1 Introduction

Let \(\Omega \subseteq {\mathbb {R}}^n\) be a bounded open set with smooth boundary, and let A be a set containing \(\Omega \). Consider the functional

$$\begin{aligned} F(A,v)=\int _{A}|\nabla v|^2\,\hbox {d}{\mathcal {L}}^n+\beta \int _{\partial A}v^2\,\hbox {d}{\mathcal {H}}^{n-1}+C_0{\mathcal {L}}^n(A), \end{aligned}$$
(1.1)

with \(v\in H^1(A)\), \(v=1\) in \(\Omega \) and \(\beta , C_0>0\) fixed positive constants. The problem of minimizing this functional arises in the environment of thermal insulation: F represents the energy of a heat configuration v when the temperature is maintained constant inside the body \(\Omega \) and there’s a bulk layer \(A\setminus \Omega \) of insulating material whose cost is represented by \(C_0\) and the heat transfer with the external environment is conveyed by convection. For simplicity’s sake in the following we will set \(C_0=1\). The variational formulation in (1.1) leads to an Euler-Lagrange equation, which is the weak form of the following problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta u=0 &{} \text {in } A\setminus \Omega , \\ \dfrac{\partial u}{\partial \nu } +\beta u=0 &{} \text {on } \partial A,\\ u= 1 &{} \text {in }\Omega , \end{array}\right. } \end{aligned}$$
(1.2)

The problems we are interested in concern the existence of a solution and its regularity. In this sense, one could be interested in studying a more general setting in which it is possible to consider possibly irregular sets A. Specifically, we could generalize the problem into the context of \({{\,\textrm{SBV}\,}}\) functions, aiming to minimize the functional

$$\begin{aligned} F(v)=\int _{{\mathbb {R}}^n}|\nabla v|^2\, \hbox {d}{\mathcal {L}}^n+\beta \int _{J_v}\left( {\underline{v}}^2 +{\overline{v}}^2\right) \,\hbox {d}{\mathcal {H}}^{n-1}+{\mathcal {L}}^n(\{v>0\}\setminus \Omega ) \end{aligned}$$

with \(v\in {{\,\textrm{SBV}\,}}({\mathbb {R}}^n)\) and \(v=1\) in \(\Omega \). This problem has been studied in [7], where the authors have proved the existence of a solution u for the problem and the regularity of its jump set. Similar two-phase problems in the linear case can be found in [1], and [3]. With regards to the nonlinear context, analogous versions of the problem have been addressed in [4], and in [6] with a boundedness constraint.

In this paper, our main aim is to generalize the problem and techniques employed in [7] to a nonlinear formulation. In detail, for \(p,q>1\) fixed, we consider the functional

$$\begin{aligned} {\mathcal {F}}(v)=\int _{{\mathbb {R}}^n}|\nabla v|^p\,\hbox {d}{\mathcal {L}}^n+\beta \int _{J_v}\left( {\underline{v}}^q +{\overline{v}}^q\right) \, \hbox {d}{\mathcal {H}}^{n-1}+{\mathcal {L}}^n(\{v>0\}\setminus \Omega ), \end{aligned}$$
(1.3)

and in the following we are going to study the problem

$$\begin{aligned} \inf \left\{ {\mathcal {F}}(v) \left| \begin{aligned}&v\in {{\,\textrm{SBV}\,}}({\mathbb {R}}^n) \\&v(x)=1 \text { in }\Omega \end{aligned}\right. \right\} . \end{aligned}$$

Notice that if \(v\in {{\,\textrm{SBV}\,}}({\mathbb {R}}^n)\) with \(v=1\) a.e. in \(\Omega \), letting \(v_0=\max \{0,\min \{v,1\}\}\) we have that \(v_0\in {{\,\textrm{SBV}\,}}({\mathbb {R}}^n)\) with \(v_0=1\) a.e. in \(\Omega \) and \({\mathcal {F}}(v_0)\le {\mathcal {F}}(v)\) so it suffices to consider the problem

$$\begin{aligned} \inf \left\{ {\mathcal {F}}(v) \left| \begin{aligned}&v\in {{\,\textrm{SBV}\,}}({\mathbb {R}}^n),\\&v(x)\in [0,1] \,{\mathcal {L}}^n\text {-a.e.}, \\&v(x)=1 \text { in }\Omega \end{aligned}\right. \right\} . \end{aligned}$$
(1.4)

In a more regular setting, problem (1.4) can be seen as a PDE. Let us fix \(\Omega ,A\) sufficiently smooth open sets, \(u\in W^{1,p}(A)\) with \(u=1\) on \(\Omega \), and let us define the functional

$$\begin{aligned} F(u,A)=\int _\Omega |\nabla u|^p\, \hbox {d}{\mathcal {L}}^n+\beta \int _{\partial \Omega }|u|^q\,\hbox {d}{\mathcal {H}}^{n-1}+ {\mathcal {L}}^n({A\setminus \Omega }). \end{aligned}$$
(1.5)

minimizers u to (1.5) solve the following boundary value problem

$$\begin{aligned} {\left\{ \begin{array}{ll} {{\,\textrm{div}\,}}\left( |\nabla u|^{p-2}\nabla u\right) =0 &{}\quad \text {in }A\setminus \Omega ,\\ |\nabla u|^{p-2}\frac{\partial u}{\partial \nu }+\beta \frac{q}{p}|u|^{q-2}u=0 &{}\quad \text {on }\partial A,\\ u=1 &{}\quad \text {in }\Omega . \end{array}\right. } \end{aligned}$$
(1.6)

In Sect. 2 we give some preliminary tools and definitions, and then we will prove the existence of a minimizer u of (1.4), under a prescribed condition on p and q. Finally, we will prove density estimates for the jump set \(J_u\).

We resume in the following theorems the main results of this paper.

Theorem 1.1

Let \(\Omega \subseteq {\mathbb {R}}^n\) be a bounded open set, and let \(p,q>1\) be exponents satisfying one of the following conditions:

  • \(1<p<n\), and \(1<q<\dfrac{p(n-1)}{n-p}:=p_*\);

  • \(n\le p<\infty \), and \(1<q<\infty \).

Then there exists a solution u to problem (1.4) and there exists a constant \(\delta _0=\delta _0(\Omega ,\beta ,p,q)>0\) such that

$$\begin{aligned} u>\delta _0 \end{aligned}$$
(1.7)

\({\mathcal {L}}^n\)-almost everywhere in \(\{u>0\}\), and there exists \(\rho (\delta _0)>0\) such that

$$\begin{aligned} {{\,\textrm{supp}\,}}u\subseteq B_{\rho (\delta _0)}. \end{aligned}$$

Theorem 1.2

Let \(\Omega \subseteq {\mathbb {R}}^n\) be a bounded open set, and let \(p,q>1\) be exponents satisfying the assumptions of Theorem 1.1. Then there exist positive constants \(C(\Omega ,\beta ,p,q)\), \(c(\Omega ,\beta ,p,q)\), \(C_1(\Omega ,\beta ,p,q)\) such that if u is a minimizer to problem (1.4), then

$$\begin{aligned} c\,r^{n-1}\le {\mathcal {H}}^{n-1}(J_u\cap B_r(x))\le C\, r^{n-1}, \end{aligned}$$

and

$$\begin{aligned} {\mathcal {L}}^n(B_r(x)\cap \{u>0\})\ge C_1\,r^n, \end{aligned}$$

for every \(x\in \overline{J_u}\) with \(B_r(x)\subseteq {\mathbb {R}}^n{\setminus } \Omega \).

In particular, this implies the essential closedness of the jump set \(J_u\) outside of \(\Omega \), namely

$$\begin{aligned} {\mathcal {H}}^{n-1}((\overline{J_u}\setminus J_u)\setminus {\bar{\Omega }})=0. \end{aligned}$$

In Sect. 3 we prove that the a priori estimate (1.7) holds for inward minimizers (see Definition 3.1), such an estimate will be crucial in the proof of Theorem 1.1 in Sect. 4. Finally, in Sect. 5 we prove Theorem 1.2.

Remark 1.3

Notice that the condition on the exponents is undoubtedly verified when \(p\ge q>1\). Furthermore, if \(\Omega \) is a set with Lipschitz boundary, the exponent \(p_*\) is the optimal exponent such that

$$\begin{aligned} W^{1,p}(\Omega )\subset \subset L^q(\partial \Omega ) \qquad \forall q\in [1,p_*). \end{aligned}$$

2 Notation and tools

In this section, we give the definition of the space \({{\,\textrm{SBV}\,}}\), and some useful notations and results that we will use in the following sections. We refer to [2, 5, 9] for a more intensive study of these topics.

