x,u_{h}(x))[\nabla
u_{h}(x)]\\!-\\!A_{2}(\Phi(x),u_{h}(x))[\nabla
u_{h}(x)]\Big{\\}}\,dx=H_{1,h}+H_{2,h}.$

The first term $H_{1,h}$ can be estimated observing that, in consequence of
$\eqref{ellipticity1}$ we have,

$|A_{2}[\xi]-A_{2}[\eta]|\leq
N|\xi+\eta||\xi-\eta|\qquad\qquad\forall\xi,\eta\in\mathbb{R}^{n}.$

We apply the last inequality to the vectors

$\xi:=\nabla u_{h}(x),\qquad\qquad\eta:=\sqrt{J\Phi(x)}[\nabla
u_{h}(x)\\!\circ\\!\nabla\Phi^{-1}(\Phi(x))],$

and use estimates $\eqref{phi}$ and $\eqref{Jphi}$ to get-

$|\xi-\eta|\leq\sigma C(n)|\nabla u_{h}(x)|\qquad\qquad|\xi+\eta|\leq
C(n)|\nabla u_{h}(x)|,$

for some constant $C(n)$ depending only on $n$. From the previous estimates we
deduce that

$|H_{1,h}|\leq\sigma NC^{2}(n)\int_{B_{r}}|\nabla u_{h}(x)|^{2}\,dx\leq\sigma
NC^{2}(n)\Theta.$ (26)

The second term $H_{2,h}$ can be estimated using the Lipschitz assumption of
$a_{i,j}$ and observing that $|x-\Phi(x)|\leq\sigma r2^{n}$. As a matter of
fact we deduce that

$|H_{2,h}|\leq\sigma r2^{n}\sup_{i,j}[a_{i,j}]_{0,1}\int_{B_{r}}|\nabla
u_{h}(x)|^{2}\,dx\leq\sigma C(n,[a_{i,j}]_{0,1})\Theta.$ (27)

In conclusion, since the other terms in $\eqref{J1h}$ can be estimated in the
same way, collecting estimates $\eqref{H1h}$ and $\eqref{H2h}$ we get

$|J_{1,h}|\leq\sigma C(n,N,L_{D})\Theta.$

The same estimate holds true for $J_{2,h}$ then we conclude that

$I_{1,h}\geq-\sigma C_{2}(n,N,L_{D})\Theta$ (28)

for some constant $C_{2}(n,N,[a_{i,j}]_{0,1})$ depending on
$n,N,\sup_{i,j}[a_{i,j}]_{0,1}$.
Step 3. Estimate of $I_{2,h}$. In order to estimate $I_{2,h}$ we can use the
area formula for maps between rectifiable set. If we denote by $T_{h,x}$ the
tangential gradient of $\Phi$ along the approximate tangent space to
$\partial^{*}E_{h}$ in $x$ and $T^{*}_{h,x}$ is the adjoint of the map
$T_{h,x}$, the $(n-1)$-dimensional jacobian of $T_{h,x}$ is given by

$J_{n-1}T_{h,x}=\sqrt{{\rm det}\bigl{(}T^{*}_{h,x}\circ T_{h,x}\bigr{)}}.$

Thereafter we can estimate

$J_{n-1}T_{h,x}\leq 1+\sigma+2^{n}(n-1)\sigma.$ (29)

We address the reader to [8] where explicit calculations are given. To
estimate $I_{2,h}$, we use the area formula for maps between rectifiable sets
([3, Theorem 2.91]), thus getting

$\displaystyle I_{2,h}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!P(E_{h},{\overline{B}}_{r})-P({\widetilde{E}}_{h},{\overline{B}}_{r})=\int_{\partial^{*}E_{h}\cap{\overline{B}}_{r}}\\!d\mathcal{H}^{n-1}-\int_{\partial^{*}E_{h}\cap{\overline{B}}_{r}}\\!J_{n-1}T_{h,x}\,d\mathcal{H}^{n-1}$
$\displaystyle=$
$\displaystyle\\!\\!\\!\int_{\partial^{*}E_{h}\cap{\overline{B}}_{r}\setminus
B_{r/2}}\\!\left(1-J_{n-1}T_{h,x}\right)\,d\mathcal{H}^{n-1}+\int_{\partial^{*}E_{h}\cap
B_{r/2}}\\!\left(1-J_{n-1}T_{h,x}\right)\,d\mathcal{H}^{n-1}\,.$

Notice that the last integral in the above formula is non-negative since
$\Phi$ is a contraction in $B_{r/2}$, hence $J_{n-1}T_{h,x}<1$ in $B_{r/2}$,
while from (29) we have

$\int_{\partial^{*}E_{h}\cap{\overline{B}}_{r}\setminus
B_{r/2}}\\!\left(1-J_{n-1}T_{h,x}\right)\,d\mathcal{H}^{n-1}\geq-2^{n}nP(E_{h},{\overline{B}}_{r})\sigma\geq-2^{n}n\Theta\sigma\,,$

thus concluding that

$I_{2,h}\geq-2^{n}n\Theta\sigma\,.$ (30)

Step 4. Estimate of $I_{3,h}$. To estimate $I_{3,h}$ we recall (19), (20),
(23), thus getting

$\displaystyle I_{3,h}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\lambda_{h}\int_{E_{h}\cap B_{r}\setminus
B_{r/2}}\\!\left(J\Phi(x)-1\right)\,dx+\lambda_{h}\int_{E_{h}\cap
B_{r/2}}\\!\left(J\Phi(x)-1\right)\,dx$ $\displaystyle\geq$
$\displaystyle\\!\\!\\!\lambda_{h}C_{1}(n)\Bigl{(}\frac{\omega_{n}}{2^{n+2}}-\varepsilon\Bigr{)}\sigma
r^{n}-\lambda_{h}\bigl{[}1-\bigl{(}1-(2^{n}-1)\sigma\bigr{)}^{n}\bigr{]}\varepsilon
r^{n}$ $\displaystyle\geq$ $\displaystyle\\!\\!\\!\lambda_{h}\sigma
r^{n}\Bigl{[}C_{1}(n)\frac{\omega_{n}}{2^{n+2}}-C_{1}(n)\varepsilon-(2^{n}-1)n\varepsilon\Bigr{]}\,.$

Therefore, if we choose $0<\varepsilon<\varepsilon(n)$, with $\varepsilon(n)$
depending only on the dimension, we have that

$I_{3,h}\geq\lambda_{h}C_{3}(n)\sigma r^{n}\,,$

for some positive $C_{3}(n)$. From this inequality, recalling (2.1), (28) and
(30) we obtain

${\mathcal{F}}_{\lambda_{h}}(u_{h},E_{h})-{\mathcal{F}}_{\lambda_{h}}({\tilde{u}}_{h},{\widetilde{E}}_{h})\geq\sigma\bigl{(}\lambda_{h}C_{3}r^{n}-\Theta(C_{2}(n,N,L_{D})+2^{n}n)\bigr{)}>0$

if $\lambda_{h}$ is sufficiently large. This contradicts the minimality of
$(u_{h},E_{h})$, thus concluding the proof.

∎

The previous theorem motivates the following definition.

######  Definition 2.5 ($\Lambda$-minimizers).

The energy pair $(E,u)$ is a $\Lambda$-minimizer in $\Omega$ of the functional
$\mathcal{F}$, defined in (1), iff for every $B_{r}(x_{0})\subset\Omega$

$\mathcal{F}(E,u;B_{r}(x_{0}))\leq\mathcal{F}(F,v;B_{r}(x_{0}))+\Lambda|F\Delta
E|,$

whenever $(F,v)$ is an admissible test pair, namely, $F$ is a set of finite
perimeter with $F\Delta E\subset\subset B_{r}(x_{0})$ and $v-u\in
H^{1}_{0}(B_{r}(x_{0}))$.

##  3\. Decay of the bulk energy

In the first part of this section we collect some preliminary results
concerning the decay estimates for local minimizers $u$ of the functional
$\eqref{intro0}$ when $E$ is fixed. We start quoting higher integrability
results both for local minimizers of functional $\eqref{intro0}$ and for
comparison functions that we will use later in the paper. It is worth
mentioning that the following lemmata can be applyed in general to minimizers
of integrals functionals of the type

$\mathcal{H}(u;\Omega):=\int_{\Omega}F(x,u,\nabla u)\,dx,$ (31)

assuming that the energy density only satisfies the structure condition
$\eqref{structure1}$ and the growth conditions $\eqref{ellipticity1}$ and
$\eqref{ellipticity2}$, without assuming any continuity on the coefficients.
Therefore functionals of the type $\eqref{intro0}$ belong to this class and in
addition the involved estimates only depend on the constants appearing in
$\eqref{ellipticity1}$ and $\eqref{ellipticity2}$ but don’t depend on $E$
accordingly.

######  Lemma 3.1.

Let $u\in H^{1}(\Omega)$ be a local minimizer of the functional $\mathcal{H}$
defined in $\eqref{H}$, where $F$ satisfies the structure condition
$\eqref{structure1}$ and the growth conditions $\eqref{ellipticity1}$ and
$\eqref{ellipticity2}$. Then for every $B_{2R}(x_{0})\subset\subset\Omega$ it
holds

$\fint_{B_{R}(x_{0})}|\nabla u|^{2}\,dx\leq
C_{1}\Bigl{(}1+\fint_{B_{2R}(x_{0})}|\nabla
u|^{2m}\,dx\Bigr{)}^{\frac{1}{m}},$ (32)

where $m=\frac{n}{n+2}$, $C_{1}=C_{1}(\nu,N,L,n)$ is a constant depending only
on $\nu,N,L,n$.

###### Proof.

Whithout loss of generality we can assume that $x_{0}=0$. Let $R<t<s<2R$ and
choose $\eta\in C^{\infty}_{0}(Q_{s})$ such that $\eta\equiv 1$ in $Q_{t}$ and
$|\nabla\eta|\leq 2/(s-t)$. We choose a test function $v=u-\phi$, where
$\phi=\eta(u-u_{s})$ and $u_{s}$ denotes the average of $u$ in $Q_{s}$,
$u_{s}=\fint_{Q_{s}}u\,dx$. From the growth conditions $\eqref{ellipticity1}$
and $\eqref{ellipticity2}$ we can deduce that

$\frac{\nu}{2}|z|^{2}-\frac{L^{2}}{\nu}\leq F(x,s,z)\leq(N+1)|z|^{2}+L(L+1),$
(33)

then testing the minimality of $u$ with $v$ we deduce that

$\displaystyle\frac{\nu}{2}\int_{Q_{s}}[|\nabla
u|^{2}-\frac{L^{2}}{\nu^{2}}]\,dx$ $\displaystyle\leq
2\int_{Q_{s}}\Bigr{[}(N+1)|\nabla
u(1-\eta)-\nabla\eta(u-u_{s})|^{2}+L(L+1)\Bigr{]}\,dx$ $\displaystyle\leq
4(N+1)\int_{Q_{s}\setminus Q_{t}}|\nabla
u|^{2}\,dx+4(N+1)\int_{Q_{s}}|u-u_{s}|^{2}|\nabla\eta|^{2}\,dx+2\int_{Q_{s}}L(L+1)\,dx.$

Adding to both sides $4(N+1)\int_{Q_{t}}|\nabla u|^{2}\,dx$ we deduce

$\displaystyle[4(N+1)+\nu/2]\int_{Q_{t}}|\nabla u|^{2}\,dx$ $\displaystyle\leq
4(N+1)\int_{Q_{s}}|\nabla
u|^{2}\,dx+4(N+1)\int_{Q_{s}}|u-u_{s}|^{2}|\nabla\eta|^{2}\,dx+\int_{Q_{s}}C(L,\nu)\,dx.$

Eventually we get

$\int_{Q_{t}}|\nabla u|^{2}\,dx\leq\theta\int_{Q_{s}}|\nabla
u|^{2}\,dx+\frac{C(\nu,N,L,n)}{(s-t)^{2}}\int_{Q_{s}}|u-u_{s}|^{2}\,dx+C(\nu,N,L,n).$

Where $\theta=4(N+1)/4(N+1+\nu/2)<1$ and the constant $C(\nu,N,L,n)$ depends
only on $\nu,N,L,n$. We can iterate the previous estimate using 2.2 to deduce
that there exist $C=C(\theta)=C(\nu,N)$ depending only on $\nu,N$ such that

$\int_{Q_{R}}|\nabla u|^{2}\,dx\leq
C(\nu,N)\biggl{[}\frac{C(\nu,N,L,n)}{(s-t)^{2}}\int_{Q_{2R}}|u-u_{s}|^{2}\,dx+C(\nu,N,L,n)\biggr{]}.$

Finally we use Sobolev-Poincaré inequality

$\int_{B_{2R}}|u-u_{2R}|^{2}\,dx\leq C(n)\Bigl{(}\int_{B_{2R}}|\nabla
u|^{2m}\Bigr{)}^{\frac{1}{m}},$

to conlude $\eqref{ReverseH}$. ∎

Starting from the previous lemma the higher integrability can be obtained in a
standard way by means of the Gehring’s Lemma (see [12, Proposition 6.1]).

######  Lemma 3.2.

Let $u\in H^{1}(\Omega)$ be a local minimizer of the functional $\mathcal{H}$
defined in (31), where $F$ satisfies the structure condition
$\eqref{structure1}$ and the growth conditions $\eqref{ellipticity1}$ and
$\eqref{ellipticity2}$. There exists an $s>1$ such that, for every ball
$B_{2R}(x_{0})\subset\subset\Omega$ it holds

$\fint_{B_{R}(x_{0})}|\nabla u|^{2s}\,dx\leq
C_{2}\Bigl{(}\fint_{B_{2R}(x_{0})}(1+|\nabla u|^{2})\,dx\Bigr{)}^{s},$

where $s$ and $C_{2}$ depend only on $\nu,N,L,n$.

In the next section we will prove some energy density estimates by using a
standard comparison argument. For this purpose we will need a reverse Hölder
inequality for the comparison function defined below.

######  Definition 3.3 (Comparison).

Let $u\in H^{1}(\Omega)$ be a local minimizer of the functional (1) and
$B_{2R}\subset\subset\Omega$. We shall denote by $v$ the solution of the
following problem

$v:=\arg\min\left\\{\int_{B_{R}}\tilde{F}(x,\nabla w)\,dx:\quad w\in
u+H^{1}_{0}(B_{R})\right\\},$ (34)

where $\tilde{F}(x,z):=F(x,u(x),z)$ satisfies the structure condition
$\eqref{structure1}$ and the growth conditions $\eqref{ellipticity1}$ and
$\eqref{ellipticity2}$.

For the comparison function $v$ defined in (34) we can state the following
reverse Hölder inequality up to the boundary of $B_{R}$.

