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# A free boundary problem in thermal insulation with a prescribed heat source

P. Acampora, E. Cristoforoni, C. Nitsch, C. Trombetti

###### Abstract

We study the thermal insulation of a bounded body
$\Omega\subset\mathbb{R}^{n}$, under a prescribed heat source $f>0$, via a
bulk layer of insulating material. We consider a model of heat transfer
between the insulated body and the environment determined by convection; this
corresponds to Robin boundary conditions on the free boundary of the layer. We
show that a minimal configuration exists and that it satisfies uniform density
estimates.

MSC 2020: 35R35, 35J25, 35A01.

Keywords: Robin, thermal insulation, free boundary, heat source

##  1 Introduction

Let $\Omega\subseteq\mathbb{R}^{n}$ be an open bounded set with smooth
boundary, let $f\in L^{2}(\Omega)$ be a positive function and let
$\beta,\,C_{0}$ be positive constants. We consider the following energy
functional

$\mathcal{F}(A,v)=\int_{A}\lvert\nabla
v\rvert^{2}\,d\mathcal{L}^{n}+\beta\int_{\partial
A}v^{2}\,d\mathcal{H}^{n-1}-2\int_{\Omega}fv\,d\mathcal{L}^{n}+C_{0}\mathcal{L}^{n}(A\setminus\Omega),$
(1.1)

and the variational problem

$\inf\Set{\mathcal{F}(A,v)}{\begin{aligned} &A\supseteq\Omega\text{ open,
bounded and Lipschitz}\\\ &v\in W^{1,2}(A),\>v\geq 0\,\text{in
}A\end{aligned}}.$ (1.2)

This problem is related to the following thermal insulation problem: for a
given heat source $f$ distributed in a conductor $\Omega$, find the best
possible configuration of insulating material surrounding $\Omega$. A similar
problem has been studied in [2] and [8] for a thin insulating layer, and in
[6], [4] and [5] for a prescribed temperature in $\Omega$.

For a fixed open set $A$ with Lipschitz boundary, we have, via the direct
methods of the calculus of variations, that there exists $u_{A}\in W^{1,2}(A)$
such that

$\mathcal{F}(A,u_{A})\leq\mathcal{F}(A,v),$

for all $v\in W^{1,2}(A)$, with $v\geq 0$ in $A$. Furthermore $u_{A}$ solves
the following stationary problem, with Robin boundary condition on $\partial
A$. Precisely

$\begin{cases}-\Delta u_{A}=f&\text{in }\Omega,\\\\[3.0pt] \dfrac{\partial
u_{A}^{+}}{\partial\nu\hphantom{\scriptstyle{+}}}=\dfrac{\partial
u_{A}^{-}}{\partial\nu\hphantom{\scriptstyle{+}}}&\text{on
}\partial\Omega,\\\\[6.0pt] \Delta u_{A}=0&\text{in
}A\setminus\Omega,\\\\[3.0pt] \dfrac{\partial u_{A}}{\partial\nu}+\beta
u_{A}=0&\text{on }\partial A,\end{cases}$

where $u_{A}^{-}$ and $u_{A}^{+}$ denote the traces of $u_{A}$ on
$\partial\Omega$ in $\Omega$ and in $A\setminus\Omega$ respectively. That is

$\int_{A}\nabla u_{A}\cdot\nabla\varphi\,d\mathcal{L}^{n}+\beta\int_{\partial
A}u_{A}\varphi\,\mathcal{H}^{n-1}=\int_{\Omega}f\varphi\,d\mathcal{L}^{n},$
(1.3)

for all $\varphi\in W^{1,2}(A)$. The Robin boundary condition represents the
case when the heat transfer with the environment is conveyed by convection.

If for any couple $(A,v)$ with $A$ an open bounded set with Lipschitz boundary
containing $\Omega$ and $v\in W^{1,2}(A)$, $v\geq 0$ in $A$, we identify $v$
with $v\chi_{A}$, where $\chi_{A}$ is the characteristic function of $A$, and
the set $A$ with the support of $v$, then the energy functional (1.1) becomes

$\mathcal{F}(v)=\int_{\mathbb{R}^{n}}\lvert\nabla
v\rvert^{2}\,d\mathcal{L}^{n}+\beta\int_{J_{v}}\left(\overline{v}^{2}+\underline{v}^{2}\right)\,d\mathcal{H}^{n-1}-2\int_{\Omega}fv\,d\mathcal{L}^{n}+C_{0}\mathcal{L}^{n}(\set{v>0}\setminus\Omega),$
(1.4)

and the minimization problem (1.2) becomes

$\inf\Set{\mathcal{F}(v)}{v\in\operatorname{SBV}^{\frac{1}{2}}(\mathbb{R}^{n})\cap
W^{1,2}(\Omega)},$ (1.5)

where $\overline{v}$ and $\underline{v}$ are respectively the approximate
upper and lower limits of $v$, $J_{v}$ is the jump set and $\nabla v$ is the
absolutely continuous part of the derivative of $v$. See Section 2 for the
definitions.

We state the main results of this paper in the two following theorems.

######  Theorem 1.1.

Let $n\geq 2$, let $\Omega\subset\mathbb{R}^{n}$ be an open bounded set with
$C^{1,1}$ boundary, let $f\in L^{2}(\Omega)$, with $f>0$ almost everywhere in
$\Omega$. Assume in addition that, if $n=2$,

$\lVert
f\rVert_{2,\Omega}^{2}<C_{0}\lambda_{\beta}(B)\mathcal{L}^{2}(\Omega),$ (1.6)

where $B$ is a ball having the same measure of $\Omega$. Then problem (1.5)
admits a solution. Moreover, if $p>n$ and $f\in L^{p}(\Omega)$, then there
exists a positive constant $C=C(\Omega,f,p,\beta,C_{0})$ such that if $u$ is a
minimizer to problem (1.5) then

$\lVert u\rVert_{\infty}\leq C.$

######  Theorem 1.2.

Let $n\geq 2$, let $\Omega\subset\mathbb{R}^{n}$ be an open bounded set with
$C^{1,1}$ boundary, let $p>n$ and let $f\in L^{p}(\Omega)$, with $f>0$ almost
everywhere in $\Omega$. Assume in addition that, if $n=2$ condition (1.6)
holds true. Then there exist positive constants
$\delta_{0}=\delta_{0}(\Omega,f,p,\beta,C_{0})$,
$c=c(\Omega,f,p,\beta,C_{0})$, $C=C(\Omega,f,p,\beta,C_{0})$ such that if $u$
is a minimizer to problem (1.5) then

$u\geq\delta_{0}\qquad\text{$\mathcal{L}^{n}$-a.e. in }\set{u>0},$

and the jump set $J_{u}$ satisfies the density estimates

$cr^{n-1}\leq\mathcal{H}^{n-1}(J_{u}\cap B_{r}(x))\leq Cr^{n-1},$

with $x\in\overline{J_{u}}$, and $0<r<d(x,\partial\Omega)$. In particular, we
have

$\mathcal{H}^{n-1}(\overline{J_{u}}\setminus J_{u})=0.$

We refer to Section 2 for the definitions of $\lambda_{\beta}(B)$ in (1.6),
and the distance $d(x,\partial\Omega)$ in Theorem 1.2.

Section 3 is devoted to the proof Theorem 1.1, while Section 4 is devoted to
the proof Theorem 1.2.

We notice that the assumptions on the function $f$ do not seem to be sharp.
Indeed, it is well known that (see for instance [10, Theorem 8.15]), in the
more regular case, the assumption $f\in L^{p}(\Omega)$ with $p>n/2$ ensures
the boundedness of solutions to equation (1.3).

##  2 Notation and tools

In this section we recall some definitions and proprieties of the spaces
$\operatorname{BV}$, $\operatorname{SBV}$, and
$\operatorname{SBV}^{\frac{1}{2}}$. We refer to [1], [3], [9] for a deep study
of the properties of these functions.

In the following, given an open set $\Omega\subseteq\mathbb{R}^{n}$ and $1\leq
p\leq\infty$, we will denote the $L^{p}(\Omega)$ norm of a function $v\in
L^{p}(\Omega)$ as $\lVert v\rVert_{p,\Omega}$, in particular when
$\Omega=\mathbb{R}^{n}$ we will simply write $\lVert v\rVert_{p}=\lVert
v\rVert_{p,\mathbb{R}^{n}}$.

######  Definition  ($\operatorname{BV}$).

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
$u\in\operatorname{BV}(\mathbb{R}^{n})$ if its distributional derivative is a
Radon measure, namely

$\int_{\Omega}u\,\frac{\partial\varphi}{\partial
x_{i}}=\int_{\Omega}\varphi\,dD_{i}u\qquad\forall\varphi\in
C^{\infty}_{c}(\mathbb{R}^{n}),$

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

$\lVert u\rVert_{\operatorname{BV}(\mathbb{R}^{n})}=\lVert u\rVert_{1}+\lvert
Du\rvert(\mathbb{R}^{n}).$

######  Definition .

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

$E^{(1)}=\Set{x\in\mathbb{R}^{n}}{\lim_{r\to
0^{+}}\dfrac{\mathcal{L}^{n}(B_{r}(x)\cap E)}{\mathcal{L}^{n}(B_{r}(x))}=1},$

and the _set of points of density 0 for $E$_ as

$E^{(0)}=\Set{x\in\mathbb{R}^{n}}{\lim_{r\to
0^{+}}\dfrac{\mathcal{L}^{n}(B_{r}(x)\cap E)}{\mathcal{L}^{n}(B_{r}(x))}=0}.$

Moreover, we define the _essential boundary_ of $E$ as

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

######  Definition  (Approximate upper and lower limits).

Let $u\colon\mathbb{R}^{n}\to\mathbb{R}$ be a measurable function. We define
the _approximate upper and lower limits_ of $u$, respectively, as

$\overline{u}(x)=\inf\Set{t\in\mathbb{R}}{\limsup_{r\to
0^{+}}\dfrac{\mathcal{L}^{n}(B_{r}(x)\cap\set{u>t})}{\mathcal{L}^{n}(B_{r}(x))}=0},$

and

$\underline{u}(x)=\sup\Set{t\in\mathbb{R}}{\limsup_{r\to
0^{+}}\dfrac{\mathcal{L}^{n}(B_{r}(x)\cap\set{u<t})}{\mathcal{L}^{n}(B_{r}(x))}=0}.$

We define the _jump set_ of $u$ as

$J_{u}=\Set{x\in\mathbb{R}^{n}}{\underline{u}(x)<\overline{u}(x)}.$

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 $\varepsilon>0$,

$\limsup_{r\to 0^{+}}\dfrac{\mathcal{L}^{n}(B_{r}(x)\cap\set{\lvert
u-l\rvert\geq\varepsilon)}}{\mathcal{L}^{n}(B_{r}(x))}=0.$

If $u\in\operatorname{BV}(\mathbb{R}^{n})$, 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 $M_{i}$. Futhermore, 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

$H^{+}_{\nu_{u}}=\set{y\in\mathbb{R}^{n}}{\nu_{u}(x)\cdot(y-x)\geq 0}$

and

$H^{-}_{\nu_{u}}=\set{y\in\mathbb{R}^{n}}{\nu_{u}(x)\cdot(y-x)\leq 0}$

respectively.

######  Definition .

Let $E\subseteq\mathbb{R}^{n}$ be a measurable set and let
$\Omega\subseteq\mathbb{R}^{n}$ be an open set. We define the _relative
perimeter_ of $E$ inside $\Omega$ as

$P(E;\Omega)=\sup\Set{\int_{E}\operatorname{div}\varphi\,d\mathcal{L}^{n}}{\begin{aligned}
\varphi\in&\>C^{1}_{c}(\Omega,\mathbb{R}^{n})\\\ &\lvert\varphi\rvert\leq
1\end{aligned}}.$

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

######  Theorem 2.1 (Relative Isoperimetric Inequality).

Let $\Omega$ be an open, bounded, connected set with Lipschitz boundary. Then
there exists a positive constants $C=C(\Omega)$ such that

$\min\Set{\mathcal{L}^{n}(\Omega\cap E),\mathcal{L}^{n}(\Omega\setminus
E)}^{\frac{n-1}{n}}\leq CP(E;\Omega),$

for every set $E$ of finite perimeter.

See for instance [13] for the proof of this theorem.

######  Theorem 2.2.

Let $\Omega$ be an open, bounded, connected set with Lipschitz boundary. Then
there exists a constant $C=C(\Omega)>0$ such that

$\mathcal{H}^{n-1}(\partial^{*}E\cap\partial\Omega)\leq
C\mathcal{H}^{n-1}(\partial^{*}E\cap\Omega)$

for every set of finite perimeter $E\subset\Omega$ with
$0<\mathcal{L}^{n}(E)\leq\mathcal{L}^{n}(\Omega)/2$.