Definition 2.1

(\({{\,\textrm{BV}\,}}\)) Let \(u\in L^1({\mathbb {R}}^n)\). We say that u is a function of bounded variation in \({\mathbb {R}}^n\) and we write \(u\in {{\,\textrm{BV}\,}}({\mathbb {R}}^n)\) if its distributional derivative is a Radon measure, namely

$$\begin{aligned} \int _{\Omega }u\,\frac{\partial \varphi }{\partial x_i}=\int _{\Omega }\varphi \, \textrm{d}D_i u\qquad \forall \varphi \in C^\infty _c({\mathbb {R}}^n), \end{aligned}$$

with Du a \({\mathbb {R}}^n\)-valued measure in \({\mathbb {R}}^n\). We denote with \(|Du|\) the total variation of the measure Du. The space \({{\,\textrm{BV}\,}}({\mathbb {R}}^n)\) is a Banach space equipped with the norm

$$\begin{aligned} \Vert u\Vert _{{{\,\textrm{BV}\,}}({\mathbb {R}}^n)}=\Vert u\Vert _{L^1({\mathbb {R}}^n)}+|Du|({\mathbb {R}}^n). \end{aligned}$$

Definition 2.2

Let \(E\subseteq {\mathbb {R}}^n\) be a measurable set. We define the set of points of density 1 for E as

$$\begin{aligned} E^{(1)}=\left\{ x\in {\mathbb {R}}^n \left| \lim _{r\rightarrow 0^+}\dfrac{{\mathcal {L}}^n(B_r(x)\cap E)}{{\mathcal {L}}^n(B_r(x))}=1\right. \right\} , \end{aligned}$$

and the set of points of density 0 for E as

$$\begin{aligned} E^{(0)}=\left\{ x\in {\mathbb {R}}^n \left| \lim _{r\rightarrow 0^+}\dfrac{{\mathcal {L}}^n(B_r(x)\cap E)}{{\mathcal {L}}^n(B_r(x))}=0\right. \right\} . \end{aligned}$$

Moreover, we define the essential boundary of E as

$$\begin{aligned} \partial ^*E={\mathbb {R}}^n \setminus (E^{(0)}\cup E^{(1)}). \end{aligned}$$

Definition 2.3

(Approximate upper and lower limits) Let \(u:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) be a measurable function. We define the approximate upper and lower limits of u, respectively, as

$$\begin{aligned} {\overline{u}}(x)=\inf \left\{ t\in {\mathbb {R}}\left| \limsup _{r\rightarrow 0^+} \dfrac{{\mathcal {L}}^n(B_r(x)\cap \{u>t\})}{{\mathcal {L}}^n(B_r(x))}=0\right. \right\} , \end{aligned}$$

and

$$\begin{aligned} {\underline{u}}(x)=\sup \left\{ t\in {\mathbb {R}}\left| \limsup _{r\rightarrow 0^+} \dfrac{{\mathcal {L}}^n(B_r(x)\cap \{u<t\})}{{\mathcal {L}}^n(B_r(x))}=0\right. \right\} . \end{aligned}$$

We define the jump set of u as

$$\begin{aligned} J_u=\left\{ x\in {\mathbb {R}}^n|{\underline{u}}(x)<{\overline{u}}(x)\right\} . \end{aligned}$$

We denote by \(K_u\) the closure of \(J_u\).

If \({\overline{u}}(x)={\underline{u}}(x)=l\), we say that l is the approximate limit of u as y tends to x, and we have that, for any \(\varepsilon >0\),

$$\begin{aligned} \limsup _{r\rightarrow 0^+}\dfrac{{\mathcal {L}}^n(B_r(x)\cap \{|u-l|\ge \varepsilon )\}}{{\mathcal {L}}^n(B_r(x))}=0. \end{aligned}$$

If \(u\in {{\,\textrm{BV}\,}}({\mathbb {R}}^n)\), the jump set \(J_u\) is a \((n-1)\)-rectifiable set, i.e., \({J_u\subseteq \bigcup _{i\in {\mathbb {N}}}M_i}\), up to a \({\mathcal {H}}^{n-1}\)-negligible set, with \(M_i\) a \(C^1\)-hypersurface in \({\mathbb {R}}^n\) for every i. We can then define \({\mathcal {H}}^{n-1}\)-almost everywhere on \(J_u\) a normal \(\nu _u\) coinciding with the normal to the hypersurfaces \(M_i\). Furthermore, the direction of \(\nu _u(x)\) is chosen in such a way that the approximate upper and lower limits of u coincide with the approximate limit of u on the half-planes

$$\begin{aligned} H^+_{\nu _u}=\{y\in {\mathbb {R}}^n|\nu _u(x)\cdot (y-x)\ge 0\} \end{aligned}$$

and

$$\begin{aligned} H^-_{\nu _u}=\{y\in {\mathbb {R}}^n|\nu _u(x)\cdot (y-x)\le 0\} \end{aligned}$$

, respectively.

Definition 2.4

Let \(\Omega \subseteq {\mathbb {R}}^n\) be an open set, and \(E\subseteq {\mathbb {R}}^n\) a measurable set. We define the relative perimeter of E inside \(\Omega \) as

$$\begin{aligned} P(E;\Omega )=\sup \left\{ \int _E {{\,\textrm{div}\,}}\varphi \,\hbox {d}{\mathcal {L}}^n\left| \begin{aligned} \varphi \in&\,C^1_c(\Omega ,{\mathbb {R}}^n) \\&|\varphi |\le 1 \end{aligned}\right. \right\} . \end{aligned}$$

If \(P(E;{\mathbb {R}}^n)<+\infty \) we say that E is a set of finite perimeter.

Theorem 2.5

(Decomposition of \({{\,\textrm{BV}\,}}\) functions) Let \(u\in {{\,\textrm{BV}\,}}({\mathbb {R}}^n)\). Then we have

$$\begin{aligned} \textrm{d}Du=\nabla u\,\textrm{d}{\mathcal {L}}^n+|{\overline{u}}-{\underline{u}}| \nu _u\,\textrm{d}{\mathcal {H}}^{n-1}\lfloor _{{J_u}}+ \textrm{d}D^c u, \end{aligned}$$

where \(\nabla u\) is the density of Du with respect to the Lebesgue measure, \(\nu _u\) is the normal to the jump set \(J_u\) and \(D^c u\) is the Cantor part of the measure Du. The measure \(D^c u\) is singular with respect to the Lebesgue measure and concentrated out of \(J_u\).

Definition 2.6

Let \(v\in {{\,\textrm{BV}\,}}({\mathbb {R}}^n)\), let \(\Gamma \subseteq {\mathbb {R}}^n\) be a \({\mathcal {H}}^{n-1}\)-rectifiable set, and let \(\nu (x)\) be the generalized normal to \(\Gamma \) defined for \({\mathcal {H}}^{n-1}\)-a.e. \(x\in \Gamma \). For \({\mathcal {H}}^{n-1}\)-a.e. \(x\in \Gamma \) we define the traces \(\gamma _\Gamma ^{\pm }(v)(x)\) of v on \(\Gamma \) by the following Lebesgue-type limit quotient relation

$$\begin{aligned} \lim _{r\rightarrow 0}\frac{1}{r^n}\int _{B_r^{\pm }(x)}|v(y) -\gamma _\Gamma ^{\pm }(v)(x)|\,\hbox {d}{\mathcal {L}}^n(y)=0, \end{aligned}$$

where

$$\begin{aligned} B_{r}^{+}(x)= & {} \{y\in B_r(x) | \nu (x)\cdot (y-x)>0\},\\ B_{r}^{-}(x)= & {} \{y\in B_r(x) | \nu (x)\cdot (y-x)<0\}. \end{aligned}$$

Remark 2.7

Notice that, by [2, Remark 3.79], for \({\mathcal {H}}^{n-1}\)-a.e. \(x\in \Gamma \), \((\gamma _\Gamma ^{+}(v)(x),\gamma _\Gamma ^-(v)(x))\) coincides with either \(({\overline{v}}(x),{\underline{v}}(x))\) or \(({\underline{v}}(x),{\overline{v}}(x))\), while, for \({\mathcal {H}}^{n-1}\)-a.e. \(x\in \Gamma \setminus J_v\), we have that \(\gamma _\Gamma ^+(v)(x)=\gamma _\Gamma ^-(v)(x)\) and they coincide with the approximate limit of v in x. In particular, if \(\Gamma =J_v\), we have

$$\begin{aligned} \gamma _{J_v}^+(v)(x)={\overline{v}}(x) \qquad \gamma _{J_v}^-(v)(x)={\underline{v}}(x) \end{aligned}$$

for \({\mathcal {H}}^{n-1}\)-a.e. \(x\in J_v\).

We now focus our attention on the \({{\,\textrm{BV}\,}}\) functions whose Cantor parts vanish.

Definition 2.8

(SBV) Let \(u\in {{\,\textrm{BV}\,}}({\mathbb {R}}^n)\). We say that u is a special function of bounded variation and we write \(u\in {{\,\textrm{SBV}\,}}({\mathbb {R}}^n)\) if \(D^c u=0\).

For \({{\,\textrm{SBV}\,}}\) functions we have the following.

Theorem 2.9

(Chain rule) Let \(g:{\mathbb {R}}\rightarrow {\mathbb {R}}\) be a differentiable function. Then if \(u\in {{\,\textrm{SBV}\,}}({\mathbb {R}}^n)\), we have

$$\begin{aligned} \nabla g(u)=g'(u)\nabla u. \end{aligned}$$

Furthermore, if g is increasing,

$$\begin{aligned} \overline{g(u)}=g({\overline{u}}),\quad \underline{g(u)}=g({\underline{u}}) \end{aligned}$$

while, if g is decreasing,

$$\begin{aligned} \overline{g(u)}=g({\underline{u}}),\quad \underline{g(u)}=g({\overline{u}}). \end{aligned}$$

We now state a compactness theorem in \({{\,\textrm{SBV}\,}}\) that will be useful in the following.

Theorem 2.10

(Compactness in \({{\,\textrm{SBV}\,}}\)) Let \(u_k\) be a sequence in \({{\,\textrm{SBV}\,}}({\mathbb {R}}^n)\). Let \(p,q>1\), and let \(C>0\) such that for every \(k\in {\mathbb {N}}\)

$$\begin{aligned} \int _{{\mathbb {R}}^n}|\nabla u_k|^p\,\textrm{d}{\mathcal {L}}^n+\Vert u_k\Vert _{\infty }+{\mathcal {H}}^{n-1}(J_{u_k})<C. \end{aligned}$$

Then there exists \(u\in {{\,\textrm{SBV}\,}}({\mathbb {R}}^n)\) and a subsequence \(u_{k_j}\) such that

  • Compactness:

    $$\begin{aligned} u_{k_j}\xrightarrow {L^1_{{{\,\textrm{loc}\,}}}({\mathbb {R}}^n)} u \end{aligned}$$

  • Lower semicontinuity: for every open set A we have

    $$\begin{aligned} \int _A |\nabla u|^p\, \textrm{d}{\mathcal {L}}^n\le \liminf _{j\rightarrow +\infty }\int _A |\nabla u_{k_j}|^p\,\textrm{d}{\mathcal {L}}^n\end{aligned}$$

    and

    $$\begin{aligned} \int _{J_u\cap A}\left( {{\overline{u}}}^q+{\underline{u}}^q\right) \, \textrm{d}{\mathcal {H}}^{n-1}\le \liminf _{j\rightarrow +\infty }\int _{J_{u_{k_j}}\cap A} \left( {{\overline{u}}}_{k_j}^q+{\underline{u}}_{k_j}^q\right) \, \textrm{d}{\mathcal {H}}^{n-1}\end{aligned}$$

We refer to [2, Theorem 4.7, Theorem 4.8, Theorem 5.22] for the proof of this theorem. We now conclude this section with the following proposition whose proof can be found in [7, Lemma 3.1].