######  Lemma 3.4.

Let $u\in H^{1}(\Omega)$ be a local minimizer of the functional (1), $v$ the
comparison function defined above and $B_{2R}\subset\subset\Omega$. Let us
consider the following extension of $v$:

$V(x):=\left\\{\begin{array}[]{ll}v(x)&\mbox{ for }x\in B_{R}\\\ u(x)&\mbox{
for }x\in\Omega\setminus B_{R}.\end{array}\right.$

Denote by $B_{\rho}(x_{0})$ a generic ball centered in $x_{0}\in B_{R}$ with
$\rho<\frac{R}{2}$. Then:

$\fint_{B_{\rho}(x_{0})}|\nabla V|^{2}\,dx\leq
C\Bigl{(}\fint_{B_{2\rho}(x_{0})}(|\nabla V|^{2m}+|\nabla
u|^{2m}+1)\,dx\Bigr{)}^{\frac{1}{2}},$ (35)

where $m=\frac{n}{n+2}$ and $C=C(n,\nu,N,L)$.

###### Proof.

Let $x_{0}\in B_{R}$ and $\rho\leq s<t\leq r<2\rho<R$, where
$r=\frac{3}{2}\rho$; then, the following alternative holds,

$\left\\{\begin{array}[]{ll}i)&B_{r}(x_{0})\subset\subset B_{R}\\\
ii)&\overline{B}_{r}(x_{0})\cap(\Omega\setminus
B_{R})\neq\emptyset.\end{array}\right.$

In the case i) we can proceed exactly as in Lemma 3 to get the desired
estimate. Let us then consider the case ii) which is slightly different.
Choose $\eta\in C^{\infty}_{0}(B_{t}(x_{0}))$, such that $0\leq\eta\leq 1$,
$\eta\equiv 1$ in $B_{s}$ and $|\nabla\eta|\leq 2/(t-s)$. Now we can use the
following function $\varphi:=\eta(V-u)$ to test the minimality of $v$ with the
aim of estimating the difference $\int_{B_{s}}|\nabla(V-u)|$. Using the growth
conditions $\eqref{ellipticity1}$ and $\eqref{ellipticity2}$ and the
minimality of $v$ we obtain

$\displaystyle\frac{\nu}{2}\int_{B_{t}}\Bigl{(}|\nabla\varphi|^{2}-\frac{2L^{2}}{\nu^{2}}\Bigr{)}\,dx\leq\int_{B_{t}\cap
B_{R}}\tilde{F}(x,\nabla\varphi)\,dx=$ $\displaystyle=$
$\displaystyle\int_{B_{t}\cap B_{R}}\tilde{F}(x,\nabla v)\,dx+\int_{B_{t}\cap
B_{R}}\Bigl{(}\tilde{F}(x,\nabla\varphi)-\tilde{F}(x,\nabla v)\Bigr{)}\,dx$
$\displaystyle\leq$ $\displaystyle\int_{B_{t}\cap
B_{R}}\tilde{F}(x,\nabla(v-\varphi))\,dx+\int_{B_{t}\cap
B_{R}}\sum_{i,j=1}^{n}a_{ij}(x)\nabla_{i}(\varphi-v)\nabla_{j}(\varphi-v)\,dx$
$\displaystyle-$ $\displaystyle 2\int_{B_{t}\cap
B_{R}}\sum_{i,j=1}^{n}a_{ij}(x)\nabla_{i}\varphi\nabla_{j}(\varphi-v)+\int_{B_{t}\cap
B_{R}}\sum_{i=1}^{n}a_{i}(x)(\nabla_{i}\varphi-\nabla_{i}v)\,dx$
$\displaystyle\leq$
$\displaystyle\frac{\nu}{4}\int_{B_{t}}|\nabla\varphi|^{2}\,dx+(3N+\frac{8N^{2}}{\nu^{2}})\int_{B_{t}}|\nabla(V-\varphi)|^{2}\,dx+\int_{B_{t}}L^{2}\,dx$

Where we used Young’s inequality in the last estimate. We can summarize the
previous estimate as follows

$\frac{\nu}{4}\int_{B_{t}}|\nabla\varphi|^{2}\,dx\leq
C(\nu,N,L)\int_{B_{t}}\Bigl{(}1+|\nabla(V-\varphi)|^{2}\Bigr{)}\,dx.$ (37)

Now we observe that $|\nabla(V-\varphi)|\leq|\nabla
u|+(1-\eta)|\nabla(V-u)|+\frac{1}{(t-s)}|V-u|$; then by (37) we deduce

$\displaystyle\int_{B_{s}}|\nabla(V-u)|^{2}\,dx\leq
C(\nu,N,L)\int_{B_{t}\setminus B_{s}}|\nabla(V-u)|^{2}\,dx$ (38)
$\displaystyle+\frac{C(\nu,N,L)}{(t-s)^{2}}\int_{B_{t}}|(V-u)|^{2}\,dx+C(\nu,N,L)\int_{B_{t}}\bigl{(}1+|\nabla
u|^{2}\bigr{)}\,dx.$

Now we use the “hole filling” technique adding
$C(\nu,N,L)\int_{B_{s}}|\nabla(V-u)|^{2}\,dx$ on both sides of (38) to get

$\begin{split}&\int_{B_{s}}|\nabla(V-u)|^{2}\,dx\leq\theta\int_{B_{t}}|\nabla(V-u)|^{2}\,dx\\\
&+C(\nu,N,L)\biggl{[}\frac{1}{(t-s)^{2}}\int_{B_{t}}|(V-u)|^{2}\,dx+\int_{B_{t}}\big{(}1+|\nabla
u|^{2}\big{)}\,dx\biggr{]},\end{split}$

where $\theta=C(\nu,N,L)/(C(\nu,N,L)+1)$. Using Lemma 2.2 we obtain

$\int_{B_{\rho}}|\nabla(V-u)|^{2}\,dx\leq\frac{C(\nu,N,L)}{(r-\rho)^{2}}\int_{B_{r}}|(V-u)|^{2}\,dx+C(\nu,N,L)\int_{B_{r}}\big{(}1+|\nabla
u|^{2}\big{)}\,dx.$

Therefore, having chosen $r=\frac{3}{2}\rho$ and by condition ii), we have

$|{B}_{2\rho}(x_{0})\setminus B_{R})|\geq C|{B}_{\rho}(x_{0})|,$

for some universal constant $C=C(n)$. We can now use Sobolev-Poincaré’s
inequality for functions vanishing on a set of positive measure (see [12,
Inequality (3.29)]) to deduce

$\fint_{B_{\rho}(x_{0})}\\!\\!\\!\\!\\!|\nabla(V-u)|^{2}\,dx\leq
C(\nu,N,L)\Biggl{[}\biggl{(}\fint_{B_{2\rho}(x_{0})}\\!\\!\\!\\!\\!(|\nabla(V-u)|^{2m}\,dx\biggr{)}^{\frac{1}{m}}+\fint_{B_{2\rho}}\\!\\!\big{(}1+|\nabla
u|^{2}\big{)}\,dx\Biggr{]}$

Finally we can apply reverse Hölder inequality $\eqref{ReverseH}$ for $u$ in
the last estimate to get $\eqref{revh2}$. ∎

Reasoning in a similar way as above, the higher integrability for $v$ can be
obtained by means of the Gehring’s Lemma (see [12, Proposition 6.1]).

######  Lemma 3.5.

Let $u\in H^{1}(\Omega)$ be a local minimizer of the functional (1), $v\in
H^{1}(B_{R}(x_{0}))$ the comparison function defined in $\eqref{Comparison}$.
Denoting by $s>1$ the same exponent given in Lemma 3.2, it holds

$\fint_{B_{R}(x_{0})}|\nabla v|^{2s}\,dx\leq
C_{3}\biggl{(}\fint_{B_{2R}(x_{0})}\big{(}1+|\nabla
u|^{2}\big{)}\,dx\biggr{)}^{s},$

where $C_{3}$ depend only on $\nu,N,L,n$.

###  3.1. Decay estimates for elastic minima

In this section we prove a decay estimate for elastic minima that will be
crucial for the proof strategy. As a matter of fact we show that if $(E,u)$ is
a $\Lambda$ minimizer of the functional $\eqref{intro0}$ and $x_{0}$ is a
point in $\Omega$, where either the density of E is close to $0$ or $1$, or
the set $E$ is asymptotically close to a hyperplane, then for sufficiently
small $\rho$ we have

$\int_{B_{\rho}(x_{0})}|\nabla u_{E}|^{2}\,dx\leq C\rho^{n-\delta},$

for any $\delta>0$. A preliminary result we want to mention, that we will use
later, provides an upper bound for $\mathcal{F}$. It is rather standard and is
related to the threshold Hölder exponent $\frac{1}{2}$ of the function $u$,
when $(E,u)$ is either a solution of the constrained problem ($P_{c}$) or a
solution of the penalized problem ($P$) defined in section 1. For the proof we
address the reader too [19, Lemma 2.3] and [11]. A detailed proof in the case
of costrained problems and for functionals satisfying general $p$-polinomial
growth is contained in [5].

######  Theorem 3.6.

Let $(E,u)$ be a $\Lambda$-minimizers of the functional ${\mathcal{F}}$
defined in (1). For every open set $U\subset\subset\Omega$ there exists a
constant $C_{1}$, depending on $U$ and $\left\lVert\nabla
u\right\rVert_{L^{2}(\Omega)}$, such that for every $B_{r}(x_{0})\subset U$ it
holds

${\mathcal{F}}(E,u;B_{r}(x_{0}))\leq C_{1}r^{n-1}.$

As a consequence of the previous theorem we can infer, using Poincaré’s
inequality and the characterization of Campanato spaces (see for example [12,
Theorem 2.9]), that $u\in C^{0,\frac{1}{2}}$. We deduce the following remark.

######  Remark 3.7.

Let $(E,u)$ be a $\Lambda$-minimizers of the functional ${\mathcal{F}}$
defined in (1). For every open set $U\subset\subset\Omega$ there exists a
constant $C$, depending on $U$ and $\left\lVert\nabla
u\right\rVert_{L^{2}(\Omega)}$, such that

$\sup_{x,y\in U}\frac{|u(x)-u(y)|}{|x-y|^{\frac{1}{2}}}\leq C\left\lVert\nabla
u\right\rVert_{L^{2}(\Omega)}.$ (39)

######  Notation 3.8.

In the sequel $E\subset\Omega$ will denote any given subset of $\Omega$ with
finite perimeter. We denote by $u_{E}$, or simply by $u$ if no confusion
arises, any local minimizers of the functional $\mathcal{F}(E,v;\Omega)$.

  * •

If $x\in\mathbb{R}^{n}$ we write $x=(x^{\prime},x_{n})$, where
$x^{\prime}\in\mathbb{R}^{n-1}$ and $x_{n}\in\mathbb{R}$.
Accordingly we denote
$\nabla^{\prime}=(\partial_{x_{1}},\dots,\partial_{x_{n-1}})$ the gradient
with respect to the first $n-1$ components.

  * •

We will denote $H=\\{x\in\Omega\,:\,x_{n}>0\\},$

In order to prove the main lemma of this section we introduce the following
preliminary result. For reader convenience we give here a sketch of the proof
which can be found in [22]. Actually we state here a weaker version that is
suitable for our aim.

######  Lemma 3.9.

Let $v\in H^{1}(B_{1})$ be a solution of

$-\textnormal{div}(A\nabla u)=\textnormal{div}\,G,\qquad\mbox{ in
}\mathcal{D}^{\prime}(B_{1}),$

where

$G^{+}:=\mathbbm{1}_{H}G\in C^{0,\alpha}(H\cap B_{1}),\qquad
G^{-}:=\mathbbm{1}_{H^{c}}G\in C^{0,\alpha}(H^{c}\cap B_{1}),$

for some $\alpha>0$ and $A$ is an elliptic matrix satisfying

$\nu|\xi|^{2}\leq A_{ij}(x)\xi_{i}\xi_{j}\leq N|\xi|^{2},$

and

$A^{+}:=\mathbbm{1}_{H}A\in C^{0,\alpha}(\overline{H}\cap B_{1}),\qquad
A^{-}:=\mathbbm{1}_{H^{c}}A\in C^{0,\alpha}(\overline{H^{c}}\cap B_{1}),$

for some $\nu,N>0$. Let us denote

$C_{A}=\max\\{\left\lVert A^{+}\right\rVert_{C^{0,\alpha}},\left\lVert
A^{-}\right\rVert_{C^{0,\alpha}}\\},\qquad C_{G}=\max\\{\left\lVert
G^{+}\right\rVert_{C^{0,\alpha}},\left\lVert
G^{-}\right\rVert_{C^{0,\alpha}}\\}.$

Then $v\in L_{loc}^{2,n}(B_{1})$. Moreover, there exist two constants
$C=C(n,\nu,N,C_{A},C_{G})$ and $r_{0}=r_{0}(n,\nu,N,\left\lVert
G\right\rVert_{L^{\infty}},C_{A},C_{G})$ such that, for any $r<r_{0}$ with
$B_{r}(x_{0})\subset B_{1}$,

$\int_{B_{\rho}(x_{0})}|\nabla v|^{2}\,dx\leq
C\Bigl{(}\frac{\rho}{r}\Bigr{)}^{n}\int_{B_{r}(x_{0})}|\nabla
v|^{2}\,dx+C\rho^{n},\quad\forall\,\rho<\frac{r}{4}.$ (40)

###### Proof.

Fix $x_{0}\in B_{1}$ and let $r$ be such that $B_{r}(x_{0})\subset B_{1}$. Let
us denote by $a^{+}$ and $a^{-}$ the averages of $A$ in $H\cap B_{r}(x_{0})$
and $H^{c}\cap B_{r}(x_{0})$ respectively. In an analogous way we define
$g^{+}$ and $g^{-}$ the averages of $G$. For $x\in B_{r}(x_{0})$ we define

$\overline{A}:=a^{+}\mathbbm{1}_{H}+a^{-}\mathbbm{1}_{H^{c}},\qquad\qquad\overline{G}:=g^{+}\mathbbm{1}_{H}+g^{-}\mathbbm{1}_{H^{c}}.$

Notice that by assumption

$|A(x)-\overline{A}(x)|\leq
C_{A}r^{\alpha}\qquad\mbox{and}\qquad|G(x)-\overline{G}(x)|\leq
C_{G}r^{\alpha},$ (41)

Let $w$ be the solution of

$\begin{cases}-\textnormal{div}(\overline{A}\nabla
w)=\textnormal{div}\overline{G}\\\ w=v\mbox{ on }\partial
B_{r}(x_{0}).\end{cases}$

The last equation can be rewritten as

$\begin{cases}-\textnormal{div}(a^{+}\nabla w^{+})=0&\text{in
}B_{r}(x_{0})\cap H,\\\ -\textnormal{div}(a^{-}\nabla w^{-})=0&\text{in
}B_{r}(x_{0})\cap H^{c},\\\ w^{+}=w^{-}&\text{on }B_{r}(x_{0})\cap\partial
H,\\\ a^{+}\nabla w^{+}\cdot e_{n}-a^{-}\nabla w^{-}\cdot e_{n}=g^{+}\cdot
e_{n}-g^{-}\cdot e_{n},&\text{on }B_{r}(x_{0})\cap\partial H\\\ \end{cases}$
(42)

where $w^{+}:=w\mathbbm{1}_{B_{r}(x_{0})\cap H}$,
$w^{-}:=w\mathbbm{1}_{B_{r}(x_{0})\cap H^{c}}$. Set