We refer to [7, Theorem 2.3] for the proof of the previous theorem, observing
that if $\Omega$ is a Lipschitz set, then it is an admissible set in the sense
defined in [7](see [15, Remark 5.10.2]).

######  Theorem 2.3 (Decomposition of $\operatorname{BV}$ functions).

Let $u\in\operatorname{BV}(\mathbb{R}^{n})$. Then we have

$dDu=\nabla
u\,d\mathcal{L}^{n}+\lvert\overline{u}-\underline{u}\rvert\nu_{u}\,d\mathcal{H}^{n-1}\lfloor_{{\Huge
J_{u}}}+dD^{c}u,$

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 .

Let $v\in\operatorname{BV}(\mathbb{R}^{n})$, let
$\Gamma\subseteq\mathbb{R}^{n}$ be a $\mathcal{H}^{n-1}$-rectificable 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

$\lim_{r\to 0}\frac{1}{r^{n}}\int_{B_{r}^{\pm}(x)}\lvert
v(y)-\gamma_{\Gamma}^{\pm}(v)(x)\rvert\,d\mathcal{L}^{n}(y)=0,$

where

$B_{r}^{+}(x)=\set{y\in B_{r}(x)}{\nu(x)\cdot(y-x)>0},$
$B_{r}^{-}(x)=\set{y\in B_{r}(x)}{\nu(x)\cdot(y-x)<0}.$

######  Remark .

Notice that, by [1, 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

$\gamma_{J_{v}}^{+}(v)(x)=\overline{v}(x)\qquad\gamma_{J_{v}}^{-}(v)(x)=\underline{v}(x)$

for $\mathcal{H}^{n-1}$-a.e. $x\in J_{v}$.

We now focus our attention on the $\operatorname{BV}$ functions whose Cantor
parts vanish.

######  Definition  ($\operatorname{SBV}$).

Let $u\in\operatorname{BV}(\mathbb{R}^{n})$. We say that $u$ is a _special
function of bounded variation_ and we write
$u\in\operatorname{SBV}(\mathbb{R}^{n})$ if $D^{c}u=0$.

For $\operatorname{SBV}$ functions we have the following.

######  Theorem 2.4 (Chain rule).

Let $g\colon\mathbb{R}\to\mathbb{R}$ be a differentiable function. Then if
$u\in\operatorname{SBV}(\mathbb{R}^{n})$, we have

$\nabla g(u)=g^{\prime}(u)\nabla u.$

Furthermore, if $g$ is increasing,

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

while, if $g$ is decreasing,

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

We now give the definition of the following class of functions.

######  Definition  ($\operatorname{SBV}^{1/2}$).

Let $u\in L^{2}(\mathbb{R}^{n})$ be a non-negative function. We say that
$u\in\operatorname{SBV}^{\frac{1}{2}}(\mathbb{R}^{n})$ if
$u^{2}\in\operatorname{SBV}(\mathbb{R}^{n})$. In addition, we define

$\displaystyle
J_{u}:=J_{u^{2}}\qquad\quad\overline{u}:=\sqrt{\,\overline{u^{2}}\,}\qquad\quad\underline{u}:=\sqrt{\,\underline{u^{2}}\,}$
$\displaystyle\nabla u:=\frac{1}{2u}\nabla(u^{2})\chi_{\set{u>0}}$

Notice that this definition extends the validity of the Chain Rule to the
functions in $\operatorname{SBV}^{\frac{1}{2}}(\mathbb{R}^{n})$. We refer to
[3, Lemma 3.2] for the coherence of this definition.

######  Theorem 2.5 (Compactness in $\operatorname{SBV}^{1/2}$).

Let $u_{k}$ be a sequence in
$\operatorname{SBV}^{\frac{1}{2}}(\mathbb{R}^{n})$ and let $C>0$ be such that
for every $k\in\mathbb{N}$

$\int_{\mathbb{R}^{n}}\lvert\nabla
u_{k}\rvert^{2}\,d\mathcal{L}^{n}+\int_{J_{u_{k}}}\left(\overline{u}_{k}^{2}+\underline{u}_{k}^{2}\right)\,d\mathcal{H}^{n-1}+\int_{\mathbb{R}^{n}}u_{k}^{2}\,d\mathcal{L}^{n}<C$

Then there exists $u\in\operatorname{SBV}^{\frac{1}{2}}(\mathbb{R}^{n})$ and a
subsequence $u_{k_{j}}$ such that

  * •

_Compactness:_

$u_{k_{j}}\xrightarrow{L^{2}_{\operatorname{loc}}(\mathbb{R}^{n})}u$

  * •

_Lower semicontinuity:_ for every open set $\Omega$ we have

$\int_{\Omega}\lvert\nabla
u\rvert^{2}\,d\mathcal{L}^{n}\leq\liminf_{j\to+\infty}\int_{\Omega}\lvert\nabla
u_{k_{j}}\rvert^{2}\,d\mathcal{L}^{n}$

and

$\int_{J_{u}\cap\Omega}\left(\overline{u}^{2}+\underline{u}^{2}\right)\,d\mathcal{H}^{n-1}\leq\liminf_{j\to+\infty}\int_{J_{u_{k_{j}}}\cap\Omega}\left(\overline{u}_{k_{j}}^{2}+\underline{u}_{k_{j}}^{2}\right)\,d\mathcal{H}^{n-1}$

######  Definition  (Robin Eigenvalue).

Let $\Omega\subseteq\mathbb{R}^{n}$ be an open bounded set with Lipschitz
boundary, let $\beta>0$. We define $\lambda_{\beta}(\Omega)$ as

$\lambda_{\beta}(\Omega)=\inf\Set{\dfrac{\displaystyle\int_{\Omega}\lvert\nabla
v\rvert^{2}\,d\mathcal{L}^{n}+\beta\int_{\partial\Omega}v^{2}\,d\mathcal{H}^{n-1}}{\displaystyle\int_{\Omega}v^{2}\,d\mathcal{L}^{n}}}{v\in
W^{1,2}(\Omega)\setminus\set{0}}.$ (2.1)

######  Remark .

Standard tools of calculus of variation ensures that the infimum in (2.1) is
achieved, see for instance.

######  Lemma .

For every $0<r<R$, the following inequality holds

$\lambda_{\beta}(B_{r})\leq\left(\dfrac{\mathcal{L}^{n}(B_{R})}{\mathcal{L}^{n}(B_{r})}\right)^{\frac{2}{n}}\lambda_{\beta}(B_{R}),$

where $B_{R}$ and $B_{r}$ are balls with radii $R$ and $r$ respectively.

###### Proof.

Let $\varphi$ be a minimum of (2.1) for $\Omega=B_{R}$ and with
$\lVert\varphi\rVert_{2,B_{R}}=1$. We define

$w(x)=\varphi\left(\frac{R}{r}x\right)\qquad\forall x\in B_{r}.$

Therefore,

$\begin{split}\lambda_{\beta}(B_{r})&\leq\dfrac{\displaystyle\int_{B_{r}}\lvert\nabla
w(x)\rvert^{2}\,d\mathcal{L}^{n}(x)+\int_{\partial
B_{r}}w(x)^{2}\,d\mathcal{H}^{n-1}(x)}{\displaystyle\int_{B_{r}}w(x)^{2}\,d\mathcal{L}^{n}(x)}\\\\[7.0pt]
&=\dfrac{\displaystyle\left(\frac{r}{R}\right)^{n-2}\int_{B_{R}}\lvert\nabla\varphi(y)\rvert^{2}\,d\mathcal{L}^{n}(y)+\left(\frac{r}{R}\right)^{n-1}\int_{\partial
B_{R}}\varphi(y)^{2}\,d\mathcal{H}^{n-1}(y)}{\left(\dfrac{r}{R}\right)^{n}}.\end{split}$

Since $r/R<1$, by minimality of $\varphi$, we get

$\lambda_{\beta}(B_{r})\leq\dfrac{\displaystyle\left(\frac{r}{R}\right)^{n-2}}{\displaystyle\left(\dfrac{r}{R}\right)^{{n}\hphantom{\\!2}}}\lambda_{\beta}(B_{R})=\left(\dfrac{\mathcal{L}^{n}(B_{r})}{\mathcal{L}^{n}(B_{R})}\right)^{-\frac{2}{n}}\lambda_{\beta}(B_{R}).$

∎

Let $\beta,m>0$, and let us denote by

$\Lambda_{\beta,m}=\inf\Set{\dfrac{\displaystyle\int_{\mathbb{R}^{n}}\\!\lvert\nabla
v\rvert^{2}\,d\mathcal{L}^{n}+\beta\int_{J_{v}}\\!\\!\left(\underline{v}^{2}+\overline{v}^{2}\right)\,d\mathcal{H}^{n-1}}{\displaystyle\int_{\mathbb{R}^{n}}v^{2}\,d\mathcal{L}^{n}}}{\begin{aligned}
&v\in\operatorname{SBV}^{\frac{1}{2}}(\mathbb{R}^{n})\setminus{\\{0\\}}\\\
&\mathcal{L}^{n}\left(\set{v>0}\right)\leq m\end{aligned}}.$

Here we state a theorem, referring to [3, Theorem 5] for the proof.

######  Theorem 2.6.

Let $B\subseteq\mathbb{R}^{n}$ be a ball of volume $m$. Then

$\Lambda_{\beta,m}=\lambda_{\beta}(B).$

We will denote by $d(x,\partial\Omega)$ the distance between
$x\in\mathbb{R}^{n}$ and the boundary $\partial\Omega$, and for every
$\varepsilon>0$ we define

$\Omega_{\varepsilon}=\Set{x\in\Omega}{d(x,\partial\Omega)>\varepsilon}.$

We will use the following result.

######  Proposition .

Let $\Omega$ be an open bounded set with $C^{1,1}$ boundary, then there exist
a constant $C=C(\Omega)>0$ and a $\varepsilon_{0}=\varepsilon_{0}(\Omega)>0$
such that

$\mathcal{L}^{n}(\Omega\setminus\Omega_{\varepsilon})\leq
C\varepsilon\qquad\qquad\forall\varepsilon<\varepsilon_{0}.$

###### Proof.

It is well known (see for instance [12, Theorem 17.5]) that there exist a
constant $C=C(\Omega)$ and $\varepsilon_{0}=\varepsilon_{0}(\Omega)>0$ such
that

$P(\Omega_{\varepsilon})=P(\Omega)+C(\Omega)\,\varepsilon+O(\varepsilon^{2}),$

for every $0<\varepsilon<\varepsilon_{0}$. Let $r(x)=d(x,\partial\Omega)$ be
the distance from the boundary of $\Omega$. By coarea formula we have

$\mathcal{L}^{n}(\Omega\setminus\Omega_{\varepsilon})=\int_{\set{0<r<\varepsilon}}\,d\mathcal{L}^{n}=\int_{0}^{\varepsilon}P(\Omega_{t})\,dt\leq
C(\Omega)\varepsilon.$

∎

##  3 Existence of minimizers

In this section we prove Theorem 1.1: in Section 3 we prove the existence of a
minimizer to problem (1.5); in Section 3 we prove the $L^{\infty}$ estimate
for a minimizer.

In this section we will assume that $\Omega\subseteq\mathbb{R}^{n}$ is an open
bounded set with $C^{1,1}$ boundary, that $f\in L^{2}(\Omega)$ is a positive
function and that $\beta,C_{0}$ are positive constants. We consider the energy
functional $\mathcal{F}$ defined in (1.4).