Proposition 2.11

Let \(u\in {{\,\textrm{BV}\,}}({\mathbb {R}}^n)\cap L^\infty ({\mathbb {R}}^n)\). Then

$$\begin{aligned} \int _0^1 P(\{u>s\};{\mathbb {R}}^n\setminus J_u)\,\textrm{d}s=|Du|({\mathbb {R}}^n\setminus J_u). \end{aligned}$$

3 Lower bound

In the following, we assume that \(\Omega \subset {\mathbb {R}}^n\) is a bounded open set and that p and q are two positive real numbers such that

$$\begin{aligned} \dfrac{q'}{p'}>1-\dfrac{1}{n} \end{aligned}$$
(3.1)

where \(p'\) and \(q'\) are the Hölder conjugates of p and q, respectively.

Definition 3.1

Let \(v\in {{\,\textrm{SBV}\,}}({\mathbb {R}}^n)\) be a function such that \(v=1\) a.e. in \(\Omega \). We say that v is an inward minimizer if

$$\begin{aligned} {\mathcal {F}}(v)\le {\mathcal {F}}(v\chi _A), \end{aligned}$$

for every set of finite perimeter A containing \(\Omega \), where \(\chi _A\) is the characteristic function of set A.

Let \(A\subset {\mathbb {R}}^n\) be a set of finite perimeter such that \(\Omega \subset A\), and let \(v\in {{\,\textrm{SBV}\,}}({\mathbb {R}}^n)\). We will make use of the following expression

$$\begin{aligned} \begin{aligned} {\mathcal {F}}(v\chi _A)&=\int _{A}|\nabla v|^p\, \hbox {d}{\mathcal {L}}^n+ \beta \int _{J_v\cap A^{(1)}}\left( {\underline{v}}^q+{\overline{v}}^{\,q}\right) \,\hbox {d}{\mathcal {H}}^{n-1}+\beta \int _{\partial ^* A\setminus J_v}v^q\,\hbox {d}{\mathcal {H}}^{n-1}\\&\quad +\beta \int _{J_v\cap \partial ^* A}\gamma _{\partial A}^-(v)^q\, \hbox {d}{\mathcal {H}}^{n-1}+{\mathcal {L}}^n\left( (\{v>0\}\cap A)\setminus \Omega \right) , \end{aligned} \end{aligned}$$
(3.2)

Let B be a ball containing \(\Omega \), then \(\chi _B\in {{\,\textrm{SBV}\,}}({\mathbb {R}}^n)\) and \(\chi _B=1\) in \(\Omega \), we will denote \({\mathcal {F}}(\chi _B)\) by \(\tilde{{\mathcal {F}}}\).

Theorem 3.2

There exists a positive constant \(\delta =\delta (\Omega ,\beta ,p,q)\) such that if u is an inward minimizer with \({\mathcal {F}}(u)\le 2\tilde{{\mathcal {F}}}\), then

$$\begin{aligned} u>\delta \end{aligned}$$

\({\mathcal {L}}^n\)-almost everywhere in \(\left\{ u>0\right\} \).

Proof

Let \(0<t<1\) and

$$\begin{aligned} f(t)=\int _{\{u\le t\}\setminus J_u} u^{q-1}|\nabla u|\,\hbox {d}{\mathcal {L}}^n=\int _0^t s^{q-1}P(\left\{ u> s\right\} ;{\mathbb {R}}^n\setminus J_u)\,\hbox {d}s. \end{aligned}$$

For every such t, we have

$$\begin{aligned} f(t)\le \left( \int _{\{u\le t\}} u^{(q-1)p'}\,\hbox {d}{\mathcal {L}}^n\right) ^{\frac{1}{p'}}\left( \int _{\{u\le t\}\setminus J_u}|\nabla u|^p\,\hbox {d}{\mathcal {L}}^n\right) ^{\frac{1}{p}}\le {\mathcal {F}}(u)\le 2\tilde{{\mathcal {F}}}. \end{aligned}$$
(3.3)

Let \(u_t=u\chi _{\{u>t\}}\). Using (3.2) we have

$$\begin{aligned} \begin{aligned} 0&\le \,{\mathcal {F}}(u_t)-{\mathcal {F}}(u)\\&=\beta \int _{\partial ^*\{u>t\}\setminus J_u}{\overline{u}}^q\,\hbox {d}{\mathcal {H}}^{n-1}-\int _{\{u\le t\}\setminus J_u}|\nabla u|^p\,\hbox {d}{\mathcal {L}}^n-\beta \int _{J_u\cap \partial ^* \{u>t\}}{\underline{u}}^q\,\hbox {d}{\mathcal {H}}^{n-1}\\&\quad -\beta \int _{J_u\cap \{u>t\}^{(0)}}\left( {\overline{u}}^q +{\underline{u}}^q\right) \,\hbox {d}{\mathcal {H}}^{n-1}-{\mathcal {L}}^n(\{0<u\le t\}), \end{aligned} \end{aligned}$$

and rearranging the terms,

$$\begin{aligned} \begin{aligned}&\int _{\{u\le t\}\setminus J_u}|\nabla u|^p\,\hbox {d}{\mathcal {L}}^n+\beta \int _{J_u\cap \partial ^*\{u>t\}}{\underline{u}}^q\,\hbox {d}{\mathcal {H}}^{n-1}+\beta \int _{J_u\cap \{u>t\}^{(0)}}\left( {\overline{u}}^q +{\underline{u}}^q\right) \,\hbox {d}{\mathcal {H}}^{n-1}\\&\quad +{\mathcal {L}}^n(\{0<u\le t\}) \le \beta t^q P(\{u>t\};{\mathbb {R}}^n\setminus J_u)=\beta t f'(t). \end{aligned} \end{aligned}$$
(3.4)

On the other hand,

$$\begin{aligned} \begin{aligned} f(t)&=\int _{\{u\le t\}\setminus J_u} u^{q-1}|\nabla u|\,\hbox {d}{\mathcal {L}}^n\\&\le \left( \int _{\{u\le t\}} u^{(q-1)p'}\,\hbox {d}{\mathcal {L}}^n\right) ^{\frac{1}{p'}} \left( \int _{\{u\le t\}\setminus J_u}|\nabla u|^p\,\hbox {d}{\mathcal {L}}^n\right) ^{\frac{1}{p}}\\&\le \Biggl ({\mathcal {L}}^n(\{0<u\le t\})\Biggr )^{\frac{1}{p'\gamma '}} \left( \int _{\{u\le t\}} u^{q 1^*}\,\hbox {d}{\mathcal {L}}^n\right) ^{\frac{1}{q'1^*}}\left( \int _{\{u\le t\}\setminus J_u}|\nabla u|^p \,\hbox {d}{\mathcal {L}}^n\right) ^{\frac{1}{p}}, \end{aligned} \end{aligned}$$

where we used

$$\begin{aligned} 1^*=\dfrac{n}{n-1}, \qquad \text {and}\qquad \gamma =\dfrac{q1^*}{(q-1)p'}, \end{aligned}$$

and \(\gamma >1\) by (3.1). By classical BV embedding in \(L^{1^*}\) applied to the function \((u\chi _{\{u\le t\}})^q\) and the estimate (3.4), we have

$$\begin{aligned} f(t)\le C(n,\beta ) \biggl (t f'(t)\biggr )^{1-\frac{n-1}{q'n}}\left( \int _{{\mathbb {R}}^n}\,\hbox {d}|*|{D (u\chi _{\{u\le t\}})^q}\right) ^{\frac{1}{q'}}. \end{aligned}$$

We can compute

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}^n}\,\hbox {d}|*|{D(u\chi _{\{u\le t\}})^q}&\le q\Biggl ({\mathcal {L}}^n(\{0<u\le t\})\Biggr )^{\frac{1}{p'}}\left( \int _{\{u\le t\}\setminus J_u}|\nabla u|^p\,\hbox {d}{\mathcal {L}}^n\right) ^{\frac{1}{p}}\\&\quad + \int _{J_u\cap \{u>t\}^{(0)}}\left( {\overline{u}}^q +{\underline{u}}^q\right) \,\hbox {d}{\mathcal {H}}^{n-1}+\!\int _{J_u\cap \partial ^*\{u>t\}}{\underline{u}}^q\,\hbox {d}{\mathcal {H}}^{n-1}\\&\quad +t^qP(\{u>t\};{\mathbb {R}}^n\setminus J_u)\le (2+q\beta )tf'(t). \end{aligned} \end{aligned}$$

We therefore get

$$\begin{aligned} f(t)\le C(n,\beta ,q) \left( tf'(t)\right) ^{1+\frac{1}{nq'}}. \end{aligned}$$

Let \(0<t_0<1\) such that \(f(t_0)>0\), then for every \(t_0<t<1\), we have \(f(t)>0\) and

$$\begin{aligned} \dfrac{f'(t)}{f(t)^{\frac{nq}{q(n+1)-1}}}\ge \dfrac{C(n,\beta ,q)}{t}, \end{aligned}$$

integrating from \(t_0\) to 1, we have

$$\begin{aligned} f(1)^{\frac{q-1}{q(n+1)-1}}-f(t_0)^{\frac{q-1}{q(n+1)-1}}\ge C(n,\beta ,q) \log \dfrac{1}{t_0}, \end{aligned}$$

so that, using (3.3),

$$\begin{aligned} f(t_0)^{\frac{q-1}{q(n+1)-1}}\le (2\tilde{{\mathcal {F}}})^{\frac{q-1}{q(n+1)-1}} + C(n,\beta ,q)\log t_0. \end{aligned}$$