$\overline{D}_{c}w:=\sum_{i=1}^{n}\overline{A}_{in}\nabla_{i}w+\overline{G}\cdot
e_{n}$

We notice that $\overline{D}_{c}w$ has no jumps on the boundary thanks to the
transmission condition in (42). This allows to prove that the distributional
gradient of $\overline{D}_{c}w$ coincides with the point-wise one.
Step 1: Tangential derivatives of $w$. Let us denote with $\tau$ the general
direction tangent to the hyperplane $\partial H$. Since $\overline{A}$ and
$\overline{G}$ are both constant along the tangential directions, the
classical difference quotient method gives that $\nabla_{\tau}w\in
W^{1,2}_{loc}(B_{r}(x_{0}))$ and

$\mbox{div}(\overline{A}\nabla(\nabla_{\tau}w))=0\qquad\mbox{ in
}B_{r}(x_{0}).$

Hence, Caccioppoli’s inequality holds,

$\int_{B_{\rho}(x)}|\nabla(\nabla_{\tau}w)|^{2}\,dy\leq\frac{c(n,\nu,N)}{\rho^{2}}\int_{B_{2\rho}(x)}|\nabla_{\tau}w-(\nabla_{\tau}w)_{x,2\rho}|^{2}\,dy,$
(43)

for all balls $B_{2\rho}(x)\subset B_{r}(x_{0})$ and, by De Giorgi’s
regularity theorem, $\nabla_{\tau}w$ is Hölder continuous and there exists
$\gamma=\gamma(n,\nu,N)>0$ such that if $B_{s}(x)\subset B_{r}(x_{0})$

$\displaystyle\int_{B_{\rho}(x)}|\nabla_{\tau}w-(\nabla_{\tau}w)_{x,\rho}|^{2}\,dy$
(44) $\displaystyle\qquad\qquad\qquad\leq
c(n,\nu,N)\bigg{(}\frac{\rho}{s}\bigg{)}^{n+2\gamma}\int_{B_{s}(x)}|\nabla_{\tau}w-(\nabla_{\tau}w)_{x,s}|^{2}\,dy,$

for any $\rho\in\big{(}0,\frac{s}{2}\big{)}$ and

$\max_{B_{\frac{\rho}{2}}(x)}|\nabla_{\tau}w|^{2}\leq\frac{c(n,\nu,N)}{\rho^{n}}\int_{B_{\rho}(x)}|\nabla_{\tau}w|^{2}\,dy.$
(45)

Step 2: Regularity of $\overline{D}_{c}w$. First of all observe that
$\nabla_{\tau}(\overline{D}_{c}w)=\overline{D}_{c}(\nabla_{\tau}w)-\overline{G}\cdot
e_{n}$. This imply by Step 1 that the tangential derivatives of
$\overline{D}_{c}w$ belong to $L^{2}_{loc}(B_{r}(x_{0}))$. Furthermore we can
estimate directly by definition of $\overline{D}_{c}w$

$|\nabla_{n}(\overline{D}_{c}w)|\leq c(n,N)|\nabla\nabla_{\tau}w|,$

which implies again by Step1

$|\nabla\overline{D}_{c}w|\leq c(n,N)|\nabla\nabla_{\tau}w|.$

We can conclude that $\overline{D}_{c}w\in W^{1,2}_{loc}(B_{r}(x_{0}))$. Using
Poincaré’s inequality and (43), we have

$\begin{split}&\int_{B_{\rho}(x)}|\overline{D}_{c}w-(\overline{D}_{c}w)_{x,\rho}|^{2}\,dy\leq
c(n)\rho^{2}\int_{B_{\rho}(x)}|\nabla(\overline{D}_{c}w)|^{2}\,dy\\\ &\leq
c(n,N)\rho^{2}\int_{B_{\rho}(x)}|\nabla(\nabla_{\tau}w)|^{2}\,dy\\\ &\leq
c(n,\nu,N)\int_{B_{2\rho}(x)}|\nabla_{\tau}w-(\nabla_{\tau}w)_{x,2\rho}|^{2}\,dy,\end{split}$

for any $B_{2\rho}(x)\subset B_{r}(x_{0})$. By (44) and (45) we infer

$\begin{split}&\int_{B_{\rho}(x)}|\overline{D}_{c}w-(\overline{D}_{c}w)_{x,\rho}|^{2}\,dy\\\
&\leq
c(n,\nu,N)\bigg{(}\frac{\rho}{r}\bigg{)}^{n+2\gamma}\int_{B_{\frac{r}{2}}(x)}|\nabla_{\tau}w-(\nabla_{\tau}w)_{x,\frac{r}{2}}|^{2}\,dy\\\
&\leq
c(n,\nu,N)\bigg{(}\frac{\rho}{r}\bigg{)}^{n+2\gamma}\int_{B_{r}(x_{0})}|\nabla_{\tau}w|^{2}\,dy,\end{split}$

for any $x\in B_{\frac{r}{4}}(x_{0})$, $\rho\leq\frac{r}{4}$. Hence by Lemma
4.2 in [22] (see also [3, Lemma 7.51]), $\overline{D}_{c}w$ is Hölder
continuous and

$\begin{split}\max_{B_{\frac{r}{4}}(x_{0})}|\overline{D}_{c}w|^{2}&\leq
c(n,\nu,N)\int_{B_{r}(x_{0})}|\nabla_{\tau}w|^{2}\,dy+\bigg{|}\fint_{B_{\frac{r}{4}}(x_{0})}\overline{D}_{c}w(y)\,dy\bigg{|}^{2}\\\
&\leq\frac{c(n,\nu,N)}{r^{n}}\int_{B_{r}(x_{0})}|\nabla
w|^{2}\,dy+2\left\lVert G\right\rVert^{2}_{L^{\infty}}.\end{split}$ (46)

Step 3: Comparison between $v$ and $w$. Subtracting the equation for $w$ from
the equation for $v$ we get

$\displaystyle\int_{B_{r}(x_{0})}\overline{A}_{ij}(x)\big{(}\nabla_{i}v-\nabla_{i}w\big{)}\nabla_{j}\varphi\,dx$
(47)
$\displaystyle=\int_{B_{r}(x_{0})}\bigl{(}\overline{A}_{ij}(x)-A_{ij}(x)\bigr{)}\nabla_{i}v\nabla_{j}\varphi\,dx+\int_{B_{r}(x_{0})}\bigl{(}\overline{G}_{i}-G_{i}\bigr{)}\nabla_{i}\varphi\,dx$

for any $\varphi\in W^{1,2}_{0}(B_{r}(x_{0}))$. Choosing $\varphi=v-w$ in the
previous equation and using assumption $\eqref{MVhoelder}$ we have

$\int_{B_{r}(x_{0})}|\nabla v-\nabla w|^{2}\,dx\leq
C_{A}r^{\alpha}\int_{B_{r}(x_{0})}|\nabla v|^{2}\,dy+C_{G}r^{n+\alpha},$ (48)

Finally we can estimate

$\begin{split}&\int_{B_{\rho}(x_{0})}|\nabla v|^{2}\,dy\leq
2\int_{B_{\rho}(x_{0})}|\nabla w|^{2}\,dy+2\int_{B_{\rho}(x_{0})}|\nabla
v-\nabla w|^{2}\,dy\\\ &\leq 2\omega_{n}\rho^{n}\sup_{B_{\frac{r}{4}}}|\nabla
w|^{2}+2\int_{B_{\rho}(x_{0})}|\nabla v-\nabla w|^{2}\,dy,\end{split}$

for any $\rho\leq\frac{r}{4}$ and observing that

$\begin{split}\sup_{B_{\frac{r}{4}}(x_{0})}|\nabla
w|^{2}&=\sup_{B_{\frac{r}{4}}(x_{0})}|\nabla_{\tau}w|^{2}+\sup_{B_{\frac{r}{4}}(x_{0})}|\nabla_{n}w|^{2}\\\
&\leq
c(n,\nu,N)\sup_{B_{\frac{r}{4}}(x_{0})}|\nabla_{\tau}w|^{2}+c(\nu)\sup_{B_{\frac{r}{4}}(x_{0})}|\overline{D}_{c}w|^{2}+c(\nu,\left\lVert
G\right\rVert_{\infty}),\end{split}$

by (45), (46), minimality of $w$ and Young’s inequality we gain

$\begin{split}&\int_{B_{\rho}(x_{0})}|\nabla v|^{2}\,dy\\\ &\leq
c(n,\nu,N)\bigg{(}\frac{\rho}{r}\bigg{)}^{n}\int_{B_{r}(x_{0})}|\nabla
w|^{2}\,dy+c(n,\nu,\left\lVert
G\right\rVert_{\infty},C_{A},C_{G})\bigg{[}r^{\alpha}\int_{B_{r}(x_{0})}|\nabla
v|^{2}\,dy+r^{n}\bigg{]}\\\ &\leq C(n,\nu,N,\left\lVert
G\right\rVert_{\infty},C_{A},C_{G})\Bigg{\\{}\Bigg{[}\bigg{(}\frac{\rho}{r}\bigg{)}^{n}+r^{\alpha}\Bigg{]}\int_{B_{r}(x_{0})}|\nabla
v|^{2}\,dy+r^{n}\Bigg{\\}}\end{split}$

which leads to our aim if we apply Lemma 2.3. ∎

The next lemma is inspired by [11, Proposition 2.4] and is the main result of
this section.

######  Lemma 3.10.

Let $(E,u)$ be a $\Lambda$-minimizers of the functional ${\mathcal{F}}$
defined in (1). There exists $0<\tau_{0}<1$ such that the following statement
is true: for all $\tau\in(0,\tau_{0})$ there exists
$\varepsilon_{0}=\varepsilon_{0}(\tau)>0$ such that if
$B_{r}(x_{0})\subset\subset\Omega$ with $r^{\frac{1}{2n}}<\tau$ and one of the
following conditions holds:

  * _(i)_

$|E\cap B_{r}(x_{0})|<\varepsilon_{0}|B_{r}|$,

  * _(ii)_

$|B_{r}(x_{0})\setminus E|<\varepsilon_{0}|B_{r}|$,

  * _(iii)_

There exists a halfspace $H$ such that $\frac{\left|(E\Delta H)\cap
B_{r}(x_{0})\right|}{|B_{r}|}<\varepsilon$,

then

$\int_{B_{\tau r}(x_{0})}|\nabla u|^{2}\,dx\leq
C_{0}\tau^{n}\int_{B_{r}(x_{0})}|\nabla u|^{2}\,dx+C_{0}r^{n},$

for some constant $C_{0}$ depending only on $n,\nu,N,L,L_{D},\left\lVert\nabla
u\right\rVert_{L^{2}(\Omega)}$.

###### Proof.

Let us fix $B_{r}(x_{0})\subset\subset\Omega$ and $0<\tau<1$. Without loss of
generality, we may assume that $\tau<1/4$ and $x_{0}=0$. We start proving
_(i)_ , being the proof of _(ii)_ similar. Let us define

$A^{0}_{ij}:=a_{ij}(x_{0},u_{r/2}(x_{0})),\quad
B^{0}_{i}:=a_{i}(x_{0},u_{r/2}(x_{0})),\quad f^{0}:=a(x_{0},u_{r/2}(x_{0})),$
$F_{0}(\xi):=(A^{0}\xi;\xi)+(B^{0};\xi)+f^{0}.$

Let us denote by $v$ the solution of the following problem

$\min\left\\{\mathcal{F}_{0}(w;B_{r/2}):\;w=u\;\mbox{ on }\partial
B_{r/2}\right\\},$

where

$\mathcal{F}_{0}(w;B_{r/2}):=\int_{B_{r/2}}F_{0}(\nabla w)\,dx$

We use now the following identity

$(A^{0}\xi;\xi)-(A^{0}\eta;\eta)=(A^{0}(\xi-\eta);(\xi-\eta))+2(A^{0}\eta;\xi-\eta),$

in order to deduce that

$\displaystyle\mathcal{F}_{0}(u)-\mathcal{F}_{0}(v)$
$\displaystyle=\int_{B_{r/2}}\bigl{[}(A^{0}\nabla u;\nabla u)-(A^{0}\nabla
v;\nabla v)\bigr{]}\,dx+\int_{B_{r/2}}(B^{0};\nabla u-\nabla v)\,dx$
$\displaystyle=\int_{B_{r/2}}(A^{0}(\nabla u-\nabla v);(\nabla u-\nabla
v))\,dx$ (49) $\displaystyle+2\int_{B_{r/2}}(A^{0}\nabla v;\nabla u-\nabla
v)\,dx+\int_{B_{r/2}}(B^{0};\nabla u-\nabla v)\,dx$

By the Euler-Lagrange equation for $v$ we deduce that the sum of the last two
integrals in the previous identity is zero being also $u=v$ on $\partial
B_{r/2}$. Therefore, using the ellipticity assumption of $A^{0}$ we finally
achieve that

$\nu\int_{B_{r/2}}|\nabla u-\nabla
v|^{2}\,dx\leq\mathcal{F}_{0}(u)-\mathcal{F}_{0}(v).$ (50)

We prove now that $u$ is an $\omega$-minimizer of $\mathcal{F}_{0}$. We start
writing

$\displaystyle\mathcal{F}_{0}(u)=\mathcal{F}(E,u)+[\mathcal{F}_{0}(u)-\mathcal{F}(E,u)]$
$\displaystyle\leq\mathcal{F}(E,v)+[\mathcal{F}_{0}(u)-\mathcal{F}(E,u)]$ (51)
$\displaystyle=\mathcal{F}_{0}(v)+[\mathcal{F}_{0}(u)-\mathcal{F}(E,u)]+[\mathcal{F}(E,v)-\mathcal{F}_{0}(v)].$

_Estimate of $\mathcal{F}_{0}(u)-\mathcal{F}(E,u)$_. We use (10), (39) and
(39) to infer

$\begin{split}&\mathcal{F}_{0}(u)-\mathcal{F}(E,u)=\int_{B_{r/2}}\bigl{(}a_{ij}(x_{0},u_{r/2}(x_{0}))-a_{ij}(x,u(x))\bigr{)}\nabla_{i}u\nabla_{j}u\,dx\\\
&+\int_{B_{r/2}}\bigl{(}a_{i}(x_{0},u_{r/2}(x_{0}))-a_{i}(x,u(x))\bigr{)}\nabla_{i}u\,dx\\\
&+\int_{B_{r/2}}\bigl{(}a(x_{0},u_{r/2}(x_{0}))-a(x,u(x))\bigr{)}\,dx-\int_{B_{r/2}\cap
E}G(x,u,\nabla u)\,dx\\\ &\leq 2L_{D}\left\lVert\nabla
u\right\rVert_{L^{2}(\Omega)}\biggl{(}r^{\frac{1}{2}}\int_{B_{r/2}}|\nabla
u|^{2}\,dx+r^{n+\frac{1}{2}}\biggr{)}+C(N,L)\int_{B_{r/2}\cap E}|\nabla
u|^{2}\,dx+2Lr^{n}\end{split}$ (52)

where we denoted $L_{D}$ the greatest lipschitz constant of the data
$a_{ij},b_{ij},a_{i},b_{i},a,b$ defined in (8). Now we use Lemma 3.2 to
estimate

$\displaystyle\int_{B_{r/2}\cap E}|\nabla u|^{2}\,dx\leq|E\cap
B_{r}|^{1-1/s}|B_{r}|^{1/s}\biggl{(}\fint_{B_{r/2}}|\nabla
u|^{2s}\biggr{)}^{1/s}$ $\displaystyle\leq C_{2}^{1/s}\biggl{(}\frac{|E\cap
B_{r}|}{|B_{r}|}\biggr{)}^{1-1/s}\int_{B_{r}}\big{(}1+|\nabla
u|^{2}\big{)}\,dx.$ (53)

Merging the last estimate in $\eqref{eqq6}$ we deduce

$\displaystyle\mathcal{F}_{0}(u)-\mathcal{F}(E,u)\leq\Bigl{(}L_{D}\left\lVert\nabla
u\right\rVert_{L^{2}(\Omega)}+C(N,L)C_{2}^{1/s}\Bigr{)}\Bigl{(}r^{\frac{1}{2}}+\varepsilon_{0}^{1-1/s}\Bigr{)}\int_{B_{r}}|\nabla
u|^{2}\,dx$ $\displaystyle+\Bigl{(}C_{2}^{1/s}+2L+L_{D}\left\lVert\nabla
u\right\rVert_{L^{2}(\Omega)}\Bigr{)}r^{n}$ (54)

Estimate of $\mathcal{F}(E,v)-\mathcal{F}_{0}(v)$.

