2.1 Notation

For a given function $v$ we denote by $v^+=\max (v,\,0)$ and by $v^-= -\min (v,\,0)$. For a fixed $k>0$, we define the truncation functions $T_{k}:\mathbb {R}\to \mathbb {R}$ as follows

\[ T_k(s):=\max ({-}k,\min (s,k)). \]

We denote by $|E|$ and by $\mathcal {H}^{N-1}(E)$ respectively the Lebesgue measure and the $(N-1)$–dimensional Hausdorff measure of a set $E$. Moreover $\chi _{E}$ stands for its characteristic function.

If no otherwise specified, we denote by $C$ several positive constants whose value may change from line to line and, sometimes, on the same line. These values will only depend on the data but they will never depend on the indexes of the sequences we introduce below.

2.2 Functional spaces

Throughout this paper, $\Omega \subset \mathbb {R}^N$ (with $N\ge 2$) stands for an open bounded set with, at least, Lipschitz boundary. The unit outward normal vector, which exists $\mathcal {H}^{N-1}$–a.e. on $\partial \Omega$, is denoted by $\nu$.

We denote by $L^q(E)$ the usual Lebesgue space of $q$–summable functions on $E$. The symbol $L^q(\partial \Omega,\, \lambda )$ stands for the Lebesgue space having weight $\lambda$.

A function $f$ belongs to the Marcinkiewicz (or weak Lebesgue) space $L^{N,\infty }(\Omega )$ when $|\{|f|>t\}|\le C t^{-N}$, for any $t>0$. We recall that $L^{N}(\Omega )\subset L^{N,\infty }(\Omega )\subset L^{N-\varepsilon }(\Omega )$, for any $\varepsilon >0$. We refer to [Reference Hunt13] for an overview on these spaces.

We will denote by $W^{1,p}(\Omega )$ the usual Sobolev space, of measurable functions having weak derivative in $L^{p}(\Omega )^N$. It is a Banach space when endowed with the usual norm. It is well-known that functions in Sobolev spaces have a trace on the boundary, this fact allows us to write $u\big |_{\partial \Omega }$. Moreover, if $u\in W^{1,1}(\Omega )$, then $u\big |_{\partial \Omega }\in L^1(\partial \Omega )$ and the embedding $W^{1,1}(\Omega )\to L^1(\partial \Omega )$ is onto. On the other hand, the Sobolev space $W^{1,1}(\Omega )$ is compactly embedded in $L^1(\Omega )$ and continuously embedded into the Lorentz space $L^{\frac {N}{N-1},\,1}(\Omega )$ (see [Reference Alvino2]). Since this Lorentz space has $L^{N,\infty }(\Omega )$ as its dual (see [Reference Hunt13]), it follows that $fu\in L^1(\Omega )$ for every $f\in L^{N,\infty }(\Omega )$ and every $u\in W^{1,1}(\Omega )$. Finally, for a nonnegative $\lambda \in L^\infty (\partial \Omega )$ not identically null, the norm defined in $W^{1,1}(\Omega )$ as

(2.1)\begin{equation} \|v\|_\lambda=\int_\Omega |\nabla v|\, {\rm d}x+\int_{\partial\Omega}\lambda(x)|v|\, {\rm d}{\mathcal{H}}^{N-1} \end{equation}

is equivalent to the usual norm in $W^{1,1}(\Omega )$ (see [Reference Nečas27, section 2.7]).

The space of functions of bounded variation is defined as

\[ BV(\Omega):=\{ u\in L^1(\Omega)\;:\;Du\;{\rm is\;a\;Radon\;measure\;with\;finite\;variation} \}, \]

which is a Banach space.

Most of the features of $W^{1,1}(\Omega )$ also hold for $BV(\Omega )$, since the proofs can easily be adapted by approximation. In this paper, we will use the following facts:

1. (1) equation (2.1) defines a norm in $BV(\Omega )$ equivalent to the usual one;

2. (2) the trace operator $BV(\Omega )\to L^1(\partial \Omega )$ is continuous and onto;

3. (3) the embedding $BV(\Omega )\to L^1(\Omega )$ is compact;

4. (4) the embedding $BV(\Omega )\to L^{\frac {N}{N-1},\,1}(\Omega )$ is continuous.

As a consequence of the last property, $fu\in L^1(\Omega )$ for every $f\in L^{N,\infty }(\Omega )$ and every $u\in BV(\Omega )$. We refer to [Reference Ambrosio, Fusco and Pallara6] for a complete account on this space.