Let

$$\begin{aligned} \delta =\exp \left( -\dfrac{(2\tilde{{\mathcal {F}}})^{\frac{q-1}{q(n+1)-1}}}{C(n,\beta ,q)}\right) , \end{aligned}$$

for every \(t_0<\delta \) we would have \(f(t_0)<0\), which is a contradiction. Therefore \(f(t)=0\) for every \(t<\delta \), from which \(u>\delta \) \({\mathcal {L}}^n\)-almost everywhere on \(\{u>0\}\). \(\square \)

Remark 3.3

From Theorem 3.2, if u is an inward minimizer with \({\mathcal {F}}(u)\le 2\tilde{{\mathcal {F}}}\), we have that

$$\begin{aligned} \partial ^*\{u>0\}\subseteq J_u\subseteq K_u. \end{aligned}$$

Indeed, on \(\partial ^*\{u>0\}\) we have that, by definition, \({\underline{u}}=0\) and that, since \(u\ge \delta \) \({\mathcal {L}}^n\)-almost everywhere in \(\{u>0\}\), \({\overline{u}}\ge \delta \).

Proposition 3.4

There exists a positive constant \(\delta _0=\delta _0(\Omega ,\beta ,p,q)<\delta \) such that if u is an inward minimizer with \({\mathcal {F}}(u)\le 2\tilde{{\mathcal {F}}}\), then u is supported on \(B_{\rho (\delta _0)}\), where \(\rho (\delta _0)=\delta _0^{1-q}\) and \(B_{\rho (\delta _0)}\) is the ball centered at the origin with radius \(\rho (\delta _0)\). Moreover there exist positive constants \(C(\Omega ,\beta ,p,q),C_1(\Omega ,\beta ,p,q)\) such that, for any \(B_r(x)\subseteq {\mathbb {R}}^n\setminus \Omega \) we have

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_u\cap B_r(x))\le C(\Omega ,p,q)r^{n-1}, \end{aligned}$$
(3.5)

and if \(x\in K_u\), then

$$\begin{aligned} {\mathcal {L}}^n(B_r(x)\cap \{u>0\})\ge C_1(\Omega ,p,q)r^n. \end{aligned}$$
(3.6)

Proof

By Theorem 3.2, if u is an inward minimizer, we have

$$\begin{aligned} \int _{J_u\cap B_r(x)}\left( {\overline{u}}^q+{\underline{u}}^q\right) \,\hbox {d}{\mathcal {H}}^{n-1}\ge \delta ^q{\mathcal {H}}^{n-1}(J_u\cap B_r(x)), \end{aligned}$$

on the other hand, using \(u\chi _{{\mathbb {R}}^n\setminus B_r(x)}\) as a competitor for u, we have

$$\begin{aligned} \int _{J_u\cap B_r(x)}\left( {\overline{u}}^q+{\underline{u}}^q\right) \,\hbox {d}{\mathcal {H}}^{n-1}\le \int _{\partial B_r(x)\cap \{u>0\}^{(1)}}\left( {\overline{u}}^q+{\underline{u}}^q\right) \,\hbox {d}{\mathcal {H}}^{n-1}\le C(n)r^{n-1}. \end{aligned}$$

Let now \(x\in K_u\) and consider \(\mu (r)={\mathcal {L}}^n\left( B_r(x)\cap \{u>0\}^{(1)}\right) \). Using the isoperimetric inequality and inequality (3.5), we have that for almost every \(r\in (0,d(x,\Omega ))\)

$$\begin{aligned} \begin{aligned} 0<\mu (r)&\le K(n)\,P\!\left( B_r(x)\cap \{u>0\}^{(1)}\right) ^{\frac{n}{n-1}}\\&\le K(\Omega ,\beta ,p,q)\,P\!\left( B_r(x);\{u>0\}^{(1)} \right) ^{\frac{n}{n-1}}. \end{aligned} \end{aligned}$$

Notice that we used Remark 3.3 in the last inequality. We have

$$\begin{aligned} \mu (r)\le K \mu '(r)^{\frac{n}{n-1}}. \end{aligned}$$

Integrating the differential inequality, we obtain

$$\begin{aligned} {\mathcal {L}}^n(B_r(x)\cap \{u>0\})\ge C_1(\Omega ,\beta ,p,q)r^n. \end{aligned}$$

Finally, let \(\delta _0>0\) and \(x\in K_u\) such that \(d(x,\Omega )>\rho (\delta _0)=\delta _0^{1-q}\). By (3.6)

$$\begin{aligned} C_1(\Omega ,\beta ,p,q)\rho (\delta _0)^{n}\le {\mathcal {L}}^n(\{u>0\}\cap \Omega )\le 2\tilde{{\mathcal {F}}}, \end{aligned}$$

which leads to a contradiction if \(\delta _0\) is too small, hence there exists a positive value \(\delta _0(\Omega ,\beta ,p,q)\) such that \(\{u>0\}\subset B_{\rho (\delta _0)}\). \(\square \)

4 Existence

In this section, we are going to prove the existence of a solution u to the problem (1.4). Let us denote

$$\begin{aligned} H_a=\left\{ u\in {{\,\textrm{SBV}\,}}({\mathbb {R}}^n) \left| \begin{aligned}&u(x)=1 \text { in }\Omega \\&u(x)\in \{0\}\cup [a,1] ~{\mathcal {L}}^n\text { -a.e.} \\&{{\,\textrm{supp}\,}}u \subseteq B_\frac{1}{a^{q-1}} \end{aligned}\right. \right\} . \end{aligned}$$

We also denote by \(H_0\) the set

$$\begin{aligned} H_0=\left\{ u\in {{\,\textrm{SBV}\,}}({\mathbb {R}}^n) \left| \begin{aligned}&u(x)=1 \text { in }\Omega \\&u(x)\in [0,1] {\mathcal {L}}^n\text {-a.e.} \end{aligned}\right. \right\} . \end{aligned}$$

Notice that if \(u\in H_0\) is an inward minimizer, by Theorem 3.2 and Corollary 3.4, then \(u\in H_{\delta _0}\).

Proposition 4.1

Let \(u\in H_0\). Then u is a minimizer for the functional (1.3) on \(H_0\) if and only if \(u\in H_{\delta _0}\) and

$$\begin{aligned} {\mathcal {F}}(u)\le {\mathcal {F}}(v) \qquad \forall v\in H_{\delta _0}. \end{aligned}$$

Proof

As we observed before, if u is a minimizer over \(H_0\) then u is in \(H_{\delta _0}\), hence it is a minimizer over \(H_{\delta _0}\). Conversely, let us take \(u\in H_{\delta _0}\) a minimizer over \(H_{\delta _0}\), and let us consider in addition \(v\in H_0\). Without loss of generality assume \({\mathcal {F}}(v)\le 2\tilde{{\mathcal {F}}}\). We will prove that there exists a sequence \(w_k\) of inward minimizers such that

$$\begin{aligned} {\mathcal {F}}(w_k)\le {\mathcal {F}}(v)+\frac{C}{k^{q-1}}. \end{aligned}$$

We first construct a family of functions \(v_a\in H_a\) such that

$$\begin{aligned} {\mathcal {F}}(v_a)\le {\mathcal {F}}(v)+r(a), \end{aligned}$$

with \(\lim _{a\rightarrow 0}r(a)=0\). Let \(0<a<1\), and let \({v_a=v\chi _{\{v\ge a\}\cap B_{\rho (a)}}}\), where \(\rho (a)=a^{1-q}\), we have

$$\begin{aligned} \begin{aligned} {\mathcal {F}}(v_a)-{\mathcal {F}}(v)&\le \beta \int _{\partial ^*(\{v\ge a\}\cap B_{\rho (a)})\setminus J_v}v^q\,\hbox {d}{\mathcal {H}}^{n-1}\\&\le \beta a^q P(\{v\ge a\})+\beta \int _ {(\partial B_{\rho (a)}\cap \{v\ge a\})\setminus J_v} v^q\,\hbox {d}{\mathcal {H}}^{n-1}\\&\le \beta a^{q}\left( P(\{v\ge a\})+\frac{1}{a^q}\int _ {(\partial B_{\rho (a)}\cap \{v\ge a\})\setminus J_v} v\,\hbox {d}{\mathcal {H}}^{n-1}\right) . \end{aligned} \end{aligned}$$
(4.1)

In order to estimate the right-hand side, fix \(t\in (0,1)\), and observe that by the coarea formula

$$\begin{aligned} \int _0^tP(\{v\ge a\})\, da\le |Dv|({\mathbb {R}}^n), \end{aligned}$$
(4.2)

while, with a change of variables,

$$\begin{aligned}{} & {} \int _0^t\frac{1}{a^q}\int _ {(\partial B_{\rho (a)}\cap \{v\ge a\})\setminus J_v} v\,\hbox {d}{\mathcal {H}}^{n-1}\,da\le (q-1)\int _0^{+\infty }\int _{\partial B_r\setminus J_v}v \,\hbox {d}{\mathcal {H}}^{n-1}\,dr\\ {}{} & {} \quad =(q-1)\Vert v\Vert _{L^1({\mathbb {R}}^n)}.\\{} & {} \int _0^t\left( P(\{v\ge a\})+\frac{1}{a^q}\int _ {(\partial B_{\rho (a)}\cap \{v\ge a\})\setminus J_v} v\,\hbox {d}{\mathcal {H}}^{n-1}\right) \,da\le q \Vert v\Vert _{{{\,\textrm{BV}\,}}}. \end{aligned}$$