$\displaystyle\mathcal{F}(E,v)-\mathcal{F}_{0}(v)=\int_{B_{r/2}}\bigl{(}a_{ij}(x,v(x))-a_{ij}(x_{0},u_{r/2}(x_{0}))\bigr{)}\nabla_{i}v\nabla_{j}v\,dx$
$\displaystyle+\int_{B_{r/2}}\bigl{(}a_{i}(x,v(x))-a_{i}(x_{0},u_{r/2}(x_{0}))\bigr{)}\nabla_{i}v\,dx$
$\displaystyle+\int_{B_{r/2}}\bigl{(}a(x,v(x))-a(x_{0},u_{r/2}(x_{0}))\bigr{)}\,dx+\int_{B_{r/2}\cap
E}G(x,v,\nabla v)\,dx.$

If we choose now $z\in\partial B_{r/2}$, recalling that $u(z)=v(z)$ we deduce

$\displaystyle\bigl{|}a_{ij}(x,v(x))-a_{ij}(x_{0},u_{r/2}(x_{0}))\bigr{|}$
$\displaystyle=\bigl{|}a_{ij}(x,v(x))-a_{ij}(x,v(z))+a_{ij}(x,u(z))-a_{ij}(x_{0},u_{r/2}(x_{0}))\bigr{|}$
$\displaystyle\leq L_{D}\big{(}|v(x)-v(z)|+r^{\frac{1}{2}}\left\lVert\nabla
u\right\rVert_{L^{2}(\Omega)}+r\big{)}$ $\displaystyle\leq
L_{D}\big{(}\text{osc}(u,\partial
B_{r/2})+C(n,\nu,N,L)r+r^{\frac{1}{2}}\left\lVert\nabla
u\right\rVert_{L^{2}(\Omega)}+r\big{)}$ $\displaystyle\leq
C(n,\nu,N,L,L_{D},\left\lVert\nabla
u\right\rVert_{L^{2}(\Omega)})r^{\frac{1}{2}}$

where we used the fact that $\text{osc}(v,B_{r/2})\leq\text{osc}(u,\partial
B_{r/2})+C(n,\nu,N,L)r$, (see [12, Lemma 8.4]). Analogously we can estimate
the other difference in $\eqref{E1}$, deducing

$\displaystyle\mathcal{F}(E,v)-\mathcal{F}_{0}(v)\leq
C(n,\nu,N,L,L_{D},\left\lVert\nabla
u\right\rVert_{L^{2}(\Omega)})r^{\frac{1}{2}}\biggl{(}\int_{B_{r/2}}|\nabla
v|^{2}\,dx+r^{n}\biggr{)}$ $\displaystyle+C(N,L)\biggl{(}\int_{B_{r/2}\cap
E}|\nabla v|^{2}\,dx+r^{n}\biggr{)},$

Reasoning in a similar way as in (3.1), we can apply the higher integrability
for $v$ given by Lemma 3.5 and infer

$\int_{B_{r/2}\cap E}|\nabla v|^{2}\,dx\leq
C(n,\nu,N,L)\varepsilon_{0}^{1-1/s}\biggl{(}\int_{B_{r}}|\nabla
u|^{2}\,dx+r^{n}\biggr{)}.$

Therefore we obtain

$\displaystyle\mathcal{F}(E,v)-\mathcal{F}_{0}(v)$ (56) $\displaystyle\leq
C(n,\nu,N,L,L_{D},\left\lVert\nabla
u\right\rVert_{L^{2}(\Omega)})\biggl{[}\Bigl{(}r^{\frac{1}{2}}+\varepsilon_{0}^{1-1/s}\Bigr{)}\int_{B_{r}}|\nabla
u|^{2}\,dx+r^{n}\biggr{]}.$

Finally, collecting $\eqref{EllF0}$, $\eqref{omegamin}$, $\eqref{E0}$ and
$\eqref{E2}$, if we choose $\varepsilon_{0}$ such that
$\varepsilon_{0}^{1-\frac{1}{s}}=\tau^{n}$ we conclude that

$\int_{B_{r/2}}|\nabla u-\nabla v|^{2}\,dx\leq C\tau^{n}\int_{B_{r}}|\nabla
u|^{2}\,dx+Cr^{n},$ (57)

for some constant $C=C(n,\nu,N,L,L_{D},\left\lVert\nabla
u\right\rVert_{L^{2}(\Omega)})$. On the other hand $v$ is the solution of a
uniform elliptic equation with constant coefficients, so we have

$\int_{B_{\tau r}}|\nabla v|^{2}\,dx\leq
C_{1}(n,\nu,N)\tau^{n}\int_{B_{r/2}}|\nabla v|^{2}\,dx\leq
C_{2}(n,\nu,N)\tau^{n}\int_{B_{r/2}}|\nabla u|^{2}\,dx.$ (58)

Hence we may estimate, using (57) and (58)

$\displaystyle\int_{B_{\tau r}}|\nabla u|^{2}\,dx$ $\displaystyle\leq$
$\displaystyle 2\int_{B_{\tau r}}|\nabla v-\nabla u|^{2}\,dx+2\int_{B_{\tau
r}}|\nabla v|^{2}\,dx$ (59) $\displaystyle\leq$ $\displaystyle
C_{0}\tau^{n}\int_{B_{r}}|\nabla u|^{2}\,dx+C_{0}r^{n},$ (60)

for some constant $C_{0}=C_{0}(n,\nu,N,L,L_{D},\left\lVert\nabla
u\right\rVert_{L^{2}(\Omega)})$.
We are left with the case _(iii)_. Let $H$ be the half space from the
assumption and let us denote accordingly

$\displaystyle A^{0}_{ij}(x):=a_{ij}(x,u(x))+\mathbbm{1}_{H}b_{ij}(x,u(x)),$
$\displaystyle B^{0}_{ij}(x):=a_{i}(x,u(x))+\mathbbm{1}_{H}b_{i}(x,u(x)),$
$\displaystyle f^{0}(x):=a(x,u(x))+\mathbbm{1}_{H}b(x,u(x)),$ $\displaystyle
F_{0}(x,\xi):=(A^{0}(x)\xi;\xi)+(B^{0}(x);\xi)+f^{0}(x).$

Let us denote by $v_{H}$ the solution of the following problem

$\min\left\\{\mathcal{F}_{0}(w;B_{r/2}):\;w=u\;\mbox{ on }\partial
B_{r/2}\right\\},$

where

$\mathcal{F}_{0}(w;B_{r/2}):=\int_{B_{r/2}}F_{0}(x,\nabla w)\,dx.$

Let us point out that $v_{H}$ solves the Euler-Lagrange equation

$-2\textnormal{div}(A^{0}\nabla v_{H})=\textnormal{div}\,B^{0}\quad\text{in
}\mathcal{D}^{\prime}(B_{r/2}).$ (61)

Therefore we are in position to apply Lemma 3.9 to the function $v_{H}$. As a
matter of fact, from the Hölder continuity of $u(x)$ (see Remark 3.7) we
deduce that the restrictions of $A^{0}$ and $B^{0}$ onto $H\cap B_{r}$ and
$B_{r}\setminus H$ respectively, are Hölder continuous. We can conclude using
also $\eqref{hoelderu}$ that there exist two constants
$C=C(n,\nu,N,L,L_{D},\left\lVert\nabla u\right\rVert_{L^{2}(\Omega)})$ and
$\tau_{0}=\tau_{0}(n,\nu,N,L,L_{D},\left\lVert\nabla
u\right\rVert_{L^{2}(\Omega)})$ such that for $\tau<\tau_{0}$,

$\int_{B_{\tau r}}|\nabla v_{H}|^{2}\,dx\leq C\tau^{n}\int_{B_{r/2}}|\nabla
v_{H}|^{2}\,dx+Cr^{n}.$ (62)

In addition, using the ellipticity condition of $A^{0}$ we can argue as in
$\eqref{EF0}$ to deduce using also the fact that $v_{H}$ satisfies
$\eqref{MV}$,

$\nu\int_{B_{r/2}}|\nabla u-\nabla
v_{H}|^{2}\,dx\leq\mathcal{F}_{0}(u)-\mathcal{F}_{0}(v_{H}).$ (63)

One more time we can prove that $u$ is an $\omega$\- minimizer of
$\mathcal{F}_{0}$. We start as above writing

$\displaystyle\mathcal{F}_{0}(u)=\mathcal{F}(E,u)+[\mathcal{F}_{0}(u)-\mathcal{F}(E,u)]$
$\displaystyle\leq\mathcal{F}(E,v_{H})+[\mathcal{F}_{0}(u)-\mathcal{F}(E,u)]$
(64)
$\displaystyle=\mathcal{F}_{0}(v_{H})+[\mathcal{F}_{0}(u)-\mathcal{F}(E,u)]+[\mathcal{F}(E,v_{H})-\mathcal{F}_{0}(v_{H})].$

We can estimate the differences $\mathcal{F}_{0}(u)-\mathcal{F}(E,u)$ and
$\mathcal{F}(E,v_{H})-\mathcal{F}_{0}(v_{H})$ exactly as before using this
time the higher integrability given in Lemma 3.5. We conclude that

$\int_{B_{r/2}}|\nabla u-\nabla v_{H}|^{2}\,dx\leq
C\tau^{n}\int_{B_{r}}|\nabla u|^{2}\,dx+Cr^{n},$ (65)

for some constant $C=C(n,\nu,N,L,L_{D},\left\lVert\nabla
u\right\rVert_{L^{2}(\Omega)})$. From the last estimate we can conclude the
proof as before using (62) and (63). ∎

##  4\. Energy density estimates

This section is devoted to prove a lower bound esimate for the functional
${\mathcal{F}}(E,u;B_{r}(x_{0}))$. Lemma 3.10 is the main tool to achieve such
result. We shall prove that the energy $\mathcal{F}$ decays “fast” if the
perimeter of $E$ is “small”.

######  Lemma 4.1.

Let $(u,E)$ be a $\Lambda$-minimizer of the functional ${\mathcal{F}}$ defined
in (1). For every $\tau\in(0,1)$ there exists
$\varepsilon=\varepsilon_{1}(\tau)>0$ such that, if
$B_{r}(x_{0})\subset\Omega$ and $P(E;B_{r}(x_{0}))<\varepsilon_{1}r^{n-1}$,
then

$\mathcal{F}(E,u;B_{\tau r}(x_{0}))\leq
C_{1}\tau^{n}\bigl{(}\mathcal{F}(E,u;B_{r}(x_{0}))+r^{n}\bigr{)},$ (66)

for some constant $C_{1}=C_{1}(n,\nu,N,L,L_{D},\left\lVert\nabla
u\right\rVert_{L^{2}(\Omega)})$ independent of $\tau$ and $r$.

###### Proof.

Without loss of generality we may assume that $\tau<\frac{1}{2}$. We can also
assume that $x_{0}=0,\ r=1$ by introducing the following rescaling,
$E_{r}=\frac{E-x_{0}}{r}$, $u_{r}=r^{-\frac{1}{2}}{u(x_{0}+ry)}$ and replacing
$\Lambda$ with $\Lambda r$. Thus, we have that $(E_{r},u_{r})$ is a $\Lambda
r$-minimizer of $\mathcal{F}$ in $\frac{\Omega-x_{0}}{r}$. For simplicity of
notation we can still denote $E_{r}$ by $E$, $u_{r}$ by $u$ and then we have
to prove that, given $0<\tau<\frac{1}{2}$, there exists
$\varepsilon_{1}=\varepsilon_{1}(\tau)$ such that, if
$P(E;B_{1})<\varepsilon_{1}$, then

$\mathcal{F}(E,u;B_{\tau})\leq
C_{1}\tau^{n}\Bigl{(}\mathcal{F}(E,u;B_{1})+r\Bigr{)}.$

Note that, since $P(E;B_{1})$ is small, holds, by the relative isoperimetric
inequality $|B_{1}\cap E|$ or $|B_{1}\setminus E|$ is small. Thus Lemma 3.10
holds. Assuming that $|B_{1}\setminus E|$ is small and using the relative
isoperimetric inequality we can deduce that,

$|B_{1}\setminus E|\leq c(n)P(E;B_{1})^{\frac{n}{n-1}}.$

If we choose as a representative of $E$ the set of points of density one, we
get, by Fubini’s theorem,

$|B_{1}\setminus E|\geq\int_{\tau}^{2\tau}\mathcal{H}^{n-1}(\partial
B_{\rho}\setminus E)\ d\rho.$

Combining these inequalities we can choose $\rho\in(\tau,2\tau)$ such that

$\mathcal{H}^{n-1}(\partial B_{\rho}\setminus
E)\leq\frac{c(n)}{\tau}P(E;B_{1})^{\frac{n}{n-1}}\leq\frac{c(n)\varepsilon_{1}^{\frac{1}{n-1}}}{\tau}P(E;B_{1}).$
(67)

Now we set $F=E\cup B_{\rho}$ and observe that

$P(F;B_{1})\leq
P(E;B_{1}\setminus\overline{B}_{\rho})+\mathcal{H}^{n-1}(\partial
B_{\rho}\setminus E).$