######  Lemma .

Let $n\geq 2$ and assume that, if $n=2$, condition (1.6) holds true. Then
there exist two positive constants $c=c(\Omega,f,\beta,C_{0})$ and
$C=C(\Omega,f,\beta,C_{0})$ such that if
$v\in\operatorname{SBV}^{\frac{1}{2}}(\mathbb{R}^{n})\cap W^{1,2}(\Omega)$,
with $\mathcal{F}(v)\leq 0$ and $\Omega\subseteq\set{v>0}$, then

$\mathcal{L}^{n}(\Set{v>0})\leq c,$ (3.1) $\lVert v\rVert_{2}\leq C.$ (3.2)

###### Proof.

Let $B^{\prime}$ be a ball with the same measure as $\Set{v>0}$. By Theorem
2.6

$\begin{split}0\geq\mathcal{F}(v)&\geq\,\lambda_{\beta}\left(B^{\prime}\right)\int_{\mathbb{R}^{n}}v^{2}\,d\mathcal{L}^{n}-2\int_{\Omega}fv\,d\mathcal{L}^{n}\\\
&\hphantom{\geq}+C_{0}\mathcal{L}^{n}\left(\set{v>0}\setminus\Omega\right).\end{split}$

By Section 2 and Hölder inequality

$\begin{split}0\geq&\lambda_{\beta}(B)\left(\frac{\mathcal{L}^{n}(\Omega)}{\mathcal{L}^{n}(\Set{v>0})}\right)^{\frac{2}{n}}\lVert
v\rVert_{2}^{2}-2\lVert f\rVert_{2,\Omega}\lVert v\rVert_{2}\\\\[3.0pt]
&+C_{0}\mathcal{L}^{n}\left(\set{v>0}\setminus\Omega\right)\end{split}$ (3.3)

where $B$ is a ball with the same measure as $\Omega$. Obviously (3.3) implies
that

$\lVert
f\rVert_{2,\Omega}^{2}-\lambda_{\beta}(B)\left(\frac{\mathcal{L}^{n}(\Omega)}{\mathcal{L}^{n}(\Set{v>0})}\right)^{\frac{2}{n}}C_{0}\mathcal{L}^{n}\left(\set{v>0}\setminus\Omega\right)\geq
0.$

Let $M=\mathcal{L}^{n}(\set{v>0})$, and notice that, since
$\Omega\subseteq\set{v>0}$,

$\mathcal{L}^{n}\left(\set{v>0}\setminus\Omega\right)=M-\mathcal{L}^{n}(\Omega),$

therefore

$\lVert f\rVert_{2,\Omega}^{2}\geq
C_{0}\lambda_{\beta}(B)\left(\mathcal{L}^{n}(\Omega)\right)^{\frac{2}{n}}\left(M^{1-\frac{2}{n}}-M^{-\frac{2}{n}}\mathcal{L}^{n}(\Omega)\right).$

This implies (taking into account (1.6) if $n=2$) that there exists
$c=c(\Omega,f,\beta,C_{0})>0$ such that

$\mathcal{L}^{n}(\set{v>0})<c.$

Finally observe that by (3.3) it follows

$\lVert v\rVert_{2}\leq C(M),$ (3.4)

where

$\begin{split}C(M)&=\frac{M^{\frac{2}{n}}\left(\lVert
f\rVert_{2,\Omega}+\sqrt{\lVert
f\rVert_{2,\Omega}^{2}-C_{0}\lambda_{\beta}(B)\left(\dfrac{\mathcal{L}^{n}(\Omega)}{M}\right)^{\frac{2}{n}}\left(M-\mathcal{L}^{n}(\Omega)\right)}\right)}{\vphantom{\big{(}}\lambda_{\beta}(B)\mathcal{L}^{n}(\Omega)}\\\\[5.0pt]
&\leq\frac{2c^{\frac{2}{n}}\lVert
f\rVert_{2,\Omega}}{\lambda_{\beta}(B)\mathcal{L}^{n}(\Omega)}\end{split}$

∎

######  Remark .

Let $v\in\operatorname{SBV}^{\frac{1}{2}}(\mathbb{R}^{n})\cap
W^{1,2}(\Omega)$, it is always possible to choose a function $v_{0}$ such that
$v_{0}=v$ in $\mathbb{R}^{n}\setminus\Omega$,
$\mathcal{F}(v_{0})\leq\mathcal{F}(v)$, and $\Omega\subseteq\set{v_{0}>0}$.
Indeed the function $v_{0}\in W^{1,2}(\Omega)$, weak solution to the following
boundary value problem

$\begin{cases}-\Delta v_{0}=f&\text{in }\Omega,\\\
v_{0}=\gamma^{-}_{\partial\Omega}(v)&\text{on }\partial\Omega,\end{cases}$
(3.5)

satisfies

$\int_{\Omega}\nabla
v_{0}\cdot\nabla\varphi\,d\mathcal{L}^{n}=\int_{\Omega}f\varphi\,d\mathcal{L}^{n}$

for every $\varphi\in W^{1,2}_{0}(\Omega)$ and
$v_{0}=\gamma^{-}_{\partial\Omega}(v)$ on $\partial\Omega$ in the sense of the
trace. Then, extending $v_{0}$ to be equal to $v$ outside of $\Omega$, we have
that $\Omega\subset\Set{v_{0}>0}$ and $\mathcal{F}(v_{0})\leq\mathcal{F}(v)$.

######  Proposition  (Existence).

Let $n\geq 2$ and, if $n=2$, assume that condition (1.6) holds true. Then
there exists a solution to problem (1.5).

###### Proof.

Let $\\{u_{k}\\}$ be a minimizing sequence for problem (1.5). Without loss of
generality we may always assume that, for all $k\in\mathbb{N}$,
$\mathcal{F}(u_{k})\leq\mathcal{F}(0)=0$, and, by Section 3,
$\Omega\subseteq\set{u_{k}>0}$. Therefore we have

$\begin{split}0\geq\mathcal{F}(u_{k})&\geq\int_{\mathbb{R}^{n}}\lvert\nabla
u_{k}\rvert^{2}\,d\mathcal{L}^{n}+\beta\int_{J_{u_{k}}}\left(\overline{u_{k}}^{2}+\underline{u_{k}}^{2}\right)\,d\mathcal{H}^{n-1}-2\int_{\Omega}fv\,d\mathcal{L}^{n}\\\
&\geq\int_{\mathbb{R}^{n}}\lvert\nabla
u_{k}\rvert^{2}\,d\mathcal{L}^{n}+\beta\int_{J_{u_{k}}}\left(\overline{u_{k}}^{2}+\underline{u_{k}}^{2}\right)\,d\mathcal{H}^{n-1}-2\lVert
f\rVert_{2,\Omega}\lVert u_{k}\rVert_{2,\Omega}\,,\end{split}$

and by (3.2),

$\int_{\mathbb{R}^{n}}\lvert\nabla
u_{k}\rvert^{2}\,d\mathcal{L}^{n}+\beta\int_{J_{u_{k}}}\left(\overline{u_{k}}^{2}+\underline{u_{k}}^{2}\right)\,d\mathcal{H}^{n-1}\leq
C\lVert f\rVert_{2,\Omega}\,.$

Then we have that there exists a positive constant still denoted by $C$,
independent on the sequence $\\{u_{k}\\}$, such that

$\int_{\mathbb{R}^{n}}\lvert\nabla
u_{k}\rvert^{2}\,d\mathcal{L}^{n}+\int_{J_{u_{k}}}\left(\overline{u}_{k}^{2}+\underline{u}_{k}^{2}\right)\,d\mathcal{H}^{n-1}+\int_{\mathbb{R}^{n}}u_{k}^{2}\,d\mathcal{L}^{n}<C.$
(3.6)

The compactness theorem in $\operatorname{SBV}^{\frac{1}{2}}(\mathbb{R}^{n})$
(Theorem 2.5), ensures that there exists a subsequence $\\{u_{k_{j}}\\}$ and a
function $u\in\operatorname{SBV}^{\frac{1}{2}}(\mathbb{R}^{n})\cap
W^{1,2}(\Omega)$, such that $u_{k_{j}}$ converges to $u$ strongly in
$L^{2}_{\operatorname{loc}}(\mathbb{R}^{n})$, weakly in $W^{1,2}(\Omega)$,
almost everywhere in $\mathbb{R}^{n}$ and

$\begin{split}\int_{\mathbb{R}^{n}}\lvert\nabla
u\rvert^{2}\,d\mathcal{L}^{n}\leq&\liminf_{j\to+\infty}\int_{\mathbb{R}^{n}}\lvert\nabla
u_{k_{j}}\rvert^{2}\,d\mathcal{L}^{n},\\\\[5.0pt]
\int_{J_{u}}\left(\overline{u}^{2}+\underline{u}^{2}\right)\,d\mathcal{H}^{n-1}\leq&\liminf_{j\to+\infty}\int_{J_{u_{k_{j}}}}\left(\overline{u}_{k_{j}}^{2}+\underline{u}_{k_{j}}^{2}\right)\,d\mathcal{H}^{n-1},\\\\[5.0pt]
\mathcal{L}^{n}(\set{u>0}\setminus\Omega)\leq&\liminf_{j\to+\infty}\mathcal{L}^{n}(\set{u_{k_{j}}>0}\setminus\Omega).\end{split}$

Finally we have

$\mathcal{F}(u)\leq\liminf_{j\to+\infty}\mathcal{F}(u_{k_{j}})=\inf\Set{\mathcal{F}(v)}{v\in\operatorname{SBV}^{\frac{1}{2}}(\mathbb{R}^{n})\cap
W^{1,2}(\Omega)},$

Therefore $u$ is a minimizer to problem (1.5).∎

######  Theorem 3.1 (Euler-Lagrange equation).

Let $u$ be a minimizer to problem (1.5), and let
$v\in\operatorname{SBV}^{1/2}(\mathbb{R}^{n})$ such that $J_{v}\subseteq
J_{u}$. Assume that there exists $t>0$ such that $\set{v>0}\subseteq\set{u>t}$
$\mathcal{L}^{n}$-a.e., and that

$\int_{J_{u}\setminus J_{v}}v^{2}\,d\mathcal{H}^{n-1}<+\infty.$

Then

$\int_{\mathbb{R}^{n}}\nabla u\cdot\nabla
v\,d\mathcal{L}^{n}+\beta\int_{J_{u}}\left(\overline{u}\gamma^{+}(v)+\underline{u}\gamma^{-}(v)\right)\,d\mathcal{H}^{n-1}=\int_{\Omega}fv\,d\mathcal{L}^{n},$
(3.7)

where $\gamma^{\pm}=\gamma_{J_{u}}^{\pm}$.

###### Proof.

Notice that since $v\in\operatorname{SBV}^{\frac{1}{2}}(\mathbb{R}^{n})$ with
$J_{v}\subseteq J_{u}$ we have that
$v\in\operatorname{SBV}^{\frac{1}{2}}(\mathbb{R}^{n})\cap W^{1,2}(\Omega)$.
Assume $v\in\operatorname{SBV}^{1/2}(\mathbb{R}^{n})\cap
L^{\infty}(\mathbb{R}^{n})$. If $s\in\mathbb{R}$, recalling that
$\set{v>0}\subseteq\set{u>t}$ $\mathcal{L}^{n}$-a.e.,

$u(x)+sv(x)=u(x)\geq 0\qquad\text{$\mathcal{L}^{n}$-a.e.}\>\forall
x\in\set{u\leq t},$

while, for $\lvert s\rvert$ small enough,

$u(x)+sv(x)\geq t-\lvert s\rvert\,\lVert v\rVert_{\infty}>0\qquad\forall
x\in\set{u>t}.$

Therefore we still have

$u+sv\in\operatorname{SBV}^{\frac{1}{2}}(\mathbb{R}^{n},\mathbb{R}^{+}).$

Moreover by minimality of $u$ we have, for every $\lvert s\rvert\leq s_{0}$

$\begin{split}\mathcal{F}(u)\leq&\mathcal{F}(u+sv)\\\\[3.0pt]
=&\int_{\mathbb{R}^{n}}\lvert\nabla u+s\nabla
v\rvert^{2}\,d\mathcal{L}^{n}+\\\\[3.0pt]
&+\int_{J_{u+sv}}\left[\left(\gamma^{+}(u)+s\gamma^{+}(v)\right)^{2}+\left(\gamma^{-}(u)+s\gamma^{-}(v)\right)^{2}\right]\,d\mathcal{H}^{n-1}+\\\\[3.0pt]
&-2\int_{\mathbb{R}^{n}}f(u+sv)\,d\mathcal{L}^{n}+C_{0}\mathcal{L}^{n}(\set{u>0}).\end{split}$
††margin:  Claim:

The set

$S:=\Set{s\in[-s_{0},s_{0}]}{\,\mathcal{H}^{n-1}(J_{u}\setminus J_{u+sv})\neq
0}$

is at most countable.