By mean value theorem, for every \(k\in {\mathbb {N}}\) we can find \(a_k\le 1/k\) such that

$$\begin{aligned} P(\{v\ge a_k\})+\frac{1}{a_k^q}\int _ {(\partial B_{\rho (a_k)}\cap \{v\ge a_k\})\setminus J_v} v\,\hbox {d}{\mathcal {H}}^{n-1}\le \frac{q\Vert v\Vert _{{{\,\textrm{BV}\,}}}}{a_k}, \end{aligned}$$

and in (4.1) we get

$$\begin{aligned} {\mathcal {F}}(v_{a_k})\le {\mathcal {F}}(v)+q\beta a_k^{q-1}\Vert v\Vert _{{{\,\textrm{BV}\,}}}\le {\mathcal {F}}(v)+q\beta \frac{\Vert v\Vert _{{{\,\textrm{BV}\,}}}}{k^{q-1}}. \end{aligned}$$

We now construct the aforementioned sequence of inward minimizers: let us consider the functional

$$\begin{aligned} {\mathcal {G}}_k(A)={\mathcal {F}}(v_{a_k}\chi _A), \end{aligned}$$

with A containing \(\Omega \) and contained in \(\{v_{a_k}>0\}\). If we consider \(A_j\) a minimizing sequence for \({\mathcal {G}}_k\), then they are certainly equibounded. Moreover,

$$\begin{aligned} \begin{aligned} {\mathcal {G}}_k(A_j)&\ge {\mathcal {L}}^n(A_j\setminus \Omega )+\beta \int _{J_{\chi _{A_j} v_{a_k}}}\left( \underline{\chi _{A_j} v_{a_k}}^{q}+\overline{\chi _{A_j} v_{a_k}}^{\,q}\right) \,\hbox {d}{\mathcal {H}}^{n-1}\\&\ge {\mathcal {L}}^n(A_j)+\beta a_k^q {\mathcal {H}}^{n-1}(J_{\chi _{A_j} v_{a_k}})-{\mathcal {L}}^n(\Omega ). \end{aligned} \end{aligned}$$

Notice in addition that since \(v_{a_k}\ge a_k\) on its support, then the jump set \(J_{\chi _{A_j} v_{a_k}}\) clearly contains \(\partial ^* A_j\). We now have that \(\chi _{A_j}\) satisfies the conditions of Theorem 2.10, and eventually extracting a subsequence we can suppose that

$$\begin{aligned} A_j\xrightarrow []{L^1} A^{(k)}, \end{aligned}$$

with a suitable \(A^{(k)}\), and moreover, letting \(w_k=\chi _{A^{(k)}} v_{a_k}\), we have

$$\begin{aligned} {\mathcal {F}}(w_k)\le \inf _{\Omega \subseteq A \subseteq \{v_{a_k}>0\}}{\mathcal {G}}_k(A)\le {\mathcal {F}}(v_{a_k})\le {\mathcal {F}}(v)+q\beta \frac{\Vert v\Vert _{{{\,\textrm{BV}\,}}}}{k^{q-1}}. \end{aligned}$$

By construction \(w_k\) is an inward minimizer, therefore we have \(w_k\in H_{\delta _0}\), and consequently, we can compare it with u, obtaining

$$\begin{aligned} {\mathcal {F}}(u)\le {\mathcal {F}}(w_k)\le {\mathcal {F}}(v)+q\beta \frac{\Vert v\Vert _{{{\,\textrm{BV}\,}}}}{k^{q-1}}. \end{aligned}$$

Letting k go to infinity we get the thesis. \(\square \)

Proposition 4.2

There exists a minimizer for problem (1.4).

Proof

By Proposition 4.1 and Theorem 3.2 it is enough to find a minimizer in \(H_{\delta _0}\). Let \(u_k\) be a minimizing sequence in \(H_{\delta _0}\), then, for k large enough, we have

$$\begin{aligned} \beta \delta _0^q{\mathcal {H}}^{n-1}(J_{u_k})+\int _{{\mathbb {R}}^n}|\nabla u_k|^p\,\hbox {d}{\mathcal {L}}^n\le {\mathcal {F}}(u_k)\le 2\tilde{{\mathcal {F}}}. \end{aligned}$$

From Theorem 2.10 we have that there exists \(u\in H_{\delta _0}\) such that, up to a subsequence, \(u_k\) converges to u in \(L^1_{{{\,\textrm{loc}\,}}}\) and

$$\begin{aligned} {\mathcal {F}}(u)\le \liminf _{k} {\mathcal {F}}(u_k), \end{aligned}$$

therefore u is a solution. \(\square \)

Proof of Theorem 1.1

The result is obtained by joining Theorems 4.2 and 3.2. \(\square \)

5 Density estimates

In this section, we prove the density estimates in Theorem 1.2 by adapting techniques used in [7] analogous to classical ones used in [8] to prove density estimates for the jump set of almost-quasi minimizers of the Mumford–Shah functional.

Definition 5.1

Let \(u\in {{\,\textrm{SBV}\,}}({\mathbb {R}}^n)\) be a function such that \(u=1\) a.e. in \(\Omega \). We say that u is a local minimizer for \({\mathcal {F}}\) on a set of finite perimeter \(E\subset {\mathbb {R}}^n\setminus \Omega \), if

$$\begin{aligned} {\mathcal {F}}(u)\le {\mathcal {F}}(v), \end{aligned}$$

for every \(v\in {{\,\textrm{SBV}\,}}({\mathbb {R}}^n)\) such that \(u-v\) has support in E.

Let E be a set of finite perimeter. We introduce the notation

$$\begin{aligned} {\mathcal {F}}(u;E)=\int _E |\nabla u|^p\,\hbox {d}{\mathcal {L}}^n+\beta \int _{J_u\cap E} \left( {\overline{u}}^q+{\underline{u}}^q\right) \,\hbox {d}{\mathcal {H}}^{n-1}+{\mathcal {L}}^n\left( \{u>0\}\cap E\right) . \end{aligned}$$

To prove Theorem 1.2 we will use the following Poincaré-Wirtinger type inequality whose proof can be found in [8, Theorem 3.1 and Remark 3.3]. Let \(\gamma _n\) be the isoperimetric constant relative to the balls of \({\mathbb {R}}^n\), i.e.,

$$\begin{aligned} \min \left\{ {\mathcal {L}}^n(E\cap B_r)^{\frac{n-1}{n}},{\mathcal {L}}^n(E\setminus B_r)^{\frac{n-1}{n}}\right\} \le \gamma _n P(E;B_r), \end{aligned}$$

for every Borel set E, then

Proposition 5.2

Let \(p\ge 1\) and let \(u\in {{\,\textrm{SBV}\,}}(B_r)\) such that

$$\begin{aligned} \left( 2\gamma _n {\mathcal {H}}^{n-1}(J_u\cap B_r)\right) {^\frac{n}{n-1}}<\dfrac{{\mathcal {L}}^n(B_r)}{2}. \end{aligned}$$
(5.1)

Then there exist numbers \(-\infty< \tau ^-\le m\le \tau ^+<+\infty \) such that the function

$$\begin{aligned} {\tilde{u}}=\max \{\min \{u,\tau ^+\},\tau ^-\}, \end{aligned}$$

satisfies

$$\begin{aligned} \Vert {\tilde{u}}-m\Vert _{L^p}\le C \Vert \nabla u\Vert _{L^p} \end{aligned}$$

and

$$\begin{aligned} {\mathcal {L}}^n(\{u\ne {\tilde{u}}\})\le C\left( {\mathcal {H}}^{n-1}(J_u\cap B_r)\right) ^{\frac{n}{n-1}}, \end{aligned}$$

where the constants depend only on n, p, and r.

Lemma 5.3

Let \(u\in H_s\) be a local minimizer on \(B_r(x)\) in the sense of definition Definition 5.1. For sufficiently small values of \(\tau \), there exist values \(r_0,\varepsilon _0\) depending only on \(n,\tau ,\beta ,p,q\) and s such that, if \(r<r_0\),

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_u\cap B_r(x))\le \varepsilon _0 r^{n-1}, \end{aligned}$$

and

$$\begin{aligned} {\mathcal {F}}(u;B_r(x))\ge r^{n-\frac{1}{2}}, \end{aligned}$$

then

$$\begin{aligned} {\mathcal {F}}(u;B_{\tau r}(x))\le \tau ^{n-\frac{1}{2}}{\mathcal {F}}(u;B_r(x)). \end{aligned}$$

Proof

Without loss of generality, assume \(x=0\). Assume by contradiction that the conclusion fails, then for every \(\tau >0\) there exists a sequence \(u_k\in H_s\) of local minimizers on \(B_{r_k}\), with \(\lim _{k}r_k=0\), such that

$$\begin{aligned} \dfrac{{\mathcal {H}}^{n-1}(J_{u_k}\cap B_{r_k})}{r_k^{n-1}}=\varepsilon _k, \end{aligned}$$

with \(\lim _k \varepsilon _k=0\),

$$\begin{aligned} {\mathcal {F}}(u_k;B_{r_k})\ge r_k^{n-\frac{1}{2}}, \end{aligned}$$
(5.2)

and yet

$$\begin{aligned} {\mathcal {F}}(u_k;B_{\tau r_{k}})>\tau ^{n-\frac{1}{2}}{\mathcal {F}}(u_k;B_{r_k}). \end{aligned}$$
(5.3)

For every \(t\in [0,1]\), we define the sequence of monotone functions

$$\begin{aligned} \alpha _k(t)=\dfrac{{\mathcal {F}}(u_k;B_{t r_{k}})}{{\mathcal {F}}(u_k,B_{r_k})}\le 1. \end{aligned}$$

By compactness of \({{\,\textrm{BV}\,}}([0,1])\) in \(L^1([0,1])\), we can assume that, up to a subsequence, \(\alpha _k\) converges \({\mathcal {L}}^1\)-almost everywhere to a monotone function \(\alpha \). Moreover, notice that, by (5.3), for every k

$$\begin{aligned} \alpha _k(\tau )>\tau ^{n-\frac{1}{2}}. \end{aligned}$$
(5.4)