If we choose $(u,F)$ to test the $\Lambda r$-minimality of $(u,E)$ we get

$\displaystyle P(E;B_{1})+\int_{B_{1}}\Bigl{(}F(x,u,\nabla
u)+\mathbbm{1}_{E}G(x,u,\nabla u)\Bigr{)}\,dx\leq$ $\displaystyle\leq$
$\displaystyle P(F;B_{1})+\int_{B_{1}}\Bigl{(}F(x,u,\nabla
u)+\mathbbm{1}_{F}G(x,u,\nabla u)\Bigr{)}\,dx+\Lambda r|F\setminus E|$
$\displaystyle\leq$ $\displaystyle
P(E;B_{1}\setminus\overline{B}_{\rho})+\mathcal{H}^{n-1}(\partial
B_{\rho}\setminus E)+\int_{B_{1}}\Bigl{(}F(x,u,\nabla
u)+\mathbbm{1}_{F}G(x,u,\nabla u)\Bigr{)}\,dx+\Lambda r|B_{\rho}|.$

Then getting rid of the common terms we obtain

$P(E;B_{\rho})\leq\mathcal{H}^{n-1}(\partial B_{\rho}\setminus
E)+\int_{B_{\rho}}G(x,u,\nabla u)\,dx+\Lambda r|B_{\rho}|.$

Now if we choose $\varepsilon_{1}$ such that
$c(n)\varepsilon_{1}^{\frac{1}{n-1}}\leq\tau^{n+1}$ we have from (67)

$P(E;B_{\rho})\leq\tau^{n}P(E;B_{1})+\int_{B_{\rho}}G(x,u,\nabla
u)\,dx+\Lambda r|B_{\rho}|.$

Then we choose $\varepsilon_{1}$ satisfying
$c(n)\varepsilon_{1}^{\frac{n}{n-1}}\leq\varepsilon_{0}(2\tau)|B_{1}|$ to
obtain using Lemma 3.10 and growth conditions (9), (10),

$\displaystyle\int_{B_{\rho}}G(x,u,\nabla u)\,dx\leq
C(N,L)\int_{B_{\rho}}(1+|\nabla u|^{2})\,dx$ $\displaystyle\leq
C(n,\nu,N,L,L_{D},\left\lVert\nabla
u\right\rVert_{L^{2}(\Omega)})\tau^{n}\int_{B_{1}}(1+|\nabla u|^{2})\,dx$
$\displaystyle\leq$ $\displaystyle C(n,\nu,N,L,L_{D},\left\lVert\nabla
u\right\rVert_{L^{2}(\Omega)})\tau^{n}\int_{B_{1}}G(x,u,\nabla u)\,dx.$

Finally, we recall that $\rho\in(\tau,2\tau)$ to get

$P(E;B_{\tau})\leq\tau^{n}P(E;B_{1})+C(n,\nu,N,L,L_{D},\left\lVert\nabla
u\right\rVert_{L^{2}(\Omega)})\tau^{n}\int_{B_{1}}G(x,u,\nabla u)\,dx+\Lambda
r|B_{2\tau}|.$

From this estimate the result easily follows. ∎

######  Theorem 4.2 (Density lower bound).

Let $(u,E)$ be a $\Lambda$-minimizer of $\mathcal{F}$ and
$U\subset\subset\Omega$. There exists a costant $C$ depending only on
$n,\nu,N,L,L_{D},\left\lVert\nabla u\right\rVert_{L^{2}(\Omega)}$, such that
for every $x_{0}\in\partial E$ and $B_{r}(x_{0})\subset U$,

$P(E,B_{r}(x_{0}))\geq Cr^{n-1}.$

Moreover, $\mathcal{H}^{n-1}((\partial E\setminus\partial^{*}E)\cap\Omega)=0$.

###### Proof.

The proof follows from Lemma 3.10 and Lemma 4.1 in a standard way, see [11,
Proposition 4.4] or [3, Theorem 7.21]. ∎

##  5\. Compactness for sequences of minimizers

In this section we basically follow the route given in [20, Part III]. We
start proving a standard compactness result.

######  Lemma 5.1 (Compactness).

Let $(E_{h},u_{h})$ be a sequence of $\Lambda_{h}$-minimizers of $\mathcal{F}$
in $\Omega$ such that $\sup_{h}\mathcal{F}(E,u_{h};\Omega)<+\infty$ and
$\Lambda_{h}\rightarrow\Lambda\in\mathbb{R}^{+}$. There exists a (not
relabelled) subsequence and a $\Lambda$-minimizer of $\mathcal{F}$, $(u,E)$,
such that for every open set $U\subset\subset\Omega$, it holds

$E_{h}\rightarrow E\mbox{ in }L^{1}(U),\;\;u_{h}\rightarrow u\mbox{ in
}H^{1}(U),\;\;P(E_{h},U)\rightarrow P(E,U).$

In addition,

$\displaystyle\mbox{if }x_{h}\in\partial E_{h}\cap U\mbox{ and
}x_{h}\rightarrow x\in U,\mbox{ then }x\in\partial E\cap U,$ (68)
$\displaystyle\mbox{if }x\in\partial E\cap U\mbox{ there exists
}x_{h}\in\partial E_{h}\cap U\mbox{ such that }x_{h}\rightarrow x.$ (69)

Finally if we assume also $\nabla u_{h}\rightharpoonup 0$ weakly in
$L^{2}_{loc}(\Omega,\mathbb{R}^{n})$ and $\Lambda_{h}\rightarrow 0$, then $E$
is a local minimizer of the perimeter, that is

$P(E,B_{r}(x_{0}))\leq P(F,B_{r}(x_{0})),$

for every $F$ such that $F\Delta E\subset\subset B_{r}(x_{0})\subset\Omega.$

###### Proof.

We start observing that, by the boundedness condition on
$\mathcal{F}(E,u_{h};\Omega)$, we can assume that, $u_{h}$ weakly converges to
$u$ in $H^{1}(U)$ and strongly in $L^{2}(U)$, and $\mathbbm{1}_{E_{h}}$
converges to $\mathbbm{1}_{E}$ in $L^{1}(U)$. Using lower semicontinuity we
are going to prove the $\Lambda$-minimality of $(u,E)$. Let us fix
$B_{r}(x_{0})\subset\subset\Omega$ and assume for simplicity of notation that
$x_{0}=0$. Let $(v,F)$ be a test pair such that $F\Delta E\subset\subset
B_{r}$ and supp$(u-v)\subset\subset B_{r}$. We can handle the perimeter term
as in [20], eventually passing to a subsequence and using Fubini’s theorem, we
can choose $\rho<r$ such that, once again, $F\Delta E\subset\subset B_{\rho}$
and supp$(u-v)\subset\subset B_{\rho}$, and in addition,

$\mathcal{H}^{n-1}(\partial^{*}F\cap\partial
B_{\rho})=\mathcal{H}^{n-1}(\partial^{*}E_{h}\cap\partial B_{\rho})=0,$

and

$\lim_{h\rightarrow 0}\mathcal{H}^{n-1}(\partial B_{\rho}\cap(E\Delta
E_{h}))=0.$ (70)

Now we choose a cut-off function $\psi\in C_{0}^{1}(B_{r})$ such that
$\psi\equiv 1$ in $B_{\rho}$, and define $v_{h}=\psi v+(1-\psi)u_{h}$,
$F_{h}:=(F\cap B_{\rho})\cup(E_{h}\setminus B_{\rho})$ to test the minimality
of $(u_{h},E_{h})$. Thanks to the $\Lambda_{h}$-minimality of $(u_{h},E_{h})$
we have

$\displaystyle\int_{B_{r}}\Bigl{(}F(x,u_{h},\nabla
u_{h})+\mathbbm{1}_{E_{h}}G(x,u_{h},\nabla
u_{h})\Bigr{)}\,dx+P(E_{h},B_{r})\leq$ (71)
$\displaystyle\leq\int_{B_{r}}\Bigl{(}F(x,v_{h},\nabla
v_{h})+\mathbbm{1}_{F_{h}}G(x,v_{h},\nabla
v_{h})\Bigr{)}\,dx+P(F_{h},B_{r})+\Lambda_{h}|F_{h}\Delta E_{h}|$
$\displaystyle\leq\int_{B_{r}}\Bigl{(}F(x,v_{h},\nabla
v_{h})+\mathbbm{1}_{F_{h}}G(x,v_{h},\nabla
v_{h})\Bigr{)}\,dx+P(F,B_{\rho})+\Lambda_{h}|F_{h}\Delta E_{h}|$
$\displaystyle+P(E_{h},B_{r}\setminus\overline{B}_{\rho})+\varepsilon_{h}.$

The mismatch term $\varepsilon_{h}=\mathcal{H}^{n-1}(\partial
B_{\rho}\cap(F^{(1)}\Delta E_{h}^{(1)}))$ appears because $F$ is not in
general a compact variation of $E_{h}$. Nevertheless we have that
$\varepsilon_{h}\rightarrow 0$ because of the assumption $\eqref{mismatch}$
(see also [20, Theorem 21.14]).
Now we use the convexity of $F$ and $G$ with respect to the $z$ variabile to
deduce

$\displaystyle\int_{B_{r}}\Bigl{(}F(x,v_{h},\nabla
v_{h})+\mathbbm{1}_{F_{h}}G(x,v_{h},\nabla v_{h})\Bigr{)}\,dx$
$\displaystyle\leq\int_{B_{r}}\Bigl{(}F(x,v_{h},\psi\nabla v+(1-\psi)\nabla
u_{h})+\mathbbm{1}_{F_{h}}G(x,v_{h},\psi\nabla v+(1-\psi)\nabla
u_{h})\Bigr{)}\,dx$
$\displaystyle+\int_{B_{r}}\left<\nabla_{z}F(x,v_{h},\nabla
v_{h}),\nabla\psi(v-u_{h})\right>\,dx$
$\displaystyle+\int_{B_{r}}\mathbbm{1}_{F_{h}}\left<\nabla_{z}G(x,v_{h},\nabla
v_{h}),\nabla\psi(v-u_{h})\right>\,dx$

where last two terms in the previous estimate tend to zero for
$h\rightarrow\infty$. As a matter of fact the term $\nabla\psi(v-u_{h})$
strongly converges to zero in $L^{2}$, being $u=v$ in $B_{r}\setminus
B_{\rho}$ and the first part in the scalar poduct weakly converges in $L^{2}$.
Then using again the convexity of $F$ and $G$ in the $z$ variable we obtain,
for some infinitesimal $\sigma_{h}$,

$\displaystyle\int_{B_{r}}\Bigl{(}F(x,v_{h},\nabla
v_{h})+\mathbbm{1}_{F_{h}}G(x,v_{h},\nabla v_{h})\Bigr{)}\,dx$ (72)
$\displaystyle\leq\int_{B_{r}}\psi\Bigl{(}F(x,v_{h},\nabla
v)+\mathbbm{1}_{F_{h}}G(x,v_{h},\nabla v)\Bigr{)}\,dx$
$\displaystyle+\int_{B_{r}}(1-\psi)\Bigl{(}F(x,v_{h},\nabla
u_{h})+\mathbbm{1}_{F_{h}}G(x,v_{h},\nabla u_{h})\Bigr{)}\,dx+\sigma_{h}.$

Finally, we connect $\eqref{min}$ and $\eqref{compact}$ and pass to the limit
with respect to $h$ using the lower semicontinuity on the left hand side. For
the right hand side we observe that
$\mathbbm{1}_{E_{h}}\rightarrow\mathbbm{1}_{E}$ and
$\mathbbm{1}_{F_{h}}\rightarrow\mathbbm{1}_{F}$ in $L^{1}(B_{r})$ and use also
the equi-integrability of $\left\\{\nabla u_{h}\right\\}$ to conclude,

$\displaystyle\int_{B_{r}}\psi\Bigl{(}F(x,u,\nabla
u)+\mathbbm{1}_{E}G(x,u,\nabla u)\Bigr{)}\,dx+P(E,B_{\rho})\leq$
$\displaystyle\leq\int_{B_{r}}\psi\Bigl{(}F(x,v,\nabla
v)+\mathbbm{1}_{F}G(x,v,\nabla v)\Bigr{)}\,dx+P(F,B_{\rho})+\Lambda|F\Delta
E|.$

Letting $\psi\downarrow\mathbbm{1}_{B_{\rho}}$ we finally get

$\displaystyle\int_{B_{\rho}}\Bigl{(}F(x,u,\nabla
u)+\mathbbm{1}_{E}G(x,u,\nabla u)\Bigr{)}\,dx+P(E,B_{\rho})\leq$ (73)
$\displaystyle\leq\int_{B_{\rho}}\Bigl{(}F(x,v,\nabla
v)+\mathbbm{1}_{F}G(x,v,\nabla v)\Bigr{)}\,dx+P(F,B_{\rho})+\Lambda|F\Delta
E|,$

and this proves the $\Lambda$-minimality of $(u,E)$.
To prove the strong convergence of $\nabla u_{h}$ to $\nabla{u}$ in
$L^{2}(B_{r})$ we start observing that by $\eqref{min}$ and $\eqref{compact}$
applied using $(u,E_{h})$ to test the minimality of $(u_{h},E_{h})$ we get

$\displaystyle\int_{B_{r}}\psi\Bigl{(}F(x,u_{h},\nabla
u_{h})+\mathbbm{1}_{E_{h}}G(x,u_{h},\nabla u_{h})\Bigr{)}\,dx$
$\displaystyle\hskip 56.9055pt\leq\int_{B_{r}}\psi\Bigl{(}F(x,u,\nabla
u)+\mathbbm{1}_{E_{h}}G(x,u,\nabla u)\Bigr{)}\,dx+\sigma_{h}$

Then from equiintegrability of $\left\\{\nabla u_{h}\right\\}$ in $L^{2}(U)$
and recalling that $\mathbbm{1}_{E_{h}}\rightarrow\mathbbm{1}_{E}$ in
$L^{1}(U)$, we obtain

$\displaystyle\limsup_{h\rightarrow+\infty}\int_{B_{r}}\psi\Bigl{(}F(x,u_{h}\nabla
u_{h})+\mathbbm{1}_{E_{h}}G(x,u_{h},\nabla u_{h})\Bigr{)}\,dx$
$\displaystyle\hskip 56.9055pt\leq\int_{B_{r}}\psi\Bigl{(}F(x,u,\nabla
u)+\mathbbm{1}_{E}G(x,u,\nabla u)\Bigr{)}\,dx.$

The opposite inequality can be obtained by semicontiuity then we can deduce,

$\displaystyle\lim_{h\rightarrow+\infty}\int_{B_{r}}\psi\Bigl{(}F(x,u_{h},\nabla
u_{h})+\mathbbm{1}_{E_{h}}G(x,u_{h},\nabla u_{h})\Bigr{)}\,dx$ (74)
$\displaystyle\hskip 85.35826pt=\int_{B_{r}}\psi\Bigl{(}F(x,u,\nabla
u)+\mathbbm{1}_{E}G(x,u,\nabla u)\Bigr{)}\,dx.$

Now from ellepticity condition $\eqref{ellipticity1}$ we infer, for some
$\sigma_{h}\rightarrow 0$,

$\displaystyle\nu\int_{B_{r}}\psi|\nabla u_{h}-\nabla
u|^{2}\,dx\leq\int_{B_{r}}\psi\Bigl{(}F(x,u_{h},\nabla u_{h})-F(x,u,\nabla
u)\Bigr{)}\,dx$
$\displaystyle+\int_{B_{r}}\psi\mathbbm{1}_{E}\Bigl{(}G(x,u_{h},\nabla
u_{h})-G(x,u,\nabla u)\Bigr{)}\,dx+\sigma_{h}$ (75)

Passing to the limit we obtain

$\lim_{h\rightarrow+\infty}\int_{B_{r}}\psi|\nabla u_{h}-\nabla u|^{2}\,dx=0.$

Finally testing the minimality of $(u_{h},E_{h})$ with the pair $(u,E)$ we
also get

$\lim_{h\rightarrow+\infty}P(E_{h},B_{\rho})=P(E,B_{\rho}).$

With an usual argument we can deduce $u_{h}\rightarrow u$ in $W^{1,2}(U)$ and
$P(E_{h},U)\rightarrow P(E,U)$ for every open set $U\subset\subset\Omega$. The
topological information stated in $\eqref{boundary1}$ and $\eqref{boundary2}$
follows as in [20, Theorem 21.14] because doesn’t depend on the presence of
the integral bulk part. ∎

##  6\. Decay of the excess

###  6.1. Excess and height bound

Now we introduce the usual quantities involved in regularity theory. Given
$x\in\partial E$, a scale $r>0$ and a direction $\nu\in\mathbbm{S}^{n-1}$ we
define the spherical excess

${\mathbf{e}}(x,r,\nu):=\frac{1}{r^{n-1}}\int_{\partial E\cap
B_{r}(x)}\frac{|\nu_{E}(y)-\nu|^{2}}{2}d\mathcal{H}^{n-1}(y),$

and

${\mathbf{e}}(x,r):=\min_{\nu\in\mathbbm{S}^{n-1}}{\mathbf{e}}(x,r,\nu).$

In addition we define the rescaled Dirichlet integral of $u$

$\mathcal{D}(x,r):=\frac{1}{r^{n-1}}\int_{B_{r}(x)}|\nabla u|^{2}dy.$

The following height bound lemma is a standard step in the proof of
regularity.