Let us define

$\displaystyle D_{0}=\Set{x\in J_{u}}{\gamma^{+}(u)(x)\neq\gamma^{-}(u)(x)},$
$\displaystyle D_{s}=\Set{x\in
J_{u}}{\gamma^{+}(u+sv)(x)\neq\gamma^{-}(u+sv)(x)},$

and notice that

$\mathcal{H}^{n-1}(J_{u}\setminus
D_{0})=0,\qquad\mathcal{H}^{n-1}(J_{u+sv}\setminus D_{s})=0.$

Then we have to prove that

$\Set{s\in[-s_{0},s_{0}]}{\,\mathcal{H}^{n-1}(D_{0}\setminus D_{s})\neq 0}$

is at most countable. Observe that if $t\neq s$,

$(D\setminus D_{t})\cap(D\setminus D_{s})=\emptyset.$

Indeed if $x\in D\setminus D_{s}$

$\displaystyle\gamma^{+}(u)(x)\neq\gamma^{-}(u)(x),$
$\displaystyle\gamma^{+}(u)+s\gamma^{+}(v)(x)=\gamma^{-}(u)+s\gamma^{-}(v)(x),$

then

$\gamma^{+}(v)(x)\neq\gamma^{-}(v)(x),$

and so

$s=\frac{\gamma^{-}(u)(x)-\gamma^{+}(u)(x)}{\gamma^{+}(v)(x)-\gamma^{-}(v)(x)}.$

If $\mathcal{H}^{0}$ denotes the counting measure in $\mathbb{R}$, we can
write

$\int_{-s_{0}}^{s_{0}}\mathcal{H}^{n-1}(D_{0}\setminus
D_{s})\,d\mathcal{H}^{0}=\mathcal{H}^{n-1}\Bigg{(}\bigcup_{\scalebox{0.6}{$(-s_{0},s_{0})$}}\\!D_{0}\setminus
D_{s}\Bigg{)}\leq\mathcal{H}^{n-1}(J_{u})<+\infty,$

then the claim is proved.

We are now able to differentiate in $s=0$ the function $\mathcal{F}(u+sv)$,
and observing that $0\notin S$ is a minimum for $\mathcal{F}(u+sv)$, we get

$\frac{1}{2}\delta\mathcal{F}(u,v)=\int_{\mathbb{R}^{n}}\nabla u\cdot\nabla
v\,d\mathcal{L}^{n}+\beta\int_{J_{u}}\left[\overline{u}\gamma^{+}(v)+\underline{u}\gamma^{-}(v)\right]\,d\mathcal{H}^{n-1}-\int_{\Omega}fv\,d\mathcal{L}^{n}=0.$

If $v\notin L^{\infty}(\mathbb{R}^{n})$, we consider $v_{h}=\min\set{v,h}$.
Then

$\delta\mathcal{F}(u,v_{h})=0\qquad\forall h>0.$

Observe that, since $\gamma^{\pm}(v_{h})=\min\set{\gamma^{\pm}(v),h}$,

$\gamma^{\pm}(v_{h})\to\gamma^{\pm}(v)\qquad\mathcal{H}^{n-1}\text{-a.e. in
}J_{u}.$

Therefore, passing to limit for $h\to+\infty$, by dominated convergence on the
term

$\int_{\mathbb{R}^{n}}\nabla u\cdot\nabla v_{h}\,d\mathcal{L}^{n},$

and by monotone convergence on the terms

$\beta\int_{J_{u}}\left[\overline{u}\gamma^{+}(v_{h})+\underline{u}\gamma^{-}(v_{h})\right]\,d\mathcal{H}^{n-1},\qquad\int_{\Omega}fv_{h}\,d\mathcal{L}^{n},$

we get

$0=\lim_{h}\delta\mathcal{F}(u,v_{h})=\delta\mathcal{F}(u,v).$

∎

We now want to use the Euler-Lagrange equation (3.7) to prove that if $f$
belongs to $L^{p}(\Omega)$ with $p>n$, and if $u$ is a minimizer to problem
(1.5) then $u$ belongs to $L^{\infty}(\mathbb{R}^{n})$. In order to prove this
we need the following

######  Lemma .

Let $m$ be a positive real number. There exists a positive constant
$C=C(m,\beta,n)$ such that, for every function
$v\in\operatorname{SBV}^{\frac{1}{2}}(\mathbb{R}^{n})$ with
$\mathcal{L}^{n}(\set{v>0})\leq m$,

$\left(\int_{\mathbb{R}^{n}}v^{2\cdot
1^{*}}\,d\mathcal{L}^{n}\right)^{\frac{1}{1^{*}}}\leq
C\left[\int_{\mathbb{R}^{n}}\lvert\nabla
v\rvert^{2}\,d\mathcal{L}^{n}+\beta\int_{J_{v}}\left(\overline{v}^{2}+\underline{v}^{2}\right)\,d\mathcal{H}^{n-1}\right],$

where $1^{*}=\dfrac{n}{n-1}$ is the Sobolev conjugate of $1$.

###### Proof.

Classical Embedding of $BV(\mathbb{R}^{n})$ in $L^{1^{*}}(\mathbb{R}^{n})$
ensures that

$\begin{split}\left(\int_{\mathbb{R}^{n}}v^{2\cdot
1^{*}}\,d\mathcal{L}^{n}\right)^{\frac{1}{1^{*}}}\leq&C(n)\left\lvert
Dv^{2}\right\rvert(\mathbb{R}^{n})\\\\[5.0pt]
=&C(n)\left[\int_{\mathbb{R}^{n}}2v\lvert\nabla
v\rvert\,d\mathcal{L}^{n}+\int_{J_{v}}\left(\overline{v}^{2}+\underline{v}^{2}\right)\,d\mathcal{H}^{n-1}\right].\end{split}$

For every $\varepsilon>0$, using Young’s and Hölder’s inequalities, we have

$\begin{split}\left(\int_{\mathbb{R}^{n}}v^{2\cdot
1^{*}}\,d\mathcal{L}^{n}\right)^{\frac{1}{1^{*}}}\leq&\frac{C(n)}{\varepsilon}\int_{\mathbb{R}^{n}}v^{2}\,d\mathcal{L}^{n}+\\\\[3.0pt]
&+C(n)\left[\varepsilon\int_{\mathbb{R}^{n}}\lvert\nabla
v\rvert^{2}\,d\mathcal{L}^{n}+\int_{J_{v}}\left(\overline{v}^{2}+\underline{v}^{2}\right)\,d\mathcal{H}^{n-1}\right]\\\\[5.0pt]
\leq&\frac{C(n)\,m^{\frac{1}{n}}}{\varepsilon}\left(\int_{\mathbb{R}^{n}}v^{2\cdot
1^{*}}\,d\mathcal{L}^{n}\right)^{\frac{1}{1^{*}}}+\\\\[3.0pt]
&+C(n)\left[\varepsilon\int_{\mathbb{R}^{n}}\lvert\nabla
v\rvert^{2}\,d\mathcal{L}^{n}+\int_{J_{v}}\left(\overline{v}^{2}+\underline{v}^{2}\right)\,d\mathcal{H}^{n-1}\right].\end{split}$

Setting $\varepsilon=2C(n)m^{\frac{1}{n}}$, we can find two constants
$C(m,n),C(m,\beta,n)>0$ such that

$\begin{split}\left(\int_{\mathbb{R}^{n}}v^{2\cdot
1^{*}}\,d\mathcal{L}^{n}\right)^{\frac{1}{1^{*}}}&\leq
C(m,n)\left[\int_{\mathbb{R}^{n}}\lvert\nabla
v\rvert^{2}\,d\mathcal{L}^{n}+\int_{J_{v}}\left(\overline{v}^{2}+\underline{v}^{2}\right)\,d\mathcal{H}^{n-1}\right]\\\\[5.0pt]
&\leq C(m,\beta,n)\left[\int_{\mathbb{R}^{n}}\lvert\nabla
v\rvert^{2}\,d\mathcal{L}^{n}+\beta\int_{J_{v}}\left(\overline{v}^{2}+\underline{v}^{2}\right)\,d\mathcal{H}^{n-1}\right].\end{split}$

∎

We refer to [14] for the following lemma.

######  Lemma .

Let $g:[0,+\infty)\to[0,+\infty)$ be a decreasing function and assume that
there exist $C,\alpha>0$ and $\theta>1$ constants such that for every $h>k\geq
0$,

$g(h)\leq C(h-k)^{-\alpha}g(k)^{\theta}.$

Then there exists a constant $h_{0}>0$ such that

$g(h)=0\qquad\forall h\geq h_{0}.$

In particular we have

$h_{0}=C^{\frac{1}{\alpha}}g(0)^{\frac{\theta-1}{\alpha}}2^{\theta(\theta-1)}.$

######  Proposition  ($L^{\infty}$ bound).

Let $n\geq 2$ and assume that, if $n=2$, condition (1.6) holds true. Let $f\in
L^{p}(\Omega)$, with $p>n$. Then there exists a constant
$C=C(\Omega,f,p,\beta,C_{0})>0$ such that if $u$ is a minimizer to problem
(1.5), then

$\lVert u\rVert_{\infty}\leq C.$

###### Proof.

Let $\gamma^{\pm}=\gamma_{J_{u}}^{\pm}$. For every
$\varphi,\psi\in\operatorname{SBV}^{\frac{1}{2}}(\mathbb{R}^{n})$ satisfying
$J_{\varphi},J_{\psi}\subseteq J_{u}$, define

$a(\varphi,\psi)=\int_{\mathbb{R}^{n}}\nabla\varphi\cdot\nabla\psi\,d\mathcal{L}^{n}+\beta\int_{J_{u}}\left[\gamma^{+}(\varphi)\gamma^{+}(\psi)+\gamma^{-}(\varphi)\gamma^{-}(\psi)\right]\,d\mathcal{H}^{n-1}.$

For every $v$ satisfying the assumptions of Theorem 3.1, it holds that

$a(u,v)=\int_{\Omega}fv\,d\mathcal{L}^{n}.$

In particular, let us fix $k\in\mathbb{R}^{+}$ and define

$\varphi_{k}(x)=\begin{cases}u(x)-k&\text{ if }u(x)\geq k,\\\ 0&\text{ if
}u(x)<k,\end{cases}$

then

$\gamma^{+}(\varphi_{k})(x)=\begin{cases}\overline{u}(x)-k&\text{ if
}\overline{u}(x)\geq k,\\\ 0&\text{ if }\overline{u}(x)<k,\end{cases}$

and analogously for $\gamma^{-}(\varphi_{k})$. Furthermore, let us define

$\mu(k)=\mathcal{L}^{n}(\set{u>k}).$

We want to prove that $\mu(k)=0$ for sufficiently large $k$. From Theorem 3.1,
we have

$a(u,\varphi_{k})=\int_{\Omega}f\varphi_{k}\,d\mathcal{L}^{n},$ (3.8)

and we can observe that

$\begin{split}a(u,\varphi_{k})&=\int_{\set{u>k}}\lvert\nabla
u\rvert^{2}\,d\mathcal{L}^{n}+\beta\int_{J_{u}\cap\set{u>k}}\left[\overline{u}(\overline{u}-k)+\underline{u}(\underline{u}-k)\right]\,d\mathcal{H}^{n-1}\\\
&\geq\int_{\set{u>k}}\lvert\nabla
u\rvert^{2}\,d\mathcal{L}^{n}+\beta\int_{J_{u}\cap\set{u>k}}\left[(\overline{u}-k)^{2}+(\underline{u}-k)^{2}\right]\,d\mathcal{H}^{n-1}\\\\[5.0pt]
&=a(\varphi_{k},\varphi_{k}).\end{split}$

Moreover, by minimality, $\mathcal{F}(u)\leq\mathcal{F}(0)=0$ and by Section
3, $\Omega\subseteq\set{u>0}$. Therefore, (3.1) holds true and we can apply
Section 3, having that there exists $C=C(\Omega,f,\beta,C_{0})>0$ such that

$\int_{\Omega}f\varphi_{k}\,d\mathcal{L}^{n}=a(u,\varphi_{k})\geq
a(\varphi_{k},\varphi_{k})\geq C\lVert\varphi_{k}\rVert_{2\cdot 1^{*}}^{2}.$
(3.9)

On the other hand

$\begin{split}\int_{\Omega}f\varphi_{k}\,d\mathcal{L}^{n}=\int_{\Omega\cap\set{u>k}}f(u-k)\,d\mathcal{L}^{n}&\leq\left(\int_{\Omega\cap\set{u>k}}f^{\frac{2n}{n+1}}\right)^{\frac{n+1}{2n}}\lVert\varphi_{k}\rVert_{2\cdot
1^{*}}\\\\[5.0pt] &\leq\lVert
f\rVert_{p,\Omega}\,\lVert\varphi_{k}\rVert_{2\cdot
1^{*}}\>\mu(k)^{\frac{n+1}{2n\sigma^{\prime}}},\end{split}$ (3.10)

where

$\sigma=\frac{p(n+1)}{2n}>1,$

since $p>n$. Joining (3.9) and (3.10), we have

$\lVert\varphi_{k}\rVert_{2\cdot 1^{*}}\leq C\lVert
f\rVert_{p,\Omega}\>\mu(k)^{\frac{n+1}{2n\sigma^{\prime}}}.$ (3.11)

Let $h>k$, then

$\begin{split}(h-k)^{2\cdot 1^{*}}\mu(h)=&\int_{\set{u>h}}(h-k)^{2\cdot
1^{*}}\,d\mathcal{L}^{n}\\\ \leq&\int_{\set{u>h}}(u-k)^{2\cdot
1^{*}}\,d\mathcal{L}^{n}\\\ \leq&\int_{\set{u>k}}(u-k)^{2\cdot
1^{*}}\,d\mathcal{L}^{n}=\lVert\varphi_{k}\rVert_{2\cdot 1^{*}}^{2\cdot
1^{*}}.\end{split}$

Using (3.11) and the previous inequality, we have

$\mu(h)\leq C(h-k)^{-2\cdot 1^{*}}\mu(k)^{\frac{n+1}{(n-1)\sigma^{\prime}}}.$

Since $p>n$, then $\sigma^{\prime}<(n+1)/(n-1)$. By Section 3, we have that
$\mu(h)=0$ for all $h\geq h_{0}$ with $h_{0}=h_{0}(\Omega,f,\beta,C_{0})>0$,
which implies

$\lVert u\rVert_{\infty}\leq h_{0}.$

∎

###### Proof of Theorem 1.1.

The result is obtained by joining Section 3 and Section 3. ∎

##  4 Density estimates for the jump set

In this section we prove Theorem 1.2: in Section 4 we prove the lower bound
for minimizers to problem (1.5); in Section 4 and Section 4 we prove the
density estimates for the jump set of a minimizer to problem (1.5).