Our final aim is to prove that there exists a p-harmonic function \(v\in W^{1,p}(B_1)\) such that for every t

$$\begin{aligned} \lim _{k\rightarrow +\infty }\alpha _k(t)=\alpha (t)=\int _{B_t}|\nabla v|^p\,\hbox {d}{\mathcal {L}}^n. \end{aligned}$$

Let

$$\begin{aligned} E_k=r_k^{p-n}{\mathcal {F}}(u_k;B_{r_k}), \qquad \qquad v_k(x)=\dfrac{u_k(r_k x)}{E_k^{1/p}}. \end{aligned}$$

Then \(v_k\in {{\,\textrm{SBV}\,}}(B_1)\), and

$$\begin{aligned} \int _{B_1}|\nabla v_k|^p\,\hbox {d}{\mathcal {L}}^n\le 1,\qquad \qquad {\mathcal {H}}^{n-1}(J_{v_k}\cap B_1)=\varepsilon _k. \end{aligned}$$

Thus, applying the Poincaré–Wirtinger type inequality in Proposition 5.2 to functions \(v_k\) we obtain truncated functions \({\tilde{v}}_k\) and values \(m_k\), such that

$$\begin{aligned} \int _{B_1}|{\tilde{v}}_k-m_k|^p\,\hbox {d}{\mathcal {L}}^n\le C \end{aligned}$$

and

$$\begin{aligned} {\mathcal {L}}^n(\{v_k\ne \tilde{v_k}\})\le C\left( {\mathcal {H}}^{n-1}(J_{v_k}\cap B_1)\right) ^{\frac{n}{n-1}}\le C\varepsilon _k^{\frac{n}{n-1}}. \end{aligned}$$
(5.5)

  • Step 1: We prove that there exists \(v\in W^{1,p}(B_1)\) such that

    $$\begin{aligned}{} & {} {\tilde{v}}_k-m_k\xrightarrow {L^p(B_1)}{v},\nonumber \\{} & {} \int _{B_\rho }|\nabla v|^p\,\hbox {d}{\mathcal {L}}^n\le \alpha (\rho ),\qquad \text {for }{\mathcal {L}}^1\text {-a.e. }\rho <1, \end{aligned}$$
    (5.6)

    and

    $$\begin{aligned} \lim _k \dfrac{r_k^{p-1}}{E_k}{\mathcal {H}}^{n-1}(\{v_k\ne {\tilde{v}}_k\}\cap \partial B_\rho )=0, \qquad \text {for }{\mathcal {L}}^1\text {-a.e. }\rho <1. \end{aligned}$$
    (5.7)

    Notice that

    $$\begin{aligned} \int _{B_1}|\nabla ({\tilde{v}}_k-m_k)|^p\,\hbox {d}{\mathcal {L}}^n\le \int _{B_1} |\nabla v_k|^p\,\hbox {d}{\mathcal {L}}^n\le 1, \end{aligned}$$

    since \({\tilde{v}}_k\) is a truncation of v. From compactness theorems in \({{\,\textrm{SBV}\,}}\) (see for instance [8, Theorem 3.5]), we have that \({\tilde{v}}_k-m_k\) converges in \(L^p(B_1)\) and \({\mathcal {L}}^n\)-almost everywhere to a function \(v\in W^{1,p}(B_1)\), since \({\mathcal {H}}^{n-1}(J_{{\tilde{v}}_k})\) goes to 0 as \(k\rightarrow +\infty \). Moreover, for every \(\rho <1\),

    $$\begin{aligned} \int _{B_\rho }|\nabla v|^p\,\hbox {d}{\mathcal {L}}^n\le \liminf _{k}\int _{B_\rho }|\nabla {\tilde{v}}_k|^p\,\hbox {d}{\mathcal {L}}^n, \end{aligned}$$

    and

    $$\begin{aligned} \int _{B_\rho }|\nabla v|^p\,\hbox {d}{\mathcal {L}}^n\le \liminf _{k}\int _{B_\rho }|\nabla {\tilde{v}}_k|^p\,\hbox {d}{\mathcal {L}}^n\le \liminf _{k}\alpha _k(\rho )=\alpha (\rho ), \end{aligned}$$

    since by definition

    $$\begin{aligned} \int _{B_\rho }|\nabla v_k|^p\,\hbox {d}{\mathcal {L}}^n=\frac{r_k^{p-n}}{E_k}\int _{B_{\rho r_k}}|\nabla u_k|^p\,\hbox {d}{\mathcal {L}}^n\le \frac{r_k^{p-n}}{E_k}{\mathcal {F}}(u_k;B_{\rho r_k})\le \alpha _k(\rho ). \end{aligned}$$

    Finally, up to a subsequence,

    $$\begin{aligned} \lim _{k} \dfrac{r_k^{p-1}}{E_k}{\mathcal {L}}^n(\{v_k\ne {\tilde{v}}_k\})=0. \end{aligned}$$

    Indeed, by (5.5),

    $$\begin{aligned} \dfrac{r_k^{p-1}}{E_k}{\mathcal {L}}^n(\{v_k\ne {\tilde{v}}_k\})\le C\dfrac{r_k^{p-1}}{E_k}\varepsilon _k^{\frac{n}{n-1}}, \end{aligned}$$

    which tends to zero as long as \(r_k^{p-1}/E_k\) is bounded. On the other hand, if \(r_k^{p-1}/E_k\) diverges, we could use the fact that \(\varepsilon _k\le s^{-q} {\mathcal {F}}(u_k;B_{r_k})r_k^{1-n}\) and get

    $$\begin{aligned} \dfrac{r_k^{p-1}}{E_k}{\mathcal {L}}^n(\{v_k\ne {\tilde{v}}_k\})\le C \dfrac{r_k^{p-1}}{E_k}\left( \dfrac{E_k}{r_k^{p-1}}\right) ^{\frac{n}{n-1}} \end{aligned}$$

    which goes to zero. Using Fubini’s theorem we have (5.7).

    Let \({\tilde{u}}_k(x)=E_k^{1/p}{\tilde{v}}_k(\frac{x}{r_k})\), and for every \(t\in [0,1]\) we define

    $$\begin{aligned} {\tilde{\alpha }}_k(t)=\dfrac{{\mathcal {F}}({\tilde{u}}_k;B_{t r_{k}})}{{\mathcal {F}}(u_k,B_{r_k})}. \end{aligned}$$

    The \({\tilde{\alpha }}_k\) functions are also monotone and bounded: the jump set of \({\tilde{u}}_k\) is contained in \(J_{u_k}\), therefore we can write

    $$\begin{aligned} {\tilde{\alpha }}_k(t)\le \alpha _k(t)+\dfrac{2\beta {\mathcal {H}}^{n-1}(J_{u_k}\cap B_{t r_{k}})}{{\mathcal {F}}(u_k;B_{r_k})}\le \left( 1+\dfrac{2}{s^q}\right) \alpha _k(t), \end{aligned}$$

    using the fact that \(u_k\in H_s\). As done for \(\alpha _k\), we can assume that \({\tilde{\alpha }}_k\) converges \({\mathcal {L}}^1\)-almost everywhere to a function \({\tilde{\alpha }}\).

  • Step 2: Let \(I\subset [0,1]\) be the set of values \(\rho \) for which (5.7) holds, \(\alpha _k\) and \({\tilde{\alpha }}_k\) converge and \(\alpha \) and \({\tilde{\alpha }}\) are continuous. Notice that \({\mathcal {L}}^1(I)=1\). Fix \(\rho ,\rho '\in I\) with \(\rho<\rho '<1\) and let

    $$\begin{aligned} {\mathcal {I}}_k(\xi )=\beta E_k^{q/p-1}r_k^{p-1}\int _{J_{\xi }\cap ( B_{\rho '}\setminus B_\rho )}\left( {\overline{\xi }}^q+{\underline{\xi }}^q\right) \,\hbox {d}{\mathcal {H}}^{n-1}, \end{aligned}$$

    with \(\xi \in {{\,\textrm{SBV}\,}}(B_1)\). Let \(w\in W^{1,p}(B_1)\) and consider \(\eta \) a smooth cutoff function supported on \(B_{\rho '}\) and identically equal to 1 in \(B_\rho \). Let

    $$\begin{aligned} \varphi _k=((w+m_k)\eta +{\tilde{v}}_k(1-\eta )) \chi _{B_{\rho '}}+v_k\chi _{B_1\setminus B_{\rho '}}. \end{aligned}$$