######  Lemma 6.1 (Height bound).

Let $(u,E)$ be a $\Lambda$-minimizer of $\mathcal{F}$ in $B_{r}(x_{0})$. There
exist two positive constants $C$ and $\varepsilon$, depending on
$\left\lVert\nabla u\right\rVert_{L^{2}(B_{r}(x_{0}))}$, such that if
$x_{0}\in\partial E$ and

${\mathbf{e}}(x,r,\nu)<\varepsilon$

for some $\nu\in\mathbbm{S}^{n-1}$ then

$\sup_{y\in\partial E\cap
B_{r/2}(x_{0})}\frac{|\langle\nu,y-x_{0}\rangle|}{r}\leq
C{\mathbf{e}}(x,r,\nu)^{\frac{1}{2(n-1)}}$

###### Proof.

The proof of this lemma is almost equal to the one of [20, Theorem 22.8]. As a
matter of fact it follows from the density lower bound (see Theorem 4.2), the
relative isoperimetric inequality and the compactness result proved above. ∎

Proceeding as in [20] we give the following Lipschitz approximation lemma
which is a consequence of the height bound lemma. Its proof follows exactly as
in [20, Theorem 23.7].

######  Theorem 6.2 (Lipschitz approximation).

Let $(u,E)$ be a $\Lambda$-minimizer of $\mathcal{F}$ in $B_{r}(x_{0})$. There
exist two positive constants $C_{3}$ and $\varepsilon_{3}$, depending on
$\left\lVert\nabla u\right\rVert_{L^{2}(B_{r}(x_{0}))}$, such that

$\mbox{ if }x_{0}\in\partial E\;\;\mbox{ and
}\;\;{\mathbf{e}}(x_{0},r,e_{n})<\varepsilon_{3}$

then there exists a Lipschitz function
$f:\mathbbm{R}^{n-1}\rightarrow\mathbbm{R}$ such that

$\sup_{x^{\prime}\in\mathbbm{R}^{n-1}}\frac{|f(x^{\prime})|}{r}\leq
C_{3}{\mathbf{e}}(x_{0},r,e_{n})^{\frac{1}{2(n-1)}},\hskip
28.45274pt\left\lVert\nabla^{\prime}f\right\rVert_{L^{\infty}}\leq 1$

and

$\frac{1}{r^{n-1}}\mathcal{H}^{n-1}((\partial E\Delta\Gamma_{f})\cap
B_{r/2}(x_{0}))\leq C_{3}{\mathbf{e}}(x_{0},r,e_{n}),$

where $\Gamma_{f}$ is the graph of $f$. Moreover $f$ is “almost harmonic” in
the sense that

$\frac{1}{r^{n-1}}\int_{B_{r/2}^{n-1}(x^{\prime}_{0})}|\nabla^{\prime}f|^{2}\,dx^{\prime}\leq
C_{3}{\mathbf{e}}(x_{0},r,e_{n}).$

Finally we shall need the following reverse Poincaré inequality which can be
proved exactly as in the case of $\Lambda$-minimizers of the perimeter (see
[20, Theorem 24.1] ).

######  Theorem 6.3 (Reverse Poincaré).

Let $(u,E)$ be a $\Lambda$-minimizer of $\mathcal{F}$ in $B_{r}(x_{0})$. There
exist two positive constants $\varepsilon_{4}$ and $C_{4}$ such that if
$x_{0}\in\partial E$ and ${\mathbf{e}}(x_{0},r,\nu)<\varepsilon_{4}$ then

${\mathbf{e}}(x_{0},r/2,\nu)\leq C_{4}\biggl{(}\frac{1}{r^{n+1}}\int_{\partial
E\cap
B_{r}(x_{0})}|\left<\nu,x-x_{0}\right>-c|^{2}d\mathcal{H}^{n-1}+\mathcal{D}(x_{0},r)+r\biggr{)},$

for every $c\in\mathbbm{R}$.

###  6.2. Weak Euler-Lagrange equation

The last ingredient to prove the excess improvement is the following Euler-
Lagrange equation that we state for $\Lambda r$-minimizers of the rescaled
functional $\mathcal{F}_{r}$ defined below. For the sake of simplicity we will
denote $A_{1}(x,s)$ the matrix whose entries are $a_{hk}(x,s)$, $A_{2}(x,s)$
the vector of components $a_{h}(x,s)$, $A_{3}(x,s)=a(x,s)$ and similarly for
$B_{i}$, $i=1,2,3$. Accordingly we define

$\begin{split}&\mathcal{F}_{r}(w,D):=\int_{B_{1}}\big{[}F_{r}(x,u,\nabla
u)+\mathbbm{1}_{D}G_{r}(x,u,\nabla u)\big{]}\,dx\\\
&=\int_{B_{1}}\big{[}\langle(A_{1r}+\mathbbm{1}_{D}B_{1r})\nabla w,\nabla
w\rangle+\sqrt{r}\langle A_{2r}+\mathbbm{1}_{D}B_{2r},\nabla
w\rangle+r(A_{3r}+\mathbbm{1}_{D}B_{3r})\big{]}\,dx,\end{split}$

where $r>0$, $x_{0}\in\Omega$, $A_{ir}:=A_{i}(x_{0}+ry,\sqrt{r}w)$,
$B_{ir}:=B_{i}(x_{0}+ry,\sqrt{r}w)$, for $i=1,2,3$. The argument used to prove
the next result is similar to the one in [3, Theorem 7.35].

######  Theorem 6.4 (Weak Euler-Lagrange equation).

Let $(u,E)$ be a $\Lambda r$-minimizer of $\mathcal{F}_{r}$ in $B_{1}$. For
every vector field $X\in C_{0}^{1}(B_{1},\mathbb{R}^{n})$ and for some
constant $\overline{C}=\overline{C}(N,L_{D},\sup|X|,\sup|\nabla X|)>0$ it
holds

$\begin{split}\int_{\partial
E}\textnormal{div}_{\tau}X\,d\mathcal{H}^{n-1}\leq\overline{C}\int_{B_{1}}\big{(}|\nabla
u|^{2}+r\big{)}\,dx+\Lambda r\int_{\partial
E}|X|\,d\mathcal{H}^{n-1},\end{split}$ (76)

where $L_{D}$ is the greatest Lipschitz constant of the data
$a_{ij},b_{ij},a_{i},b_{i},a,b$, and $\textnormal{div}_{\tau}$ denotes the
tangential divergence on $\partial E$.

###### Proof.

Let us fix $X\in C_{0}^{1}(B_{1},\mathbb{R}^{n})$. We set
$\Phi_{t}(x)=x+tX(x)$, for any $t>0$, $E_{t}=\Phi_{t}(E)$ and
$u_{t}=u\circ\Phi_{t}^{-1}$. From the $\Lambda r$-minimality it follows

$\begin{split}&[P(E_{t},B_{1})-P(E,B_{1})]+\Lambda r|E_{t}\Delta E|\\\
&+\int_{B_{1}}[F_{r}(y,u_{t},\nabla
u_{t})+\mathbbm{1}_{E_{t}}(y)G_{r}(y,u_{t},\nabla u_{t})]\,dy\\\
&-\int_{B_{1}}[F_{r}(x,u,\nabla u)+\mathbbm{1}_{E}(x)G_{r}(x,u,\nabla
u)]\,dx\geq 0.\end{split}$ (77)

In order to obtain (76) we will divide by $t$ and pass to the upper limit as
$t\rightarrow 0^{+}$. Let us study these terms separately. The first variation
of the area gives

$\lim_{t\rightarrow
0^{+}}\frac{1}{t}[P(E_{t},B_{1})-P(E,B_{1})]=\int_{\partial
E}\textnormal{div}_{\tau}X\,d\mathcal{H}^{n-1}.$ (78)

We can deal with the second term observing that

$\lim_{t\rightarrow 0^{+}}\frac{\lvert E_{t}\Delta
E\rvert}{t}\leq\int_{\partial E}\lvert
X\cdot\nu_{E}\rvert\,d\mathcal{H}^{n-1},$ (79)

(see for instance [14, Theorem 3.2]). In the first bulk term we make the
change of variables $y=\Phi_{t}(x)$ with $x\in B_{1}$ and $t>0$, taking into
account that

$\nabla\Phi_{t}^{-1}(\Phi_{t}(x))=I-t\nabla
X(x)+o(t),\quad\textnormal{J}\Phi_{t}(x)=1+t\textnormal{div}X(x)+o(t).$

Thus we gain

$\begin{split}&\int_{B_{1}}\big{[}F_{r}(y,u_{t},\nabla
u_{t})+\mathbbm{1}_{E_{t}}(y)G_{r}(y,u_{t},\nabla u_{t})\big{]}\,dy\\\
&=\int_{B_{1}}\big{[}F_{r}(\Phi_{t}(x),u,\nabla
u)+\mathbbm{1}_{E}(x)G_{r}(\Phi_{t}(x),u,\nabla
u)\big{]}(1+t\textnormal{div}X)\,dx\\\
&-t\int_{B_{1}}\big{[}2\big{\langle}C_{1}\nabla u\nabla X,\nabla
u\big{\rangle}+\sqrt{r}\big{\langle}C_{2},\nabla u\nabla
X\big{\rangle}\big{]}\,dx+o(t),\end{split}$

where we set

$C_{i}:=\tilde{A}_{ir}+\mathbbm{1}_{E}\tilde{B}_{ir}=A_{ir}(\Phi_{t}(x),u)+\mathbbm{1}_{E}(x)B_{ir}(\Phi_{t}(x),u),$

for $i=1,2,3$. By simple calculations we obtain

$\begin{split}&\int_{B_{1}}\big{[}F_{r}(y,u_{t},\nabla
u_{t})+\mathbbm{1}_{E_{t}}(y)G_{r}(y,u_{t},\nabla u_{t})\big{]}\,dy\\\
&-\int_{B_{1}}\big{[}F_{r}(x,u,\nabla u)+\mathbbm{1}_{E}(x)G_{r}(x,u,\nabla
u)\big{]}\,dx\\\ &=\int_{B_{1}}\big{\\{}F_{r}(\Phi_{t}(x),u,\nabla
u)+\mathbbm{1}_{E}(x)G_{r}(\Phi_{t}(x),u,\nabla u)-[F_{r}(x,u,\nabla
u)+\mathbbm{1}_{E}(x)G_{r}(x,u,\nabla u)]\big{\\}}\,dx\\\
&+t\bigg{[}\int_{B_{1}}\big{[}F_{r}(\Phi_{t}(x),u,\nabla
u)+\mathbbm{1}_{E}(x)G_{r}(\Phi_{t}(x),u,\nabla
u)\big{]}\textnormal{div}X\,dx\\\
&-\int_{B_{1}}\big{[}2\big{\langle}C_{1}\nabla u\nabla X,\nabla
u\big{\rangle}+\sqrt{r}\big{\langle}C_{2},\nabla u\nabla
X\big{\rangle}\big{]}\,dx\bigg{]}+o(t).\end{split}$ (80)

Let us estimate the first of the three terms. By Lipschitz continuity and
Young’s inequality we get

$\begin{split}&\int_{B_{1}}\Big{\\{}\big{\langle}F_{r}(\Phi_{t}(x),u,\nabla
u)+\mathbbm{1}_{E}(x)G_{r}(\Phi_{t}(x),u,\nabla u)-[F_{r}(x,u,\nabla
u)+\mathbbm{1}_{E}(x)G_{r}(x,u,\nabla u)]\Big{\\}}\,dx\\\ &\leq
c(L_{D})t\int_{B_{1}}|X|[|\nabla u|^{2}+\sqrt{r}|\nabla u|+r]\,dx\leq
c(L_{D})t\int_{B_{1}}|X|[|\nabla u|^{2}+r]\,dx.\end{split}$

Finally, dividing by $t$ and passing to the upper limit as $t\rightarrow
0^{+}$ we infer

$\begin{split}&\limsup_{t\rightarrow
0^{+}}\frac{1}{t}\bigg{[}\int_{B_{1}}[F_{r}(y,u_{t},\nabla
u_{t})+\mathbbm{1}_{E_{t}}(y)G_{r}(y,u_{t},\nabla u_{t})]\,dy\\\
&-\int_{B_{1}}[F_{r}(x,u,\nabla u)+\mathbbm{1}_{E}G_{r}(x,u,\nabla
u)]\,dx\bigg{]}\\\ &\leq c(L_{D})\int_{B_{1}}|X|[|\nabla
u|^{2}+r]\,dx+\int_{B_{1}}[F_{r}(x,u,\nabla u)+\mathbbm{1}_{E}G_{r}(x,u,\nabla
u)]\textnormal{div}X\,dx\\\
&-\int_{B_{1}}\big{[}2\big{\langle}(A_{1r}+\mathbbm{1}_{E}B_{1r})\nabla
u\nabla X,\nabla
u\big{\rangle}+\sqrt{r}\big{\langle}(A_{2r}+\mathbbm{1}_{E}B_{2r}),\nabla
u\nabla X\big{\rangle}\big{]}\,dx.\end{split}$ (81)

Passing to the upper limit as $t\rightarrow 0^{+}$ and putting (78), (79),
(81) together we get

$\begin{split}&\int_{\partial
E}\textnormal{div}_{\tau}X\,d\mathcal{H}^{n-1}\\\ &\leq
c(L_{D})\int_{B_{1}}|X|[|\nabla
u|^{2}+r]\,dx+\bigg{|}\int_{B_{1}}[F_{r}(x,u,\nabla
u)+\mathbbm{1}_{E}G_{r}(x,u,\nabla u)]\textnormal{div}X\,dx\bigg{|}\\\
&+\int_{B_{1}}\big{|}2\big{\langle}(A_{1r}+\mathbbm{1}_{E}B_{1r})\nabla
u\nabla X,\nabla
u\big{\rangle}+\sqrt{r}\big{\langle}(A_{2r}+\mathbbm{1}_{E}B_{2r}),\nabla
u\nabla X\big{\rangle}\big{|}\,dx+\Lambda r\int_{\partial E}\lvert
X\rvert\,d\mathcal{H}^{n-1}\\\ &\leq\overline{C}\int_{B_{1}}\big{(}|\nabla
u|^{2}+r\big{)}\,dx+\Lambda r\int_{\partial
E}|X|\,d\mathcal{H}^{n-1},\end{split}$

where $\overline{C}=\overline{C}(N,L_{D},\sup|X|,\sup|\nabla X|)$.