In this section we will assume that $\Omega\subseteq\mathbb{R}^{n}$ is an open
bounded set with $C^{1,1}$ boundary, that $f\in L^{p}(\Omega)$, with $p>n$, is
a positive function, and that $\beta,C_{0}$ are positive constants. We
consider the energy functional $\mathcal{F}$ defined in (1.4).

In order to show that if $u$ is a minimizer to problem (1.5) then $u$ is
bounded away from 0, we will first prove that there exists a positive constant
$\delta$ such that $u>\delta$ almost everywhere in $\Omega$, and then we will
show that this implies the existence of a positive constant $\delta_{0}$ such
that $u>\delta_{0}$ almost everywhere in the set $\set{u>0}$. In the following
we will denote by $U_{t}:=\Set{u<t}\cap\Omega$.

######  Remark .

Let $u$ be a minimizer to (1.5), by Section 3, $u$ is a solution to

$\begin{cases}-\Delta u=f&\text{in }\Omega,\\\ u\geq 0&\text{on
}\partial\Omega\end{cases}$

Let $u_{0}\in W^{1,2}_{0}(\Omega)$ be the solution to the following boundary
value problem

$\begin{cases}-\Delta u_{0}=f&\text{in }\Omega,\\\ u_{0}=0&\text{on
}\partial\Omega.\end{cases}$ (4.1)

Then, by maximum principle,

$u\geq u_{0}\quad\text{in
$\Omega\subseteq\set{u>0}$}\quad\text{and}\quad\set{u<t}\cap\Omega=U_{t}\subseteq\set{u_{0}<t}\cap\Omega.$

######  Lemma .

There exist two positive constants $t_{0}=t_{0}(\Omega,f)$ and $C=C(\Omega,f)$
such that if $u$ is a minimizer to (1.5) then for every $t\in[0,t_{0}]$ it
results

$\mathcal{L}^{n}(U_{t})\leq C\,t.$ (4.2)

###### Proof.

Let $u_{0}$ be the solution to (4.1), fix $\varepsilon>0$ such that the set

$\Omega_{\varepsilon}=\Set{x\in\Omega}{d(x,\partial\Omega)>\varepsilon}$

is not empty. Since $u_{0}$ is superharmonic and non-negative in $\Omega$, by
maximum principle we have that

$\alpha=\inf_{\Omega_{\varepsilon}}u_{0}>0.$

then $u_{0}$ solves

$\begin{cases}-\Delta u_{0}=f&\text{in }\Omega,\\\ u_{0}=0&\text{on
}\partial\Omega,\\\ u_{0}\geq\alpha&\text{on
}\partial\Omega_{\varepsilon}.\end{cases}$

Therefore, if we consider the solution $v$ to the following boundary value
problem

$\begin{cases}-\Delta v=0&\text{in
}\Omega\setminus\bar{\Omega}_{\varepsilon},\\\ v=0&\text{on
}\partial\Omega,\\\ v=\alpha&\text{in }\bar{\Omega}_{\varepsilon},\end{cases}$

we have that $u\geq u_{0}\geq v$ almost everywhere in $\Omega$ and

$\Set{u<t}\cap\Omega=U_{t}\subseteq\Set{u_{0}<t}\cap\Omega\subseteq\Set{v<t}\cap\Omega.$

Hopf Lemma implies that there exists a constant $\tau=\tau(\Omega)>0$ such
that

$\frac{\partial v}{\partial\nu}<-\tau\qquad\text{on }\partial\Omega.$

Let $x\in\bar{\Omega}$, and let $x_{0}$ be a projection of $x$ onto the
boundary $\partial\Omega$, then

$\lvert x-x_{0}\rvert=d(x,\partial\Omega),\qquad\qquad\frac{x-x_{0}}{\lvert
x-x_{0}\rvert}=-\nu_{\Omega}(x_{0}),$

where $\nu_{\Omega}$ denotes the exterior normal to $\partial\Omega$. We can
write

$\begin{split}v(x)&=\underbrace{v(x_{0})}_{=0}+\nabla
v(x_{0})\cdot(x-x_{0})+o\left(\lvert x-x_{0}\rvert\right)\\\\[7.0pt]
&=-\frac{\partial v}{\partial\nu}(x_{0})\lvert x-x_{0}\rvert+o\left(\lvert
x-x_{0}\rvert\right)\\\\[5.0pt] &\geq\tau\lvert x-x_{0}\rvert+o\left(\lvert
x-x_{0}\rvert\right)\\\\[5.0pt] &>\frac{\tau}{2}\lvert
x-x_{0}\rvert=\frac{\tau}{2}d(x,\partial\Omega)\end{split}$ (4.3)

for every $x$ such that $d(x,\partial\Omega)<\sigma_{0}$ for a suitable
$\sigma_{0}=\sigma_{0}(\Omega,f)>0$. Notice that if
$\bar{x}\in\overline{\Omega}$ and $\lim_{x\to\bar{x}}v(x)=0$ then necessarily
$\bar{x}\in\partial\Omega$. Therefore, there exists a
$t_{0}=t_{0}(\Omega,f)>0$ such that $v(x)<t_{0}$ implies
$d(x,\partial\Omega)<\sigma_{0}$. Consequently, if $t<t_{0}$, we have that

$\set{v<t}\subseteq\set{d(x,\partial\Omega)<\sigma_{0}},$

and by (4.3), we get

$\mathcal{L}^{n}\left(U_{t}\right)\leq\mathcal{L}^{n}\left(\Set{v<t}\right)\leq\mathcal{L}^{n}\left(\Set{x\in\Omega}{d(x,\partial\Omega)\leq\frac{2}{\tau}\,t}\right).$

Since $\Omega$ is $C^{1,1}$, by Section 2, we conclude the proof. ∎

######  Lemma .

Let $g:[0,t_{1}]\to[0,+\infty)$ be an increasing, absolutely continuous
function such that

$g(t)\leq Ct^{\alpha}\left(g^{\prime}(t)\right)^{\sigma}\qquad\forall
t\in[0,t_{1}],$ (4.4)

with $C>0$ and $\alpha>\sigma>1$. Then there exists $t_{0}>0$ such that

$g(t)=0\qquad\forall t\leq t_{0}.$

Precisely,

$t_{0}=\left(\frac{C(\alpha-\sigma)}{\sigma-1}g(t_{1})^{\frac{\sigma-1}{\sigma}}+t_{1}^{\frac{\sigma-\alpha}{\sigma}}\right)^{\frac{\sigma}{\sigma-\alpha}}.$

###### Proof.

Assume by contradiction that $g(t)>0$ for every $t>0$. Inequality (4.4)
implies

$\frac{g^{\prime}}{g^{\frac{1}{\sigma}}}\geq\frac{1}{C}t^{-\frac{\alpha}{\sigma}}.$

Integrating between $t_{0}$ and $t_{1}$, we have

$\frac{\sigma}{\sigma-1}\left(g(t_{1})^{\frac{\sigma-1}{\sigma}}-g(t_{0})^{\frac{\sigma-1}{\sigma}}\right)\geq\frac{1}{C}\frac{\sigma}{\sigma-\alpha}\left(t_{1}^{\frac{\sigma-\alpha}{\sigma}}-t_{0}^{\frac{\sigma-\alpha}{\sigma}}\right).$

Since $\alpha>\sigma>1$, we have

$0\leq
g(t_{0})^{\frac{\sigma-1}{\sigma}}\leq\frac{\sigma-1}{C\left(\alpha-\sigma\right)}\left(t_{1}^{\frac{\sigma-\alpha}{\sigma}}-t_{0}^{\frac{\sigma-\alpha}{\sigma}}\right)+g(t_{1})^{\frac{\sigma-1}{\sigma}},$

which is a contradiction if

$t_{0}\leq\left(\frac{C(\alpha-\sigma)}{\sigma-1}g(t_{1})^{\frac{\sigma-1}{\sigma}}+t_{1}^{\frac{\sigma-\alpha}{\sigma}}\right)^{\frac{\sigma}{\sigma-\alpha}}.$

∎

######  Remark .

Let $g$ be as in Section 4 and assume that $g(t_{1})\leq K$, then $g(t)=0$ for
all $0<t<\tilde{t}$ where

$\tilde{t}=\left(\frac{C(\alpha-\sigma)}{\sigma-1}K^{\frac{\sigma-1}{\sigma}}+t_{1}^{\frac{\sigma-\alpha}{\sigma}}\right)^{\frac{\sigma}{\sigma-\alpha}}.$

We now have the tools to prove the lower bound inside $\Omega$.

######  Proposition .

There exists a positive constant $\delta=\delta(\Omega,f,p,\beta,C_{0})>0$
such that if $u$ is a minimizer to problem (1.5) then

$u\geq\delta$

almost everywhere in $\Omega$.

###### Proof.

Assume that $\Omega$ is connected and define the function

$u_{t}(x)=\begin{cases}\max\set{u,t}&\text{in }\Omega,\\\ u&\text{in
}\mathbb{R}^{n}\setminus\Omega\end{cases}$

Recalling that $U_{t}=\set{u<t}\cap\Omega$, we have

$J_{u_{t}}\setminus\partial^{*}U_{t}=J_{u}\setminus\partial^{*}U_{t},$

and on this set $\underline{u_{t}}=\underline{u}$ and
$\overline{u_{t}}=\overline{u}$. Then we get by minimality of $u$, and using
the fact that $J_{u_{t}}\cap\partial^{*}U_{t}\subseteq\partial\Omega$,

$\begin{split}0&\geq\mathcal{F}(u)-\mathcal{F}(u_{t})\\\\[7.0pt]
&=\int_{U_{t}}\lvert\nabla
u\rvert^{2}\,d\mathcal{L}^{n}-2\int_{U_{t}}f(u-t)\,d\mathcal{L}^{n}+\beta\int_{\partial^{*}U_{t}\cap
J_{u}}\left(\underline{u}^{2}+\overline{u}^{2}\right)\,d\mathcal{H}^{n-1}+\\\\[5.0pt]
&\hphantom{=}-\beta\int_{J_{u_{t}}\cap\partial^{*}U_{t}\cap
J_{u}}\left[t^{2}+(\gamma_{\partial\Omega}^{+}(u))^{2}\right]\,d\mathcal{H}^{n-1}-\beta\int_{\left(J_{u_{t}}\cap\partial^{*}U_{t}\right)\setminus
J_{u}}\left(t^{2}+u^{2}\right)\,d\mathcal{H}^{n-1}\\\\[7.0pt]
&\geq\int_{U_{t}}\lvert\nabla u\rvert^{2}\,d\mathcal{L}^{n}-2\beta
t^{2}\mathcal{H}^{n-1}\left(\partial^{*}U_{t}\cap\partial\Omega\right)\end{split}$

where we ignored all the non-negative terms except the integral of
$\lvert\nabla u\rvert^{2}$, and we used that $u\leq t$ in
$\partial^{*}U_{t}\setminus J_{u}$. By Section 4, we can choose $t$ small
enough to have $\mathcal{L}^{n}(U_{t})\leq\mathcal{L}^{n}(\Omega)/2$, then
applying the isoperimetric inequality in Theorem 2.2 to the set $E=U_{t}$, we
get