    We want to prove that

    $$\begin{aligned} {\tilde{\alpha }}_k(\rho ')-{\tilde{\alpha }}_k(\rho )\ge \int _{B_{\rho '}\setminus B_\rho } |\nabla {\tilde{v}}_k|^p\,\hbox {d}{\mathcal {L}}^n+{\mathcal {I}}_k({\tilde{v}}_k), \end{aligned}$$
    (5.8)

    and

    $$\begin{aligned} \alpha _k(\rho ')\le R_k+\int _{B_{\rho '}}|\nabla \varphi _k|^p\,\hbox {d}{\mathcal {L}}^n+{\mathcal {I}}_k(\varphi _k), \end{aligned}$$
    (5.9)

    where \(R_k\) goes to zero as k goes to infinity. We immediately compute

    $$\begin{aligned} \begin{aligned} {\tilde{\alpha }}_k(\rho ')-{\tilde{\alpha }}_k(\rho )&={\mathcal {F}}(u_k;B_{r_k})^{-1}\left[ \int _{B_{\rho ' r_{k}}\cap B_{\rho r_{k}}}|\nabla {\tilde{u}}_k|^p\,\hbox {d}{\mathcal {L}}^n\right. \\&\quad \left. +\beta \int _{J_{{\tilde{u}}_k}\cap (B_{\rho ' r_{k}}\setminus B_{\rho r_{k}})}\left( \overline{{\tilde{u}}_k}^q +\underline{{\tilde{u}}_k}^q\right) \,\hbox {d}{\mathcal {H}}^{n-1}\right] \\&\quad +{\mathcal {F}}(u_k;B_{r_k})^{-1}{\mathcal {L}}^n(\{{\tilde{u}}_k>0\}\cap (B_{\rho ' r_k}\setminus B_{\rho r_k}))\\&\ge \int _{B_{\rho '}\setminus B_\rho } |\nabla {\tilde{v}}_k|^p\,\hbox {d}{\mathcal {L}}^n+E_k^{q/p-1}r_k^{p-1} \beta \int _{J_{{\tilde{v}}_k}\cap ( B_{\rho '}\setminus B_\rho )} \left( \overline{{\tilde{v}}_k}^q +\underline{{\tilde{v}}_k}^q\right) \,\hbox {d}{\mathcal {H}}^{n-1}, \end{aligned} \end{aligned}$$

    and then we have (5.8). Now let \(\psi _k=E_k^{1/p}\varphi _k(x/r_k)\) and observe that \(\psi _k\) coincides with \(u_k\) outside \(B_{\rho ' r_k}\). We get from the local minimality of \(u_k\) that

    $$\begin{aligned} \begin{aligned} {\mathcal {F}}(u_k;B_{r_k})\le {\mathcal {F}}(\psi _k;B_{r_k})&={\mathcal {F}}(\psi _k;B_{\rho ' r_k})+\beta \int _{\{u_k\ne {\tilde{u}}_k\}\cap \partial B_{\rho ' r_k}}\left( \underline{\psi _k}^{q}+\overline{\psi _k}^{\,q}\right) \,\hbox {d}{\mathcal {H}}^{n-1}\\&\quad +{\mathcal {F}}(u_k;B_{r_k}\setminus \overline{B_{\rho ' r_k}}\,)\\&\le {\mathcal {F}}(\psi _k;B_{\rho ' r_k}) +2\beta r_k^{n-1}{\mathcal {H}}^{n-1}(\{v_k\ne {\tilde{v}}_k\}\cap \partial B_{\rho '})\\&\quad +{\mathcal {F}}(u_k;B_{r_k}\setminus \overline{B_{\rho ' r_k}}\,). \end{aligned} \end{aligned}$$
    (5.10)

    So, in particular, we have

    $$\begin{aligned} \begin{aligned} {\mathcal {F}}(u_k;B_{\rho 'r_k})&={\mathcal {F}}(u_k;B_{r_k}) -{\mathcal {F}}(u_k;B_{r_k}\setminus \overline{B_{\rho 'rfcfc_k}}) -\beta \int _{J_{u_k}\cap \partial B_{\rho 'r_k}} \left( \overline{u_k}^q+\underline{u_k}^q\right) \,\hbox {d}{\mathcal {H}}^{n-1}\\&\le \,2\beta r_k^{n-1}{\mathcal {H}}^{n-1}(\{v_k\ne {\tilde{v}}_k\}\cap \partial B_{\rho '})+{\mathcal {F}}(\psi _k;B_{\rho ' r_k}). \end{aligned} \end{aligned}$$

    Dividing by \({\mathcal {F}}(u_k;B_{r_k})\) and using (5.7) we get

    $$\begin{aligned} \alpha _k(\rho ')\le R_k +r_k^{p-n}E_k^{-1}{\mathcal {F}}(\psi _k;B_{\rho 'r_k}). \end{aligned}$$

    With appropriate rescalings we have

    $$\begin{aligned} \begin{aligned} r_k^{p-n}E_k^{-1}{\mathcal {F}}(\psi _k;B_{\rho ' r_k})&= \int _{B_{\rho '}}|\nabla \varphi _k|^p\,\hbox {d}{\mathcal {L}}^n+ {\mathcal {I}}_k(\varphi _k)+r_k^p E_k^{-1}{\mathcal {L}}^n(\{\varphi _k>0\}\cap B_{\rho '}). \end{aligned} \end{aligned}$$

    From (5.2) and the definition of \(E_k\), we have

    $$\begin{aligned} r_k^p E_k^{-1}{\mathcal {L}}^n(\{\varphi _k>0\}\cap B_{\rho '})\le \omega _n r_k^{1/2}, \end{aligned}$$

    and then we get (5.9).

  • Step 3: We want to prove that for every \(\varphi \in W^{1,p}(B_1)\) such that \(v-\varphi \) is supported on \(B_{\rho }\), we have

    $$\begin{aligned} \alpha (\rho ')\le \int _{B_\rho }|\nabla \varphi |^p\,\hbox {d}{\mathcal {L}}^n+ C\left[ {{\tilde{\alpha }}}(\rho ')-{{\tilde{\alpha }}}(\rho )\right] + C\int _{B_{\rho '}\setminus B_\rho }|\nabla \varphi |^p\,\hbox {d}{\mathcal {L}}^n, \end{aligned}$$
    (5.11)

    where C does not depend on either \(\rho \) or \(\rho '\). From the definition of \(\varphi _k\), we have that on \(B_\rho \)

    $$\begin{aligned} \nabla \varphi _k=\nabla w \end{aligned}$$

    and on \(B_{\rho '}\setminus B_\rho \)

    $$\begin{aligned} \nabla \varphi _k=\eta \nabla w+(w+m_k-{\tilde{v}}_k)\nabla \eta +\nabla {\tilde{v}}_k(1-\eta ), \end{aligned}$$

    so that

    $$\begin{aligned} \begin{aligned} \int _{B_{\rho '}}|\nabla \varphi _k|^p\,\hbox {d}{\mathcal {L}}^n&\le \int _{B_\rho }|\nabla w|^p\,\hbox {d}{\mathcal {L}}^n\\&\quad +C\left[ \int _{B_{\rho '}\setminus B_\rho }|\nabla {\tilde{v}}_k|^p\,\hbox {d}{\mathcal {L}}^n+\int _{B_{\rho '}\setminus B_\rho }(|\nabla w|^p+|w+m_k-{\tilde{v}}_k|^p|\nabla \eta |^p)\,\hbox {d}{\mathcal {L}}^n\right] .\end{aligned}\nonumber \\ \end{aligned}$$
    (5.12)

    We split the proof into two cases: either

    $$\begin{aligned} \limsup _{k}E_k>0 \end{aligned}$$
    (5.13)

    or

    $$\begin{aligned} \lim _{k}E_k=0. \end{aligned}$$
    (5.14)

    Assume (5.13) occurs. Notice that \(s\le u_k\le 1\) for every k, then by definition we have that, for every k, \(s\le E_k^{1/p} {\tilde{v}}_k\le 1\) and, since \(m_k\) is a median of \(v_k\), \(0\le E_k^{1/p} m_k\le 1\). In particular we have that

    $$\begin{aligned} |{\tilde{v}}_k-m_k|\le \frac{2}{E_k^{1/p}}, \end{aligned}$$

    passing to the limit when k goes to infinity we have that

    $$\begin{aligned} \Vert v\Vert _{\infty }\le \liminf _{k}\frac{2}{E_k^{1/p}}<+\infty \quad {\mathcal {L}}^n\text {-a.e.} \end{aligned}$$

    then v belongs to \(L^\infty (B_1)\) and there exists a positive constant C independent of k, and a natural number \({\overline{k}}\) such that

    $$\begin{aligned} |v+m_k-{\tilde{v}}_k|\le \frac{C}{E_k^{1/p}}\le \frac{C}{s} {\tilde{v}}_k\quad {\mathcal {L}}^n\text {-a.e.} \end{aligned}$$

    for all \(k>{\overline{k}}\). Let \(\varphi \in W^{1,p}(B_1)\) with \(v-\varphi \) supported on \(B_{\rho }\), and let \(w=\varphi \) in the definition of \(\varphi _k\), then, for every \(k>{\overline{k}}\), we have

    $$\begin{aligned} |\varphi _k|=|{\tilde{v}}_k+(v+m_k- {\tilde{v}}_k)\eta |\le C {\tilde{v}}_k \end{aligned}$$
    (5.15)

    \({\mathcal {L}}^n\)-a.e. on \(B_{\rho '}\setminus B_\rho \). From (5.15) we have that

    $$\begin{aligned} {\mathcal {I}}_k(\varphi _k)\le C{\mathcal {I}}_k({\tilde{v}}_k). \end{aligned}$$
    (5.16)

    Notice, in addition, that (5.12) reads as

    $$\begin{aligned} \begin{aligned} \int _{B_{\rho '}}|\nabla \varphi _k|^p\,\hbox {d}{\mathcal {L}}^n&\le \int _{B_\rho }|\nabla \varphi |^p\,\hbox {d}{\mathcal {L}}^n\\&\quad +C\int _{B_{\rho '}\setminus B_\rho }|\nabla {\tilde{v}}_k|^p\,\hbox {d}{\mathcal {L}}^n+C\int _{B_{\rho '}\setminus B_\rho }|\nabla \varphi |^p\,\hbox {d}{\mathcal {L}}^n+ R_k. \end{aligned} \end{aligned}$$
    (5.17)

    finally joining  (5.9), (5.17), (5.16), and (5.8), we have

    $$\begin{aligned} \alpha _k(\rho ')\le \int _{B_\rho }|\nabla \varphi |^p\,\hbox {d}{\mathcal {L}}^n+C\left[ {\tilde{\alpha }}_k(\rho ')-{\tilde{\alpha }}_k(\rho )\right] +C\int _{B_{\rho '}\setminus B_\rho } |\nabla \varphi |^p\,\hbox {d}{\mathcal {L}}^n+R_k. \end{aligned}$$

    Letting k go to infinity we get (5.11).