∎

##  7\. Excess improvement

######  Theorem 7.1 (Excess improvement).

For every $\tau\in\big{(}0,\frac{1}{2}\big{)}$ and $M>0$ there exists a
constant $\varepsilon_{5}=\varepsilon_{5}(\tau,M)\in(0,1)$ such that if
$(u,E)$ is a $\Lambda$-minimizer of $\mathcal{F}$ in $B_{r}(x_{0})$ with
$x_{0}\in\partial E$ and

${\mathbf{e}}(x_{0},r)\leq\varepsilon_{5}\quad\text{and}\quad\mathcal{D}(x_{0},r)+r\leq
M{\mathbf{e}}(x_{0},r)$ (82)

then there exists a positive constant $C_{5}$, depending on $\left\lVert\nabla
u\right\rVert_{L^{2}(B_{r}(x_{0}))}$, such that

${\mathbf{e}}(x_{0},\tau r)\leq
C_{5}(\tau^{2}{\mathbf{e}}(x_{0},r)+\mathcal{D}(x_{0},2\tau r)+\tau r).$

###### Proof.

Without loss of generality we may assume that $\tau<\frac{1}{8}$. Let us
rescale and assume by contradiction that there exists an infinitesimal
sequence $\\{\varepsilon_{h}\\}_{h\in\mathbb{N}}\subseteq\mathbb{R}^{+}$, a
sequence $\\{r_{h}\\}_{h\in\mathbb{N}}\subseteq\mathbb{R}^{+}$ and a sequence
$\\{(u_{h},E_{h})\\}_{h\in\mathbb{N}}$ of $\Lambda r_{h}$-minimizers of
$\mathcal{F}_{r_{h}}$ in $B_{1}$, with equibounded energies, such that,
denoting by ${\mathbf{e}}_{h}$ the excess of $E_{h}$ and by $\mathcal{D}_{h}$
the rescaled Dirichlet integral of $u_{h}$, we have

${\mathbf{e}}_{h}(0,1)=\varepsilon_{h},\quad\mathcal{D}_{h}(0,1)+r_{h}\leq
M\varepsilon_{h}$

and

${\mathbf{e}}_{h}(0,\tau)>C_{5}(\tau^{2}{\mathbf{e}}(0,1)+\mathcal{D}(0,2\tau)+\tau
r_{h}),$

with some positive constant $C_{5}$ to be chosen. Up to rotating each $E_{h}$
we may also assume that for all $h\in\mathbb{N}$

${\mathbf{e}}_{h}(0,1)=\frac{1}{2}\int_{\partial E_{h}\cap
B_{1}}\lvert\nu_{E_{h}}-e_{n}\rvert^{2}\,d\mathcal{H}^{n-1}.$

Step 1. Thanks to the Lipschitz approximation theorem, for $h$ sufficiently
large, there exists a 1-Lipschitz function
$f_{h}\colon\mathbb{R}^{n-1}\rightarrow\mathbb{R}$ such that

$\sup_{\mathbb{R}^{n-1}}\lvert f_{h}\rvert\leq
C_{3}\varepsilon_{h}^{\frac{1}{2(n-1)}},\quad\mathcal{H}^{n-1}((\partial
E_{h}\Delta\Gamma_{f_{h}})\cap B_{\frac{1}{2}})\leq
C_{3}\varepsilon_{h},\quad\int_{B_{\frac{1}{2}}^{n-1}}\lvert\nabla^{\prime}f_{h}\rvert^{2}\,dx^{\prime}\leq
C_{3}\varepsilon_{h}.$ (83)

We define

$g_{h}(x^{\prime}):=\frac{f_{h}(x^{\prime})-a_{h}}{\sqrt{\varepsilon_{h}}},\quad\text{where}\quad
a_{h}=\int_{B_{\frac{1}{2}}^{n-1}}f_{h}\,dx^{\prime}$

and we assume up to a subsequence that $\\{g_{h}\\}_{h\in\mathbb{N}}$
converges weakly in $H^{1}(B_{\frac{1}{2}}^{n-1})$ and strongly in
$L^{2}(B_{\frac{1}{2}}^{n-1})$ to a function $g$.
We prove that $g$ is harmonic in $B_{\frac{1}{2}}^{n-1}$. It’s enough show
that

$\lim_{h\rightarrow+\infty}\frac{1}{\sqrt{\varepsilon_{h}}}\int_{B_{\frac{1}{2}}^{n-1}}\frac{\langle\nabla^{\prime}f_{h},\nabla^{\prime}\phi\rangle}{\sqrt{1+\lvert\nabla^{\prime}f_{h}\rvert^{2}}}\,dx^{\prime}=0,$
(84)

for all $\phi\in C_{0}^{1}(B_{\frac{1}{2}}^{n-1})$; indeed, if $\phi\in
C_{0}^{1}(B_{\frac{1}{2}}^{n-1})$, by weak convergence we have

$\begin{split}&\int_{B_{\frac{1}{2}}^{n-1}}\langle\nabla^{\prime}g,\nabla^{\prime}\phi\rangle\,dx^{\prime}=\lim_{h\rightarrow+\infty}\frac{1}{\sqrt{\varepsilon_{h}}}\int_{B_{\frac{1}{2}}^{n-1}}\langle\nabla^{\prime}f_{h},\nabla^{\prime}\phi\rangle\,dx^{\prime}\\\
&=\lim_{h\rightarrow+\infty}\frac{1}{\sqrt{\varepsilon_{h}}}\bigg{\\{}\int_{B_{\frac{1}{2}}^{n-1}}\frac{\langle\nabla^{\prime}f_{h},\nabla^{\prime}\phi\rangle}{\sqrt{1+\lvert\nabla^{\prime}f_{h}\rvert^{2}}}\,dx^{\prime}+\int_{B_{\frac{1}{2}}^{n-1}}\bigg{[}\langle\nabla^{\prime}f_{h},\nabla^{\prime}\phi\rangle-\frac{\langle\nabla^{\prime}f_{h},\nabla^{\prime}\phi\rangle}{\sqrt{1+\lvert\nabla^{\prime}f_{h}\rvert^{2}}}\bigg{]}\,dx^{\prime}\bigg{\\}}.\end{split}$

Using the lipschitz-continuity of $f_{h}$ and the third equation in (83) we
infer that the second term in the previous equality is infinitesimal:

$\begin{split}&\limsup_{h\rightarrow+\infty}\frac{1}{\sqrt{\varepsilon_{h}}}\bigg{|}\int_{B_{\frac{1}{2}}^{n-1}}\bigg{[}\langle\nabla^{\prime}f_{h},\nabla^{\prime}\phi\rangle-\frac{\langle\nabla^{\prime}f_{h},\nabla^{\prime}\phi\rangle}{\sqrt{1+\lvert\nabla^{\prime}f_{h}\rvert^{2}}}\bigg{]}\,dx^{\prime}\bigg{|}\\\
&\leq\limsup_{h\rightarrow+\infty}\frac{1}{\sqrt{\varepsilon_{h}}}\int_{B_{\frac{1}{2}}^{n-1}}\lvert\nabla^{\prime}f_{h}\rvert\lvert\nabla^{\prime}\phi\rvert\frac{\sqrt{1+\lvert\nabla^{\prime}f_{h}\rvert^{2}}-1}{\sqrt{1+\lvert\nabla^{\prime}f_{h}\rvert^{2}}}\,dx^{\prime}\\\
&\leq\limsup_{h\rightarrow+\infty}\frac{1}{2\sqrt{\varepsilon_{h}}}\int_{B_{\frac{1}{2}}^{n-1}}\lvert\nabla^{\prime}\phi\rvert\lvert\nabla^{\prime}f_{h}\rvert^{2}\,dx^{\prime}\leq\lim_{h\rightarrow+\infty}\frac{C_{3}\left\lVert\nabla^{\prime}\phi\right\rVert_{\infty}\sqrt{\varepsilon_{h}}}{2}=0.\end{split}$

Therefore, we can prove (84). We fix $\delta>0$ so that
spt$\phi\times[-2\delta,2\delta]\subseteq B_{\frac{1}{2}}$, choose a cut-off
function $\psi\colon\mathbb{R}\rightarrow[0,1]$ with
spt$\psi\subseteq(-2\delta,2\delta)$, $\psi=1$ in $(-\delta,\delta)$ and apply
to $E_{h}$ the weak Euler-Lagrange equation with $X=\phi\psi e_{n}$. By the
height bound, for $h$ sufficiently large it holds that $\partial E_{h}\cap
B_{\frac{1}{2}}\subseteq B_{\frac{1}{2}}^{n-1}\times(-\delta,\delta)$.
Plugging $X$ in the weak Euler-Lagrange equation and using the assumption in
(82), we have

$\begin{split}&-\gamma\int_{\partial E_{h}\cap
B_{\frac{1}{2}}}\langle\nu_{E_{h}},e_{n}\rangle\langle\nabla^{\prime}\phi,\nu_{E_{h}}^{\prime}\rangle\,d\mathcal{H}^{n-1}\\\
&\leq c(N,L_{D},\phi,\psi)\int_{B_{\frac{1}{2}}}\big{(}|\nabla
u_{h}|^{2}+r_{h}\big{)}\,dx+\Lambda r_{h}\int_{\partial E_{h}\cap
B_{\frac{1}{2}}}\lvert\phi\psi\rvert\,d\mathcal{H}^{n-1}\\\ &\leq
c(n,N,\Lambda,L_{D},\phi,\psi)\varepsilon_{h}.\end{split}$

Therefore, if we replace $\phi$ by $-\phi$, we infer

$\lim_{h\rightarrow+\infty}\frac{1}{\sqrt{\varepsilon_{h}}}\int_{\partial
E_{h}\cap
B_{\frac{1}{2}}}\langle\nu_{E_{h}},e_{n}\rangle\langle\nabla^{\prime}\phi,\nu_{E_{h}}^{\prime}\rangle\,d\mathcal{H}^{n-1}=0.$
(85)

Decomposing $\partial E_{h}\cap
B_{\frac{1}{2}}=\big{(}[\Gamma_{f_{h}}\cup(\partial
E_{h}\setminus\Gamma_{f_{h}})]\setminus(\Gamma_{f_{h}}\setminus\partial
E_{h})\big{)}\cap B_{\frac{1}{2}}$, we deduce

$\begin{split}&-\frac{1}{\sqrt{\varepsilon_{h}}}\int_{\partial E_{h}\cap
B_{\frac{1}{2}}}\langle\nu_{E_{h}},e_{n}\rangle\langle\nabla^{\prime}\phi,\nu_{E_{h}}^{\prime}\rangle\,d\mathcal{H}^{n-1}=\frac{1}{\sqrt{\varepsilon_{h}}}\bigg{[}-\int_{\Gamma_{f_{h}}\cap
B_{\frac{1}{2}}}\langle\nu_{E_{h}},e_{n}\rangle\langle\nabla^{\prime}\phi,\nu_{E_{h}}^{\prime}\rangle\,d\mathcal{H}^{n-1}\\\
&-\int_{(\partial E_{h}\setminus\Gamma_{f_{h}})\cap
B_{\frac{1}{2}}}\langle\nu_{E_{h}},e_{n}\rangle\langle\nabla^{\prime}\phi,\nu_{E_{h}}^{\prime}\rangle\,d\mathcal{H}^{n-1}+\int_{(\Gamma_{f_{h}}\setminus\partial
E_{h})\cap
B_{\frac{1}{2}}}\langle\nu_{E_{h}},e_{n}\rangle\langle\nabla^{\prime}\phi,\nu_{E_{h}}^{\prime}\rangle\,d\mathcal{H}^{n-1}\bigg{]}.\end{split}$
(86)

Since by the second inequality in (83) we have

$\bigg{|}\frac{1}{\sqrt{\varepsilon_{h}}}\int_{(\partial
E_{h}\setminus\Gamma_{f_{h}})\cap
B_{\frac{1}{2}}}\langle\nu_{E_{h}},e_{n}\rangle\langle\nabla^{\prime}\phi,\nu_{E_{h}}^{\prime}\rangle\,d\mathcal{H}^{n-1}\bigg{|}\leq
C_{3}\sqrt{\varepsilon_{h}}\sup_{\mathbb{R}^{n-1}}\lvert\nabla^{\prime}\phi\rvert,$
$\bigg{|}\frac{1}{\sqrt{\varepsilon_{h}}}\int_{(\Gamma_{f_{h}}\setminus\partial
E_{h})\cap
B_{\frac{1}{2}}}\langle\nu_{E_{h}},e_{n}\rangle\langle\nabla^{\prime}\phi,\nu_{E_{h}}^{\prime}\rangle\,d\mathcal{H}^{n-1}\bigg{|}\leq
C_{3}\sqrt{\varepsilon_{h}}\sup_{\mathbb{R}^{n-1}}\lvert\nabla^{\prime}\phi\rvert,$

then by (85) and the area formula, we infer

$0=\lim_{h\rightarrow+\infty}\frac{-1}{\sqrt{\varepsilon_{h}}}\int_{\Gamma_{f_{h}}\cap
B_{\frac{1}{2}}}\langle\nu_{E_{h}},e_{n}\rangle\langle\nabla^{\prime}\phi,\nu_{E_{h}}^{\prime}\rangle\,d\mathcal{H}^{n-1}=\lim_{h\rightarrow+\infty}\frac{1}{\sqrt{\varepsilon_{h}}}\int_{B_{\frac{1}{2}}^{n-1}}\frac{\langle\nabla^{\prime}f_{h},\nabla^{\prime}\phi\rangle}{\sqrt{1+\lvert\nabla^{\prime}f_{h}\rvert^{2}}}\,dx^{\prime}.$