$\int_{U_{t}}\lvert\nabla u\rvert^{2}\,d\mathcal{L}^{n}\leq
2\beta\,C\,t^{2}\,P(U_{t};\Omega).$ (4.5)

Let us define

$p(t)=P(U_{t};\Omega),$

and consider the absolutely continuous function

$g(t)=\int_{U_{t}}u\,\lvert\nabla
u\rvert\,d\mathcal{L}^{n}=\int_{0}^{t}sp(s)\,ds.$

By minimality of $u$ we can apply the a priori estimates (3.6) to prove the
equiboundedness of $g$, i.e. there exists $K=K(\Omega,f,\beta,C_{0})>0$ such
that $g(t)\leq K$ for all $t>0$. Using the Hölder inequality and the estimate
(4.5) we have

$g(t)\leq\left(\int_{U_{t}}u^{2}\,d\mathcal{L}^{n}\right)^{\frac{1}{2}}\left(\int_{U_{t}}\lvert\nabla
u\rvert^{2}\,d\mathcal{L}^{n}\right)^{\frac{1}{2}}\leq\sqrt{2\beta
C}\,t\,\mathcal{L}^{n}(U_{t})^{\frac{1}{2}}(t^{2}p(t))^{\frac{1}{2}}.$

Fix $1>\varepsilon>0$. Then we can write
$\mathcal{L}^{n}(U_{t})=\mathcal{L}^{n}(U_{t})^{\varepsilon}\,\mathcal{L}^{n}(U_{t})^{1-\varepsilon}$,
and by Section 4 there exists a constant $C=C(\Omega,f,\beta)>0$ such that

$g(t)\leq
C\,t^{2+\frac{1-\varepsilon}{2}}\,\mathcal{L}^{n}(U_{t})^{\frac{\varepsilon}{2}}\,p(t)^{\frac{1}{2}}.$

By the relative isoperimetric inequality in Theorem 2.1, we can estimate

$\mathcal{L}^{n}(U_{t})^{\frac{\varepsilon}{2}}\leq
C(\Omega,n)p(t)^{\frac{\varepsilon n}{2(n-1)}},$

and, noticing that $p(t)=g^{\prime}(t)/t$, we get

$g(t)\leq Ct^{\alpha}(g^{\prime}(t))^{\sigma},$

where

$\alpha=2-\frac{\varepsilon}{2}\left(1+\frac{n}{n-1}\right),\qquad\qquad\sigma=\frac{1}{2}+\frac{\varepsilon}{2}\frac{n}{n-1}.$

In particular, if we choose

$\varepsilon\in\left(\frac{n-1}{n},\frac{3n-3}{3n-1}\right),$

we have that $\alpha>\sigma>1$, and then, using Section 4 and Section 4, there
exists a $\delta=\delta(\Omega,f,p,\beta,C_{0})>0$ such that $g(t)=0$ for
every $t<\delta$. Then $\mathcal{L}^{n}(\set{u<t}\cap\Omega)=0$ for every
$t<\delta$, hence

$u\geq\delta$

almost everywhere in $\Omega$.

When $\Omega$ is not connected, then

$\Omega=\Omega_{1}\cup\dots\cup\Omega_{N},$

with $\Omega_{i}$ pairwise disjoint connected open sets. Using $u_{t}$ as the
function $u$ truncated inside a single $\Omega_{i}$, we find constants
$\delta_{i}>0$ such that

$u(x)\geq\delta_{i}$

almost everywhere in $\Omega_{i}$. Therefore choosing
$\delta=\min\set{\delta_{1},\dots,\delta_{N}}$ we have $u(x)>\delta$ almost
everywhere in $\Omega$. ∎

Finally, following the approach in [6], we have

######  Proposition  (Lower Bound).

There exists a positive constant
$\delta_{0}=\delta_{0}(\Omega,f,p,\beta,C_{0})$ such that if $u$ is a
minimizer to problem (1.5) then

$u\geq\delta_{0}$

almost everywhere in $\Set{u>0}$.

###### Proof.

Let $\delta$ be the constant in Section 4. For every $0<t\leq\delta$ let us
define the absolutely continuous function

$h(t)=\int_{\set{u\leq t}\setminus J_{u}}u\lvert\nabla
u\rvert\,\mathcal{L}^{n}=\int_{0}^{t}sP(\set{u>s};\mathbb{R}^{n}\setminus
J_{u})\,ds.$

By minimality of $u$ we can apply the a priori estimates (3.6) to prove the
equiboundedness of $h$, i.e. there exists $K=K(\Omega,f,\beta,C_{0})>0$ such
that $h(t)\leq K$ for all $t>0$. We will show that $h$ satisfies a
differential inequality. For any $0<t<\delta$, let us consider
$u^{t}=u\chi_{\set{u>t}}$, where $\chi_{\set{u>t}}$ is the characteristic
funtion of the set $\set{u>t}$, as a competitor for $u$. We observe that, by
Section 4, $\Omega\subseteq\set{u>t}$, so we have that

$\begin{split}0\geq&\mathcal{F}(u)-\mathcal{F}(u^{t})\\\\[5.0pt]
=&\int_{\set{u\leq t}\setminus J_{u}}\lvert\nabla
u\rvert^{2}\,\mathcal{L}^{n}+\beta\int_{J_{u}\cap\set{u>t}^{(0)}}\left(\underline{u}^{2}+\overline{u}^{2}\right)\,d\mathcal{H}^{n-1}+\\\\[5.0pt]
&+\beta\int_{J_{u}\cap\partial^{*}\set{u>t}}\underline{u}^{2}\,d\mathcal{H}^{n-1}-\beta\int_{\partial^{*}\set{u>t}\setminus
J_{u}}{u}^{2}\,d\mathcal{H}^{n-1}+\\\\[5.0pt]
&+C_{0}\mathcal{L}^{n}\left(\Set{0<u\leq t}\right).\end{split}$

Rearranging the terms,

$\begin{split}&\int_{\set{u\leq t}\setminus J_{u}}\lvert\nabla
u\rvert^{2}\,\mathcal{L}^{n}+\beta\int_{J_{u}\cap\set{u>t}^{(0)}}\left(\underline{u}^{2}+\overline{u}^{2}\right)\,d\mathcal{H}^{n-1}+\\\\[5.0pt]
&+\beta\int_{J_{u}\cap\partial^{*}\set{u>t}}\underline{u}^{2}\,d\mathcal{H}^{n-1}+C_{0}\mathcal{L}^{n}\left(\Set{0<u\leq
t}\right)\\\\[5.0pt] &\leq\beta t^{2}P(\set{u>t};\mathbb{R}^{n}\setminus
J_{u})=\beta th^{\prime}(t).\end{split}$ (4.6)

On the other hand using Hölder’s inequality, we have

$h(t)\leq\left(\int_{\set{u\leq t}}\lvert\nabla
u\rvert^{2}\,\mathcal{L}^{n}\right)^{\frac{1}{2}}\mathcal{L}^{n}\left(\Set{0<u\leq
t}\right)^{\frac{1}{2n}}\left(\int_{\set{u\leq t}}u^{2\cdot
1^{*}}\,\mathcal{L}^{n}\right)^{\frac{1}{2\cdot 1^{*}}}.$ (4.7)

Classical Embedding of $\operatorname{BV}$ in $L^{1^{*}}$ applied to
$u^{2}\chi_{\set{u\leq t}}$, ensures

$\left(\int_{\set{u\leq t}}u^{2\cdot
1^{*}}\,\mathcal{L}^{n}\right)^{\frac{1}{1^{*}}}\leq C(n)\left\lvert
D\big{(}u^{2}\chi_{\set{u\leq t}}\big{)}\right\rvert(\mathbb{R}^{n}),$

and, using (4.6),

$\begin{split}\left\lvert D\big{(}u^{2}\chi_{\set{u\leq
t}}\big{)}\right\rvert(\mathbb{R}^{n})=&2\int_{\set{u\leq t}}u\lvert\nabla
u\rvert\,\mathcal{L}^{n}+\int_{J_{u}\cap\set{u>t}^{(0)}}\left(\underline{u}^{2}+\overline{u}^{2}\right)\,d\mathcal{H}^{n-1}+\\\\[5.0pt]
&+\int_{J_{u}\cap\partial^{*}\set{u>t}}\underline{u}^{2}\,d\mathcal{H}^{n-1}+\int_{\partial^{*}\set{u>t}\setminus
J_{u}}{u}^{2}\,d\mathcal{H}^{n-1}\\\\[5.0pt]
\leq&2t\left(\mathcal{L}^{n}\left(\Set{0<u\leq t}\right)\int_{\set{u\leq
t}\setminus J_{u}}\lvert\nabla
u\rvert^{2}\,\mathcal{L}^{n}\right)^{\frac{1}{2}}+3th^{\prime}(t)\\\\[5.0pt]
\leq&\left(2\frac{\delta\beta}{\sqrt{C_{0}}}+3\right)th^{\prime}(t).\end{split}$
(4.8)

Therefore, joining (4.7), (4.6), and (4.8), we have

$h(t)\leq C_{3}\left(th^{\prime}(t)\right)^{1+\frac{1}{2n}},$ (4.9)

where

$C_{3}=\beta^{\frac{1}{2}}\left(\frac{\beta}{C_{0}}\right)^{\frac{1}{2n}}C(n)^{\frac{1}{2}}\left(2\frac{\delta\beta}{\sqrt{C_{0}}}+3\right)^{\frac{1}{2}}.$

By (4.9) we now want to show that there exists
$\delta_{0}=\delta_{0}(\Omega,f,p,\beta,C_{0})>0$ such that $h(t)=0$ for every
$0\leq t<\delta_{0}$. Indeed assume by contradiction that $h(t)>0$ for every
$0<t\leq\delta$. We have

$\frac{h^{\prime}(t)}{h(t)^{\frac{2n}{2n+1}}}\geq\frac{C_{3}^{-\frac{2n}{2n+1}}}{t}.$

Integrating from $t_{0}>0$ to $\delta$, we get

$\left(h(\delta)^{\frac{1}{2n+1}}-h(t_{0})^{\frac{1}{2n+1}}\right)\geq
C_{4}\log\left(\frac{\delta}{t_{0}}\right),$

where

$C_{4}=\dfrac{C_{3}^{-\frac{2n}{2n+1}}}{2n+1}.$

Then

$h(t_{0})^{\frac{1}{2n+1}}\leq
h(\delta)^{\frac{1}{2n+1}}+C_{4}\log\left(\frac{t_{0}}{\delta}\right).$

Finally, for any

$0<t_{0}\leq\tilde{\delta}=\delta\exp\left(-h(\delta)^{\frac{1}{2n+1}}/C_{4}\right),$

we have $h(t_{0})<0$, which is a contradiction. Then, setting
$\delta_{0}=\delta\exp\left(-K^{\frac{1}{2n+1}}/C_{4}\right)\leq\tilde{\delta}$,
we conclude that $h(t)=0$ for any $0<t<\delta_{0}$, from which we have

$u\geq\delta_{0}$

almost everywhere in $\set{u>0}$. ∎

######  Remark .

From Section 4, if $u$ is a minimizer to problem (1.5), we have that

$\partial^{*}\set{u>0}\subseteq J_{u}\subseteq K_{u}.$ (4.10)

Indeed, on $\partial^{*}\set{u>0}$ we have that, by definition,
$\underline{u}=0$ and that, since $u\geq\delta_{0}$ $\mathcal{L}^{n}$-a.e. in
$\set{u>0}$, $\overline{u}\geq\delta_{0}$.