Suppose now that (5.14) occurs. The functions \(|{\tilde{v}}_k-m_k|^p\), \(|v|^p\) are uniformly integrable, namely for every \(\varepsilon >0\) there exists a \(\sigma =\sigma _\varepsilon <\varepsilon \) such that if A is a measurable set with \(|A|<\sigma \), then

$$\begin{aligned} \int _{A}|{\tilde{v}}_k-m_k|^p\,\hbox {d}{\mathcal {L}}^n+\int _{A}|v|^p\,\hbox {d}{\mathcal {L}}^n<\varepsilon . \end{aligned}$$
(5.18)

Since \(v\in L^p(B_1)\), we can find \(M>1/\varepsilon \) such that

$$\begin{aligned} |\{|v|>M\}|<\sigma . \end{aligned}$$
(5.19)

Setting \(w=\varphi _M=\max \{-M,\min \{\varphi ,M\}\}\), then (5.12) reads as

$$\begin{aligned} \begin{aligned} \int _{B_{\rho '}}|\nabla \varphi _k|^p\,\hbox {d}{\mathcal {L}}^n&\le \int _{B_\rho \cap \{|\varphi |\le M\}}|\nabla \varphi |^p\,\hbox {d}{\mathcal {L}}^n+ C\int _{\left( B_{\rho '}\setminus B_\rho \right) \cap \{|\varphi |\le M\}}|\nabla \varphi |^p\,\hbox {d}{\mathcal {L}}^n\\&\quad +C\left[ \int _{B_{\rho '}\setminus B_\rho } |\nabla {\tilde{v}}_k|^p\,\hbox {d}{\mathcal {L}}^n+\int _{B_{\rho '}\setminus B_\rho }|\varphi _M+m_k-{\tilde{v}}_k|^p|\nabla \eta |^p\,\hbox {d}{\mathcal {L}}^n\right] .\end{aligned}\nonumber \\ \end{aligned}$$
(5.20)

We can estimate the last integral as follows

$$\begin{aligned} \begin{aligned} \int _{B_{\rho '}\setminus B_\rho }|\varphi _M+m_k-{\tilde{v}}_k|^p|\nabla \eta |^p\,\hbox {d}{\mathcal {L}}^n&\le C\varepsilon + \int _{\left( B_{\rho '}\setminus B_\rho \right) \cap \{|v|\le M\}}|v+m_k-{\tilde{v}}_k|^p|\nabla \eta |^p\,\hbox {d}{\mathcal {L}}^n].\\&=C\varepsilon +R_k, \end{aligned}\nonumber \\ \end{aligned}$$
(5.21)

where we used (5.19) and (5.18), and C only depends on \(\rho \) and \(\rho '\). Furthermore, we have

$$\begin{aligned} {\mathcal {I}}_k(\varphi _k)\le R_k + C{\mathcal {I}}_k({\tilde{v}}_k). \end{aligned}$$
(5.22)

Indeed, as before, \(|{\tilde{v}}_k-m_k|\le C {\tilde{v}}_k\), while

$$\begin{aligned} \begin{aligned} E_k^{q/p-1}r_k^{p-1}\int _{J_{{\tilde{v}}_k}\cap \left( B_{\rho '}\setminus B_\rho \right) }|\varphi _M|^q\,\hbox {d}{\mathcal {H}}^{n-1}&\le M^q E_k^{q/p-1}r_k^{p-1}{\mathcal {H}}^{n-1}\left( {J_{{\tilde{v}}_k}\cap \left( B_{\rho '}\setminus B_\rho \right) }\right) \\&\le M^q E_k^{\frac{q}{p}}\frac{r_k^{p-1}\varepsilon _k}{E_k}\\&\le \frac{M^q}{s^q}E_k^{\frac{q}{p}}, \end{aligned} \end{aligned}$$

which goes to 0 as \(k\rightarrow \infty \). Finally, joining (5.9), (5.20), (5.21), (5.22), and (5.8), we have

$$\begin{aligned} \alpha _k(\rho ')\le & {} R_k + \int _{B_\rho \cap \{|\varphi |\le M\}}|\nabla \varphi |^p+C\left[ {{\tilde{\alpha }}}(\rho ') -{{\tilde{\alpha }}}(\rho )\right] \\ {}{} & {} +C\int _{\left( B_{\rho '}\setminus B_\rho \right) \cap \{|\varphi |\le M\}}|\nabla \varphi |^p\,\hbox {d}{\mathcal {L}}^n+C\varepsilon . \end{aligned}$$

Taking the limit as k tends to infinity, and then the limit as \(\varepsilon \) tends to 0, we get (5.11).

We are now in a position to prove that v is p-harmonic: taking the limit as \(\rho '\) tends to \(\rho \) in (5.11), we have that if \(\varphi \in W^{1,p}(B_1)\), with \(v-\varphi \) supported on \(B_\rho \),

$$\begin{aligned} \int _{B_{\rho }}|\nabla v|^p\,\hbox {d}{\mathcal {L}}^n\le \alpha (\rho )\le \int _{B_{\rho }}|\nabla \varphi |^p\,\hbox {d}{\mathcal {L}}^n, \end{aligned}$$

for every \(\rho \in I\), therefore v is p-harmonic in \(B_1\). Notice that this implies that v is a locally Lipschitz function (see [2, Theorem 7.12]). Moreover, if \(\varphi =v\), we have

$$\begin{aligned} \int _{B_{\rho }}|\nabla v|^p\,\hbox {d}{\mathcal {L}}^n=\alpha (\rho ) \end{aligned}$$

for every \(\rho \in I\), so that \(\alpha \) is continuous on the whole interval [0, 1], \(\alpha (1)=1\) and \(\alpha (\tau )=\lim _k\alpha _k(\tau )\ge \tau ^{n-1/2}\). Nevertheless, if \(\tau \) is sufficiently small this contradicts the fact that v is locally Lipschitz, since

$$\begin{aligned} \tau ^{n-\frac{1}{2}}\le \int _{B_{\tau }}|\nabla v|^p\,\hbox {d}{\mathcal {L}}^n\le C\, \tau ^{n}, \end{aligned}$$

where C is a positive constant depending only on n and p. \(\square \)

Proof of Theorem 1.2

Let u be a minimizer for the problem (1.4). By Corollary 3.4 there exist two positive constants \(C(\Omega ,\beta ,p,q), C_1(\Omega ,\beta ,p,q)\) such that if \(B_r(x)\subseteq {\mathbb {R}}^n\setminus \Omega \), then

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_u\cap B_r(x))\le C(\Omega ,\beta ,p,q)r^{n-1}, \end{aligned}$$

and if \(x\in K_u\)

$$\begin{aligned} {\mathcal {L}}^n(B_r(x)\cap \{u>0\})\ge C_1(\Omega ,\beta ,p,q) r^n. \end{aligned}$$

We now prove that there exists a positive constant \(c=c(\Omega ,\beta ,p,q)\) such that

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_u\cap B_r(x))\ge c(\Omega ,\beta ,p,q) r^{n-1} \end{aligned}$$
(5.23)

for every \(x\in K_u\) and \(B_r(x)\subset {\mathbb {R}}^n\setminus \Omega \). Assume by contradiction that there exists \(x\in J_u\) such that, for \(r>0\) small enough,

$$\begin{aligned} {\mathcal {H}}^{n-1}\left( J_u\cap B_r(x)\right) \le \varepsilon _0 r^{n-1}, \end{aligned}$$

where \(\varepsilon _0\) is the one in Lemma 5.3. Iterating Lemma 5.3 it can be proven (see [7, Theorem 5.1]) that

$$\begin{aligned} \lim _{r\rightarrow 0^+} r^{1-n}{\mathcal {F}}(u;B_r)=0, \end{aligned}$$

which, in particular, implies

$$\begin{aligned} \lim _{r\rightarrow 0^+} r^{1-n}\left[ \int _{B_r(x)}|\nabla u|^p\,\hbox {d}{\mathcal {L}}^n+{\mathcal {H}}^{n-1}\left( J_u\cap B_r(x)\right) \right] =0. \end{aligned}$$
(5.24)

By [8, Theorem 3.6], (5.24) implies that \(x\notin J_u\), which is a contradiction. Finally, if \(x\in K_u\) and

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_u\cap B_{2r}(x))\le \varepsilon _0 r^{n-1}, \end{aligned}$$

there exists \(y\in J_u\cap B_r(x)\) such that

$$\begin{aligned} {\mathcal {H}}^{n-1}\left( J_u\cap B_r(y)\right) \le \varepsilon _0 r^{n-1} \end{aligned}$$

which, again, is a contradiction. Then the assertion is proved. The density estimate (5.23) implies in particular that

$$\begin{aligned} K_u\setminus {\bar{\Omega }}\subset \left\{ x\in {\mathbb {R}}^n \left| \limsup _{r\rightarrow 0^+}\,r^{1-n}\left[ \int _{B_r(x)}|\nabla u|^p\,\hbox {d}{\mathcal {L}}^n+{\mathcal {H}}^{n-1}\left( J_u\cap B_r(x)\right) \right] >0\right. \right\} , \end{aligned}$$

hence \({\mathcal {H}}^{n-1}((K_u\setminus J_u)\setminus {\bar{\Omega }})=0\) (see for instance [8, Lemma 2.6]). \(\square \)

Remark 5.4

Let u be a minimizer for problem (1.4), then from Theorem 3.2 we have that the function \(u^*=(\beta \delta ^q)^{-1/p}u\) is an almost-quasi minimizer for the Mumford–Shah functional

$$\begin{aligned} MS(v)=\int _{{\mathbb {R}}^n}|\nabla v|^p\,\hbox {d}{\mathcal {L}}^n+{\mathcal {H}}^{n-1}(J_v) \end{aligned}$$

with the Dirichlet condition \(u^*=(\beta \delta ^q)^{-1/p}\) on \(\Omega \). If \(\Omega \) is sufficiently smooth we can apply the results in [4] to have that the density estimate for the jump set of minimizers holds up to the boundary of \(\Omega \).