This proves that $g$ is harmonic.
Step 2. The proof of this step now follows exactly as in [FJ] using the eight
bound lemma and the revers Poincaré inequality. We give it here to be
thorough. By the mean value property of harmonic functions, Lemma 25.1 in
[20], Jensen inequality, semicontinuity and the third inequality in (83) we
deduce that

$\begin{split}&\lim_{h\rightarrow+\infty}\frac{1}{\varepsilon_{h}}\int_{B_{2\tau}^{n-1}}\lvert
f_{h}(x^{\prime})-(f_{h})_{2\tau}-\langle(\nabla^{\prime}f_{h})_{2\tau},x^{\prime}\rangle\rvert^{2}\,dx^{\prime}\\\
&=\int_{B_{2\tau}^{n-1}}\lvert
g(x^{\prime})-(g)_{2\tau}-\langle(\nabla^{\prime}g)_{2\tau},x^{\prime}\rangle\rvert^{2}\,dx^{\prime}\\\
&=\int_{B_{2\tau}^{n-1}}\lvert
g(x^{\prime})-g(0)-\langle\nabla^{\prime}g(0),x^{\prime}\rangle\rvert^{2}\,dx^{\prime}\\\
&\leq c(n)\tau^{n-1}\sup_{x^{\prime}\in B_{2\tau}^{n-1}}\lvert
g(x^{\prime})-g(0)-\langle\nabla^{\prime}g(0),x^{\prime}\rangle\rvert^{2}\\\
&\leq
c(n)\tau^{n+3}\int_{B_{\frac{1}{2}}^{n-1}}\lvert\nabla^{\prime}g\rvert^{2}\,dx^{\prime}\leq
c(n)\tau^{n+3}\liminf_{h\rightarrow+\infty}\int_{B_{\frac{1}{2}}^{n-1}}\lvert\nabla^{\prime}g_{h}\rvert^{2}\,dx^{\prime}\\\
&\leq\tilde{C}(n,C_{3})\tau^{n+3}.\end{split}$

On one hand, using the area formula, the mean value property, the previous
inequality and setting

$c_{h}:=\frac{(f_{h})_{2\tau}}{\sqrt{1+\lvert(\nabla^{\prime}f_{h})_{2\tau}\rvert^{2}}},\quad\nu_{h}:=\frac{(-(\nabla^{\prime}f_{h})_{2\tau},1)}{\sqrt{1+\lvert(\nabla^{\prime}f_{h})_{2\tau}\rvert^{2}}},$

we have

$\begin{split}&\limsup_{h\rightarrow+\infty}\frac{1}{\varepsilon_{h}}\int_{\partial
E_{h}\cap\Gamma_{f_{h}}\cap B_{2\tau}}\lvert\langle\nu_{h},x\rangle-
c_{h}\rvert^{2}\,d\mathcal{H}^{n-1}\\\
&=\limsup_{h\rightarrow+\infty}\frac{1}{\varepsilon_{h}}\int_{\partial
E_{h}\cap\Gamma_{f_{h}}\cap
B_{2\tau}}\frac{\lvert\langle-(\nabla^{\prime}f_{h})_{2\tau},x^{\prime}\rangle+f_{h}(x^{\prime})-(f_{h})_{2\tau}\rvert}{1+\lvert(\nabla^{\prime}f_{h})_{2\tau}\rvert^{2}}^{2}\sqrt{1+\lvert\nabla^{\prime}f_{h}(x^{\prime})\rvert^{2}}\,dx^{\prime}\\\
&\leq\lim_{h\rightarrow+\infty}\frac{1}{\varepsilon_{h}}\int_{B_{2\tau}^{n-1}}\lvert
f_{h}(x^{\prime})-(f_{h})_{2\tau}-\langle(\nabla^{\prime}f_{h})_{2\tau},x^{\prime}\rangle\rvert^{2}\,dx^{\prime}\leq\tilde{C}(n,C_{3})\tau^{n+3}.\end{split}$

On the other hand, arguing as in Step 1, we immediately get from the height
bound and the first two inequalities in (83) that

$\lim_{h\rightarrow+\infty}\frac{1}{\varepsilon_{h}}\int_{(\partial
E_{h}\setminus\Gamma_{f_{h}})\cap B_{2\tau}}\lvert\langle\nu_{h},x\rangle-
c_{h}\rvert^{2}\,d\mathcal{H}^{n-1}=0.$

Hence we conclude that

$\limsup_{h\rightarrow+\infty}\frac{1}{\varepsilon_{h}}\int_{\partial
E_{h}\cap B_{2\tau}}\lvert\langle\nu_{h},x\rangle-
c_{h}\rvert^{2}\,d\mathcal{H}^{n-1}\leq\tilde{C}(n,C_{3})\tau^{n+3}.$ (87)

We claim that the sequence
$\\{{\mathbf{e}}_{h}(0,2\tau,\nu_{h})\\}_{h\in\mathbb{N}}$ is infinitesimal;
indeed, by the definition of excess, Jensen’2 inequality and the third
inequality in (83) we have

$\begin{split}&\limsup_{h\rightarrow+\infty}\int_{\partial E_{h}\cap
B_{2\tau}}\lvert\nu_{E_{h}}-\nu_{h}\rvert^{2}\,d\mathcal{H}^{n-1}\\\
&\leq\limsup_{h\rightarrow+\infty}\bigg{[}2\int_{\partial E_{h}\cap
B_{2\tau}}\lvert\nu_{E_{h}}-e_{n}\rvert^{2}\,d\mathcal{H}^{n-1}+2\lvert
e_{n}-\nu_{h}\rvert^{2}\mathcal{H}^{n-1}(\partial E_{h}\cap
B_{2\tau})\bigg{]}\\\
&\leq\limsup_{h\rightarrow+\infty}\bigg{[}4\varepsilon_{h}+2\mathcal{H}^{n-1}(B_{2\tau})\frac{\lvert((\nabla^{\prime}f_{h})_{2\tau},\sqrt{1+\lvert(\nabla^{\prime}f_{h})_{2\tau}\rvert^{2}}-1)\rvert^{2}}{1+\lvert(\nabla^{\prime}f_{h})_{2\tau}\rvert^{2}}\bigg{]}\\\
&\leq\limsup_{h\rightarrow+\infty}\big{[}4\varepsilon_{h}+4\mathcal{H}^{n-1}(B_{2\tau})\lvert(\nabla^{\prime}f_{h})_{2\tau}\rvert^{2}\big{]}\\\
&\leq\limsup_{h\rightarrow+\infty}\bigg{[}4\varepsilon_{h}+4\int_{B_{\frac{1}{2}}^{n-1}}\lvert\nabla^{\prime}f_{h}\rvert^{2}\,dx^{\prime}\bigg{]}\leq\lim_{h\rightarrow+\infty}[4\varepsilon_{h}+4C_{3}\varepsilon_{h}]=0.\end{split}$

Therefore applying the reverse Poicaré’s inequality and (87) we have for $h$
large that

$\begin{split}{\mathbf{e}}_{h}(0,\tau)\leq{\mathbf{e}}_{h}(0,\tau,\nu_{h})\leq
C_{4}(\tilde{C}\tau^{2}{\mathbf{e}}_{h}(0,1)+\mathcal{D}(0,2\tau)+2\tau
r_{h}),\end{split}$

which is a contradiction if we choose $C_{5}>C_{4}\max\\{\tilde{C},2\\}$. ∎

##  8\. Proof of the main theorem

The proof works exactly as in [11]. We give here some details to emphasize the
dependence of $\varepsilon$ appearing in the statement of Theorem 1.3 from the
structural data of the functional. The proof is divided in four steps.
Step 1. For every $\tau\in(0,1)$ there exists
$\varepsilon_{6}=\varepsilon_{6}(\tau)>0$ such that if
${\mathbf{e}}(x,r)\leq\varepsilon_{6}$ then

$\mathcal{D}(x,\tau r)\leq C_{0}\tau\mathcal{D}(x,r),$

where $C_{0}$ is from Lemma 3.10. Assume by contradiction that for some
$\tau\in(0,1)$ there exist two positive sequences $\varepsilon_{h}$ and
$r_{h}$ and a sequence $(u_{h},E_{h})$ of $\Lambda r_{h}-$ minimizers of
$\mathcal{F}_{r_{h}}$ in $B_{1}$ with equibounded energies, such that,
denoting by ${\mathbf{e}}_{h}$ the excess of $E_{h}$ and by $\mathcal{D}_{h}$
the rescaled Dirichlet integral of $u_{E_{h}}$, we have that $0\in\partial
E_{h}$,

${\mathbf{e}}_{h}(0,1)=\varepsilon_{h}\rightarrow
0\quad\text{and}\quad\mathcal{D}_{h}(0,\tau)>C_{0}\tau\mathcal{D}_{h}(0,1).$
(88)

Thanks to the energy upper bound (Theorem 3.6) and the compactness lemma
(Lemma 5.1) we may assume that $E_{h}\rightarrow E$ in $L^{1}(B_{1})$ and
$0\in\partial E$. Since, by lower semicontinuity, the excess of $E$ at 0 is
null, $E$ is a half space in $B_{1}$, say $H$. In particular, for $h$ large it
holds

$|(E_{h}\Delta H)\cap B_{1}|<\varepsilon_{0}(\tau)|B_{1}|,$

where $\varepsilon_{0}$ is from Lemma 3.10, which gives a contradiction with
the inequality (88).
Step 2. Let $U\subset\subset\Omega$ be an open set. Prove that for every
$\tau\in(0,1)$ there exist two positive constants
$\varepsilon=\varepsilon(\tau,U)$ and $C_{6}$ such that if $x_{0}\in\partial
E$, $B_{r}(x_{0})\subset U$ and
${\mathbf{e}}(x_{0},r)+\mathcal{D}(x_{0},r)+r<\varepsilon$ then

${\mathbf{e}}(x_{0},\tau r)+\mathcal{D}(x_{0},\tau r)+\tau r\leq
C_{6}\tau({\mathbf{e}}(x_{0},r)+\mathcal{D}(x_{0},r)+r).$ (89)

Fix $\tau\in(0,1)$ and assume without loss of generality that
$\tau<\frac{1}{2}$ We can distinguish two cases.
Case 1: $\mathcal{D}(x_{0},r)+r\leq\tau^{-n}{\mathbf{e}}(x_{0},r)$. If
${\mathbf{e}}(x_{0},r)<\min\\{\varepsilon_{5}(\tau,\tau^{-n}),\varepsilon_{6}(2\tau)\\}$
it follows from Theorem 7.1 and Step 1 that

$\begin{split}{\mathbf{e}}(x_{0},\tau r)\leq
C_{5}(\tau^{2}{\mathbf{e}}(x_{0},r)+\mathcal{D}(x_{0},2\tau r)+\tau r)\leq
C_{5}(\tau^{2}{\mathbf{e}}(x_{0},r)+2C_{0}\mathcal{D}(x_{0},\tau r)+\tau
r)\end{split}.$

Case 2: ${\mathbf{e}}(x_{0},r)\leq\tau^{n}(\mathcal{D}(x_{0},r)+r)$. By the
property of the excess at different scales, we infer

${\mathbf{e}}(x_{0},\tau
r)\leq\tau^{1-n}{\mathbf{e}}(x_{0},r)\leq\tau^{n}(\mathcal{D}(x_{0},r)+r).$

We conclude that choosing
$\varepsilon=\min\\{\varepsilon_{5}(\tau,\tau^{-n}),\varepsilon_{6}(2\tau),\varepsilon_{6}(\tau)\\}$,
inequality (89) is verified.
Step 3. Fix $\sigma\in(0,\frac{1}{2})$ and choose $\tau_{0}\in(0,1)$ such that
$C_{6}\tau_{0}\leq\tau_{0}^{2\sigma}$. Let $U\subset\subset\Omega$ be an open
set. We define

$\begin{split}\Gamma\cap U:=&\\{x\in\partial E\cap
U\,:\,{\mathbf{e}}(x,r)+\mathcal{D}(x,r)+r<\varepsilon(\tau_{0},U)\\\ &\text{
for some }r>0\text{ such that }B_{r}(x_{0})\subset U\\}.\end{split}$

Note that $\Gamma\cap U$ is relatively open in $\partial E$. We show that
$\Gamma\cap U$ is a $C^{1,\sigma}$-hypersurface. Indeed inequality (89)
implies via standard iteration argument that if $x_{0}\in\Gamma\cap U$ there
exist $r_{0}>0$ and a neighborhood $V$ of $x_{0}$ such that for every
$x\in\partial E\cap V$ it holds

${\mathbf{e}}(x,\tau_{0}^{k}r_{0})+\mathcal{D}(x,\tau_{0}^{k}r_{0})+\tau_{0}^{k}r_{0}\leq\tau_{0}^{2\sigma
k},\quad\text{for }k\in\mathbb{N}_{0}.$

In particular ${\mathbf{e}}(x,\tau_{0}^{k}r_{0})\leq\tau_{0}^{2\sigma k}$ and,
arguing as in [11], we obtain that for every $x\in\partial E\cap V$ and
$0<s<t<r_{0}$ it holds

$|(\nu_{E})_{s}(x)-(\nu_{E})_{t}(x)|\leq ct^{\sigma},$

for some constant $c=c(n,\tau_{0},r_{0})$, where

$(\nu_{E})_{t}(x)=\fint_{\partial E\cap
B_{t}(x)}\nu_{E}(y)\,d\mathcal{H}^{n-1}.$

The previous estimate first implies that $\Gamma\cap U$ is $C^{1}$. By a
standard argument we then deduce again from the same estimate that $\Gamma\cap
U$ is a $C^{1,\sigma}$-hypersurface. Finally we define
$\Gamma:=\cup_{i}(\Gamma\cap U_{i})$, where $(U_{i})_{i}$ is an increasing
sequence of open sets such that $U_{i}\subset\subset\Omega$ and
$\Omega=\cup_{i}U_{i}$.
Step 4. Finally we prove that there exists $\eta>0$ such that

$\mathcal{H}^{n-1-\eta}(\partial E\setminus\Gamma)=0.$

Setting $\Sigma=\\{x\in\partial E\setminus\Gamma\,:\,\lim\limits_{r\rightarrow
0}\mathcal{D}(x,r)=0\\}$, by Lemma 3.2, $\nabla u\in L^{2s}_{loc}(\Omega)$ for
some $s>1$, depending only on $\nu,N,L,n$, and we have that

$\text{dim}_{\mathcal{H}}\big{(}\\{x\in\Omega\,:\,\limsup_{r\rightarrow
0}\mathcal{D}(x,r)>0\\}\big{)}\leq n-s.$

The conclusion follows as in [11] showing that $\Sigma=\emptyset$ when $n\leq
7$ and $\text{dim}_{\mathcal{H}}(\Sigma)\leq n-8$ for $n\geq 8$.

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