######  Proposition  (Density Estimates).

There exist positive constants $C=C(\Omega,f,p,\beta,C_{0})$,
${c=c(\Omega,f,p,\beta,C_{0})}$ and
$\delta_{1}=\delta_{1}(\Omega,f,p,\beta,C_{0})$ such that if $u$ is a
minimizer to problem (1.5) then for every $B_{r}(x)$ such that
$B_{r}(x)\cap\Omega=\emptyset$, we have:

  1. (a)

For every $x\in\mathbb{R}^{n}\setminus\Omega$,

$\mathcal{H}^{n-1}(J_{u}\cap B_{r}(x))\leq Cr^{n-1};$ (4.11)

  2. (b)

For every $x\in K_{u}$,

$\mathcal{L}^{n}(B_{r}(x)\cap\set{u>0})\geq cr^{n};$ (4.12)

  3. (c)

The function $u$ has bounded support, namely

$\set{u>0}\subseteq B_{1/\delta_{1}}.$

###### Proof.

This theorem is a consequence of Section 4, since we immediately have††margin:
- (a)

$\int_{J_{u}\cap
B_{r}(x)}\left(\underline{u}^{2}+\overline{u}^{2}\right)\,d\mathcal{H}^{n-1}\geq\delta_{0}^{2}\mathcal{H}^{n-1}(J_{u}\cap
B_{r}(x)),$

and by minimality of $u$ we have

$\begin{split}0&\geq\mathcal{F}(u)-\mathcal{F}(u\chi_{\mathbb{R}^{n}\setminus
B_{r}(x)})\\\ &\geq\int_{J_{u}\cap
B_{r}(x)}\left(\underline{u}^{2}+\overline{u}^{2}\right)\,d\mathcal{H}^{n-1}-\int_{\partial
B_{r}(x)\setminus J_{u}}{u}^{2}\,d\mathcal{H}^{n-1}\\\ &\geq\int_{J_{u}\cap
B_{r}(x)}\left(\underline{u}^{2}+\overline{u}^{2}\right)\,d\mathcal{H}^{n-1}-\int_{\partial
B_{r}(x)\cap\set{u>0}^{(1)}}\left(\underline{u}^{2}+\overline{u}^{2}\right)\,d\mathcal{H}^{n-1}\\\
&\geq\int_{J_{u}\cap
B_{r}(x)}\left(\underline{u}^{2}+\overline{u}^{2}\right)\,d\mathcal{H}^{n-1}-2\lVert
u\rVert_{\infty}^{2}\mathcal{H}^{n-1}\left(\partial
B_{r}(x)\cap\set{u>0}^{(1)}\right),\end{split}$

where, in the second inequality, we have used (4.10). Thus we have

$\mathcal{H}^{n-1}\left(J_{u}\cap B_{r}(x)\right)\leq\frac{2\lVert
u\rVert_{\infty}^{2}}{\delta_{0}^{2}}\mathcal{H}^{n-1}(\partial
B_{r}(x)\cap\set{u>0}^{(1)})\leq Cr^{n-1},$ (4.13)

where $C=C(\Omega,f,p,\beta,C_{0})>0$.

We now††margin:  (b) want to use the estimate (4.11) together with the
relative isoperimetric inequality in order to get a differential inequality
for the volume of $B_{r}(x)\cap\set{u>0}^{(1)}$. Let $x\in K_{u}$, then for
almost every $r$ we have

$\begin{split}0<V(r):=\mathcal{L}^{n}(B_{r}(x)\cap\set{u>0}^{(1)})&\leq
k\,P(B_{r}(x)\cap\set{u>0}^{(1)})^{\frac{n}{n-1}}\\\ &\leq
k\,\mathcal{H}^{n-1}(\partial
B_{r}(x)\cap\set{u>0}^{(1)})^{\frac{n}{n-1}},\end{split}$

where $k=k(\Omega,f,p,\beta,C_{0})>0$, and in the last inequality we used that
(4.10) and (4.13) imply

$\begin{split}P\left(B_{r}(x)\cap\set{u>0}^{(1)}\right)&\leq\mathcal{H}^{n-1}\left(\partial
B_{r}(x)\cap\set{u>0}^{(1)}\right)+\mathcal{H}^{n-1}\left(J_{u}\cap
B_{r}(x)\right)\\\ &\leq\left(1+\frac{2\lVert
u\rVert_{\infty}^{2}}{\delta_{0}^{2}}\right)P(B_{r}(x);\set{u>0}^{(1)}).\end{split}$

Then we have

$\frac{V^{\prime}(r)}{V(r)^{\frac{n-1}{n}}}\geq\frac{1}{k},$

which implies

$\mathcal{L}^{n}(B_{r}(x)\cap\set{u>0}^{(1)})\geq c\,r^{n}.$

Finally, let††margin:  (c) $x\in K_{u}$ such that $d(x,\partial\Omega)\geq
1/\delta_{1}$. From (4.12), noticing that
$\mathcal{F}(u)\leq\mathcal{F}(0)=0$, we have that

$c\,\delta_{1}^{-n}\leq\mathcal{L}^{n}(\set{u>0}\setminus\Omega)\leq\frac{2\lVert
u\rVert_{\infty}}{C_{0}}\int_{\Omega}f\,d\mathcal{L}^{n},$

which is a contradiction if $\delta_{1}$ is sufficiently small. Then the
thesis is given by (4.10). ∎

Finally, we have

######  Proposition  (Lower Density Estimate).

There exists a positive constant $c=c(\Omega,f,p,\beta,C_{0})$ such that if
$u$ is a minimizer to problem (1.5) then

  1. 1.

For any $x\in K_{u}$ and $B_{r}(x)\subseteq\mathbb{R}^{n}\setminus\Omega$,

$\mathcal{H}^{n-1}(J_{u}\cap B_{r}(x))\geq cr^{n-1};$

  2. 2.

$J_{u}$ is essentially closed, namely

$\mathcal{H}^{n-1}(K_{u}\setminus J_{u})=0;$

The proof of Section 4 relies on classical techniques used in [11] to prove
density estimates for the jump set of almost-quasi minimizers of the Mumford-
Shah functional. We refer to [6, Theorem 5.1] and [6, Corollary 5.4] for the
details of the proof.

###### Proof of Theorem 1.2.

The result is obtained by joining Section 4, Section 4, and Section 4. ∎

######  Remark .

Given the summability assumption on the function $f$, we have that minimizers
to (1.2) are quasi-minimizers of the functional $\mathcal{G}$, defined on
$\operatorname{SBV}^{\frac{1}{2}}(\mathbb{R}^{n})\cap W^{1,2}(\Omega)$ as

$\mathcal{G}(v)=\int_{\mathbb{R}^{n}}\lvert\nabla
v\rvert^{2}\,d\mathcal{L}^{n}+\beta\int_{J_{v}}(\overline{v}^{2}+\underline{v}^{2})\,d\mathcal{H}^{n-1},$

that is, there exists $C(\Omega,f,p,\beta,C_{0})>0$ and $\alpha(n,p)>n-1$ such
that, if $B_{r}(x)$ is a ball of radius $r\leq 1$, and
$v\in\operatorname{SBV}^{\frac{1}{2}}(\mathbb{R}^{n})\cap W^{1,2}(\Omega)$,
with $\set{u\neq v}\subset B_{r}(x)$, then

$\mathcal{G}(u;B_{r}(x))\leq\mathcal{G}(v;B_{r}(x))+Cr^{\alpha},$

where

$\mathcal{G}(v;B_{r}(x)):=\int_{B_{r}(x)}\\!\lvert\nabla
v\rvert^{2}\,d\mathcal{L}^{n}+\beta\\!\int_{J_{v}\cap\overline{B_{r}(x)}}(\overline{v}^{2}+\underline{v}^{2})\,d\mathcal{H}^{n-1}.$

Indeed, let $u$ be a minimizer to (1.2), let $B_{r}(x)$ be a ball of radius
$r\leq 1$, and let $v\in\operatorname{SBV}^{\frac{1}{2}}(\mathbb{R}^{n})\cap
W^{1,2}(\Omega)$, with $\set{u\neq v}\subset B_{r}(x)$; by the minimality of
$u$ we have that

$\begin{split}\mathcal{G}(u;B_{r}(x))-2\\!\int_{\Omega\cap
B_{r}(x)}fu\,d\mathcal{L}^{n}&\leq\mathcal{G}(v;B_{r}(x))+C_{0}\mathcal{L}^{n}(\set{v>0}\cap
B_{r}(x)\setminus\Omega)\\\\[3.0pt]
&\leq\mathcal{G}(v;B_{r}(x))+Cr^{n},\end{split}$

while

$\int_{\Omega\cap B_{r}(x)}fu\,d\mathcal{L}^{n}\leq\lVert
f\rVert_{p,\Omega}\lVert
u\rVert_{\infty}\mathcal{L}^{n}(B_{r})^{1/p^{\prime}}=C(\Omega,f,p,\beta,C_{0})r^{\alpha},$

where

$n>\alpha=\frac{n}{p^{\prime}}>n-1.$

Finally, we have

$\mathcal{G}(u;B_{r}(x))\leq\mathcal{G}(v;B_{r}(x))+Cr^{\alpha}.$

######  Remark .

Let $u$ be a minimizer to (1.5) and let $A=\set{\overline{u}>0}\setminus
K_{u}$, then the boundary of $A$ is equal to $K_{u}$: in first place, assume
by contradiction that there exists an $x\in(\partial A)\setminus K_{u}$, then
$u$ is superharmonic in a small ball centered in $x$ with radius $r$.
Therefore, being

$\set{u>0}\cap B_{r}(x)\neq\emptyset,$

it is necessary that $u>0$ in the entire ball, and then $x\notin\partial A$,
which is a contradiction. In other words,

$\partial A\subseteq K_{u}$

By the same argument we also have that $A$ is open, and moreover
$J_{u}\subseteq\partial A$, then

$K_{u}\subseteq\partial A.$

In particular, the pair $(A,u)$ is a minimizer for the functional

$\mathcal{F}(E,v)=\int_{E}\lvert\nabla
v\rvert^{2}\,d\mathcal{L}^{n}-2\int_{\Omega}fv\,d\mathcal{L}^{n}+\int_{\partial
E}\left(\underline{v}^{2}+\overline{v}^{2}\right)\,d\mathcal{H}^{n-1}+C_{0}\mathcal{L}^{n}(E\setminus\Omega)$

over all pairs $(E,v)$ with $E$ open set of finite perimeter containing
$\Omega$ and $v\in W^{1,2}(E)$.

## References

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Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli
studi di Napoli Federico II, Via Cintia, Complesso Universitario Monte S.
Angelo, 80126 Napoli, Italy.

E-mail address, P. Acampora<EMAIL_ADDRESS>
E-mail address, C. Nitsch<EMAIL_ADDRESS>
E-mail address, C. Trombetti<EMAIL_ADDRESS>
Mathematical and Physical Sciences for Advanced Materials and Technologies,
Scuola Superiore Meridionale, Largo San Marcellino 10, 80126, Napoli, Italy.

E-mail address, E. Cristoforoni<EMAIL_ADDRESS>


$A_{t+1}$ in the symbols) and the corresponding numbers $\gamma_{i}(k)$,
$\gamma^{(i)}(k),\gamma_{r}(k)$ for the polynomials $G_{k,A_{t+1}}$ agree with
those for the lattice $A$ given in (5.7.1) and 7.4.

  3. $\mathrm{(iii)}$

We have $\gamma_{A_{t+1}}(k)=k-\sum_{0\leq i\leq
t}\gamma^{(i)}(k)\geq\gamma_{A}(k)$.

###### Proof.

  1. (i)

As the proof of Lemma 7.3 shows, the term $a_{-k-\mu_{t+1}(k)}$ dominates in
the Laurent expansion of $C_{k,A_{t+1}}$ on the sphere $\mathbf{S}(q^{r})$, if
$r\geq t+1$ (actually, if $r>t+1-\varepsilon$ for some small $\varepsilon$);
whence there are no zeroes $z$ with $\log z\geq t+1$.

Items (ii) and (iii) follow from the foresaid. ∎

######  Corollary 9.3:

Assume that $k$ is $(t+1)$-regular and $\Lambda^{\prime}=\sum_{0\leq i\leq
t}\mathds{F}\lambda_{i}$ is an arbitrary $(t+1)$-dimensional separable
$\mathds{F}$-lattice with $\lvert\lambda_{i}\rvert=q^{r_{i}}$.

  1. $\mathrm{(i)}$

All the zeroes $z$ of $C_{k,\Lambda^{\prime}}$ satisfy $\log z\leq r_{t}$, and
lie on critical spheres $\mathbf{S}(q^{r_{i}})$ with $0\leq i\leq t$.

  2. $\mathrm{(ii)}$

The numbers $\tilde{\gamma}_{i}$ and the corresponding $\gamma_{i}$ agree with
those of $A$, given in Theorem 5.7.

  3. $\mathrm{(iii)}$

We have $\gamma_{\Lambda^{\prime}}(k)=k-\sum_{0\leq i\leq
t}\gamma_{i}(k)\geq\gamma_{A}(k)$.

###### Proof.

As in 9.2, the term $a_{-k-\mu_{t+1}(k)}$ dominates on $\mathbf{S}(q^{r})$ for
$r>r_{t}$ (where the larger domain of domination compared to 9.2 comes from
the additional $(t+1)$-regularity assumption). The rest is now obvious. ∎

### 9.4.

Now we present a handful of examples that exhibit phenomena like irregular
zeroes and exceptional zero numbers $\gamma(k)$ due to Case 2 in Theorem 8.9.
In what follows, we assume first that $q=4$. The relevant numbers $k-1$,
$\kappa=-k$, $\mu_{i}(k)$, $M(k)$ are displayed as the $0$-$1$-strings
(infinite for $\kappa$) of their dyadic expansions,

$n=\sum_{j\geq 0}n_{j}q^{j},\qquad n_{j}=n_{j,0}+2n_{j,1},\qquad n_{i,j}\in
P=\\{0,1\\};$

empty spaces correspond to zeroes.

######  Example 9.5 ($q=4$, $k=21$):

$n$\$j$ | 0 | 1 | $\alpha=2$ | 3 | 4
---|---|---|---|---|---
$k-1=20$ | 0 0 | 1 0 | 1 0 |  |
$\kappa$ | 1 1 | 0 1 | 0 1 | 1 1 | 1 1
$M=171$ | 1 1 | 0 1 | 0 1 | 0 1 |
$\mu_{1}=3$ | 1 1 |  |  |  |
$\mu_{2}=75$ | 1 1 | 0 1 | 0 0 | 1 0 |
$\mu_{3}=363$ | 1 1 | 0 1 | 0 1 | 1 0 | 1 0

Here $\alpha=2$, $\kappa_{\alpha}=2$, $R=2$, $H=2$, $\bar{s}=1$,
$\overline{R}=2$, $\underline{R}=0$. $k$ is regular as
$\mu_{1}\varprec\mu_{2}$, and Case 2 of Theorem 8.9 holds. We have

(9.5.1) $\gamma_{0}(k)=15,\qquad\gamma_{1}(k)=0,\qquad\gamma(k)=6.$

Hence $G_{21,\Lambda}(X)=P(X)X^{6}$, where $P$ is a monic polynomial of degree
15, $\lvert\text{coefficients}\rvert\leq 1$ and $\lvert P(0)\rvert=1$. In
fact, one may calculate $P(X)$ by hand (and with some pain), which gives

(9.5.2)
$P(X)=X^{15}+\alpha_{1}X^{12}+\alpha_{1}^{2}X^{9}+\alpha_{1}^{3}X^{6}+(\alpha_{1}^{5}+\alpha_{2}).$

Here the $\alpha_{i}=\alpha_{i}(\Lambda)$ are the coefficients of
$e_{\Lambda}$ (see 1.1.3), $\lvert\alpha_{2}\rvert<\lvert\alpha_{1}\rvert=1$.
As $k=21$ is regular, the lattice $\Lambda$ may be arbitrary separable with
$\lambda_{0}=1$.

######  Example 9.6 ($q=4$, $k=69$):

$n$\$j$ | 0 | 1 | 2 | $\alpha=3$ | 4 | 5
---|---|---|---|---|---|---
$k-1=68$ | 0 0 | 1 0 | 0 0 | 1 0 |  |
$\kappa$ | 1 1 | 0 1 | 1 1 | 0 1 | 1 1 | 1 1
$M=699$ | 1 1 | 0 1 | 1 1 | 0 1 | 0 1 |
$\mu_{1}=3$ | 1 1 |  |  |  |  |
$\mu_{2}=27$ | 1 1 | 0 1 | 1 0 |  |  |
$\mu_{3}=315$ | 1 1 | 0 1 | 1 1 | 0 0 | 1 0 |
$\mu_{4}=1467$ | 1 1 | 0 1 | 1 1 | 0 1 | 1 0 | 1 0

Here $\alpha=3$, $k_{\alpha}=2$, $R=2$, $H=3$, $\bar{s}=1$, $\overline{R}=2$,
$\underline{R}=0$. $k$ is regular as $\mu_{1}\varprec\mu_{2}\varprec\mu_{3}$,
and Case 2 holds. We have

(9.6.1)
$\gamma_{0}(k)=51,\qquad\gamma_{1}(k)=12,\qquad\gamma_{2}(k)=0,\qquad\gamma(k)=6.$

The Newton polygon of $G_{69,A}$ has the break points $(6,48)$, $(18,0)$,
$(69,0)$. (Here $48=-12\times L(1)$ with the function $L$ of 1.10.) For
arbitrary $\Lambda\neq A$ with $\lambda_{0}=1$, the Newton polygon has the
same shape, but the ordinate of the first break point will differ.

######  Example 9.7 ($q=4$, $k=75$):

$n$\$j$ | 0 | 1 | 2 | $\alpha=3$ | 4
---|---|---|---|---|---
$k-1=74$ | 0 1 | 0 1 | 0 0 | 1 0 |
$\kappa$ | 1 0 | 1 0 | 1 1 | 0 1 | 1 1
$M=\mu_{3}=693$ | 1 0 | 1 0 | 1 1 | 0 1 | 0 1
$\mu_{1}=21$ | 1 0 | 1 0 | 1 0 |  |
$\mu_{2}=165$ | 1 0 | 1 0 | 0 1 | 0 1 |
$\vartheta=309$ | 1 0 | 1 0 | 1 1 | 0 0 | 1 0

Here $\alpha=3$, $\kappa_{\alpha}=2$, $R=2$, $\bar{s}=0$, Case 1 of Theorem
8.9. As $\mu_{1}\not\varprec\mu_{2}$, $k$ is irregular. The set
$\boldsymbol{\Theta}(k,1)$ of 7.7 is
$\boldsymbol{\Theta}(k,1)=\\{\mu_{2},\vartheta\\}$ with $\vartheta=309$,
$\mathbf{Sh}(\vartheta)=(21,288)$, $\mathbf{Sh}(\mu_{2})=(33,132)$. Then
$\mathrm{ct}_{2}(\vartheta)=21<33=\mathrm{ct}_{2}(\mu_{2})$, and so

(9.7.1)
$\rho=1+\frac{\mathrm{ct}(\mu_{2})-\mathrm{ct}_{2}(\vartheta)}{\vartheta-\mu_{2}}=\frac{13}{12}$

equals $\log z$ for some irregular zeroes $z$ of $C_{75,A}$.

Our formulas yield $\gamma_{0}(k)=51$, $\gamma_{1}(k)=0$,
$\gamma^{(1)}(k)=\gamma_{\rho}(k)=9$, $\gamma_{2}(k)=3$, $\gamma(k)=12$. To
the values $\log z=0,13/12,2$ correspond $L(\log z)=0,-16/3,-20$ for the $\log
x$, where $x=t_{A}(z)$ is the corresponding zero of $G_{75,A}$. There result
the break points $(12,108)$, $(15,48)$, $(24,0)$ and $(75,0)$ of
$\mathrm{NP}(G_{75,A}(X))$.

Next we generalize the preceding example and produce irregular $k$ for
arbitrar non-prime $q=p^{f}$. For ease of presentation, we restrict to $p=2$,
but one easily sees how the construction generalizes to arbitrary $p$.

######  Example 9.8:

Let $q=2^{f}$ with $f>2$. We first write down an $M=M(k)$ with a very specific
$q$-expansion, from which we derive the ($q$-expansions of the) relevant
numbers $k$, $\mu_{1}$, $\mu_{2}$,
$\mathrm{Sh}(\mu_{2})=\mathrm{Sh}_{2}(\mu_{2})$,
${}^{1}\mu_{2}=\mathrm{Sh}_{1}(\mu_{2})$, $\mu_{1}^{(1)}$. This will show that
$\mu_{1}\not\varprec\mu_{2}$.

In the following table, the $(n,j)$-entry is $n_{j}$, where $n$ has
$q$-expansion $\sum n_{j}q^{j}$. As usual, empty spaces denote zeroes. We
assume that $f>2$ since for $f=2$ several columns overlap, which changes some
formulas and gives Example 9.7.

$n$\$j$ | 0 | … | $q-4$ | $q-3$ | $q-2$ | $q-1$ | $q$ | … | $\alpha=2q-5$ | $2q-4$ | …
---|---|---|---|---|---|---|---|---|---|---|---
$k-1$ | $q-2$ | … | … | $q-2$ | 0 | 1 | … | … | 1 |  |
$\kappa$ | 1 | … | … | 1 | $q-1$ | $q-2$ | … | … | $q-2$ | $q-1$ | …
$M$ | 1 | … | … | 1 | $q-1$ | $q-2$ | … | … | $q-2$ | $q-2$ |
$\mu_{1}$ | 1 | … | … | 1 | 1 |  |  |  |  |  |
$\mu_{2}$ | 1 | … | … | 1 | $q-2$ | 2 |  |  |  |  |
$\mathrm{Sh}(\mu_{2})$ |  |  |  | 1 | $q-4$ | 2 |  |  |  |  |
${}^{1}\mu_{2}$ | 1 | … | 1 | 0 | 2 |  |  |  |  |  |
$\mu_{1}^{(1)}$ | 1 | … | … | 1 | $q-1$ | $q-2$ | 2 |  |  |  |

Start with $M=\sum_{0\leq j\leq q-3}q^{j}+(q-1)q^{q-2}+\sum_{q-1\leq j\leq
2q-4}(q-2)q^{j}$, which gives the $M$-th row in the table, as well as $k-1$
and $\kappa=-k$. (As $M_{2q-4}<q-1$, $\alpha=\deg_{q}(k-1)=2q-5$, and the
coefficients of $k-1$ are complementary to those of $\kappa$.) Summing up the
geometric series gives

(9.8.1)
$\begin{split}k-1&=(q-2)(q^{q-2}-1)/(q-1)+q^{q-1}(q^{q-3}-1)/(q-1)\quad\text{and}\\\
k&=(q^{q-2}-1)^{2}/(q-1).\end{split}$

In particular, the quantity $R=\text{representative of $k-1$ in $Q^{\prime}$}$
equals $q-2$. The row for $\mu_{1}$ is then obvious. Since $\ell(\sum_{0\leq
j\leq q-2}\kappa_{j}q^{j})=2q-3<2(q-1)$, $\mu_{2}$ must have some entry
$0\neq(\mu_{2})_{j}$ with $j\geq q-1$, which must be $\geq 2$, since
$1\lneq_{p}q-2$. Therefore, $\mu_{2}$ as given is the least possible value.
Again, the Sheats factors $\mathrm{Sh}_{2}(\mu_{2})=\mathrm{Sh}(\mu_{2})$ and
$\mathrm{Sh}_{1}(\mu_{2})={}^{1}\mu_{2}$ are then obvious. Also, it is not
difficult to show that $\mu_{1}^{(1)}$ is as indicated. We find that
$\mu_{1}\not\varprec\mu_{2}$ as intended, so $k$ is irregular.

A view to the table shows that $\ell_{i}(M)=q-1$ for $0\leq i<f$, so
$\ell^{\sigma}(M)=(q-1)\sum_{0\leq i<f}2^{i}=(q-1)^{2}$ for each
$\sigma\in\mathbf{C}$, and therefore

(9.8.2) $H=\mathrm{ht}(M)=q-1.$

In particular, Case 1 of Theorem 8.9 holds, which gives

(9.8.3) $\gamma(k)=(R+1)q^{[\ell(k-1)/(q-1)]}=(q-1)q^{q-3}.$

Some admissible $n$ with $\mathrm{ht}(n)=1$ and $n<{}^{1}\mu_{2}$ must satisfy
$n_{q-2}<2$, and must agree with $\mu_{1}$. Therefore, with the considerations
of Section 7, we see that the set $\mathbf{R}=\mathbf{R}(k,1)$ contains
exactly one element $\rho$ which appears as $\rho=\log z$ for an irregular
zero $z$ of $C_{k,A}$. It is given by

(9.8.4)
$\rho=1+\frac{{}^{1}\mu_{2}-\mu_{1}}{\mu_{1}^{(1)}-\mu_{2}}=1+\frac{1}{q(3q-1)},$

where the last equality comes from calculation.

######  Concluding Remarks 9.9:

As we just saw, irregular numbers $k$ exists for arbitrary prime powers
$q=p^{f}$ with $f\geq 2$, but by construction, these numbers are large.
Therefore, we expect the share of irregular $k$ to be small; more explicitly,
that for given $q$,

$\limsup_{N\to\infty}\frac{\\#\\{k\leq N\mid k\text{ is irregular}\\}}{N}\leq
c(q)$

with a constant $c(q)<1$ that tends to 0 for $q\to\infty$. Finding good
estimates for $c(q)$ is an interesting challenge. Similarly, the question of
relative frequencies of the Cases 1 and 2 in Theorem 8.9 needs further
investigation. Our tiny database of the cases $(q,k)$ with $q=4$ and $k<100$
odd (we may restrict to $p\nmid k$ in view of the relation
$G_{kp}=(G_{k})^{p}$) gives that

  * •

$k$ is irregular only if $k=75$, see Example 9.7;

  * •

Case 2 holds if and only if $k\in\\{21,69,81,87,93\\}$, see Examples 9.5 and
9.6.

But this is certainly not impressive enough to draw any conclusion …

## References