2.3 $L^\infty$-divergence vector fields

We briefly present the $L^\infty$-divergence-measure vector fields theory (see [Reference Anzellotti5] and [Reference Chen and Frid8]). We denote

\[ X(\Omega):=\{ {\bf z}\in L^\infty(\Omega, \mathbb{R}^N) : \operatorname{div}{\bf z} \in L^{N,\infty}(\Omega)\}. \]

In [Reference Anzellotti5] the distribution $({\bf z},\,Dv): C^1_c(\Omega )\to \mathbb {R}$ is defined as

\[ \langle({\bf z},Dv),\varphi\rangle:={-}\int_\Omega v\varphi\operatorname{div}{\bf z}-\int_\Omega v{\bf z}\cdot\nabla\varphi,\quad \varphi\in C_c^1(\Omega), \]

which is well defined if $v\in BV(\Omega )$ and ${\bf z}$ is a bounded vector field such that its divergence belongs to $L^N(\Omega )$. Moreover $({\bf z},\, Dv)$ is a Radon measure satisfying

\[ \left| \int_B ({\bf z}, Dv) \right| \le \int_B \left|({\bf z}, Dv)\right| \le ||{\bf z}||_{L^\infty(U,\mathbb{R}^N)} \int_{B} |Dv|\,, \]

for all Borel sets $B$ and for all open sets $U$ such that $B\subset U \subset \Omega$.

Let us also remark that, in [Reference Anzellotti5], it is shown the existence of a weak trace on $\partial \Omega$ for the normal component of a bounded vector field ${\bf z}$ such that $\operatorname {div}{\bf z} \in L^1(\Omega )$. This is denoted by $[{\bf z},\, \nu ]$ where $\nu (x)$ is the outward normal unit vector. Then it is proven that

\[ ||[{\bf z},\nu]||_{L^\infty(\partial\Omega)}\le ||{\bf z}||_{\infty}\,. \]

Finally a Green formula holds:

\[ \int_{\Omega} v \operatorname{div}{\bf z} + \int_{\Omega} ({\bf z}, Dv) = \int_{\partial \Omega} v[{\bf z}, \nu] \ {\rm d}{\mathcal{H}}^{N-1}, \]

where ${\bf z} \in L^\infty (\Omega,\,\mathbb {R}^N)$, $\operatorname {div}{\bf z} \in L^N(\Omega )$ and $v\in BV(\Omega )$. Let us stress that all previous results can be easily extended to the case where ${\bf z}\in X(\Omega )$ and $u\in BV(\Omega )$ thanks to the continuous embedding of $BV(\Omega )$ into $L^{\frac {N}{N-1},\,1}(\Omega )$.

4.1 The case $\partial \Omega \in C^{1}$

In this section $\Omega$ is a bounded open set of $\mathbb {R}^{N}$ with $C^{1}$ boundary.

The main result of this section is the following:

Clearly we will prove theorem 4.4 by means of approximation through problems (3.1) and using the information already gained on $u_p$. Henceforth ${\bf z}$ and $\beta$ are the ones found in lemmas 3.5 and 3.6 respectively.

Hence we just need to show the identification of both ${\bf z}$ and $\beta$ by proving (4.3) and (4.5).

We start by proving the identification of $\beta$; we first show that the assumption on $M$ can be read as an assumption connecting $\lambda$ and $g$ in an explicit way.

Lemma 4.6 Under the assumptions of theorem 4.4 let $u_p$ be a solution of (3.1). If $M(f,\,g,\,\lambda )\le 1,$ then $|g|\le \lambda +1$. As a consequence,

(4.6)\begin{equation} T_1(\lambda\,{\it sign}\; (r)-g)r\le \lambda|r|-gr \end{equation}

holds for all $r\in \mathbb {R}$.

Proof. It follows from lemma 3.5 that if $M(f,\,g,\,\lambda )\le 1$ then $\|{\bf z}\|_\infty \le 1$ and $\|\beta \chi _{\{\lambda >0\}}\|_\infty \le 1$, so that $-1\le [{\bf z},\,\nu ]\le 1$ and $-1\le \beta \chi _{\{\lambda >0\}}\le 1$. These facts and the identity $[{\bf z},\,\nu ]+\lambda \beta =g$ yield the desired inequality. Indeed,

\begin{align} \lambda-g\ge\lambda\beta-g={-}[{\bf z},\nu]\ge-1\nonumber \end{align}

\begin{align} -\lambda-g\le \lambda\beta-g={-}[{\bf z},\nu]\le1\nonumber \end{align}

wherewith $g\le \lambda +1$ and $-g\le \lambda +1$ hold.

It is enough to analyse two possibilities since (4.6) trivially holds when $r=0$.

If $r>0$, since we have already proven $-1\le \lambda -g$, then $T_1(\lambda -g)r\le \lambda r-gr$.

If $r<0$, since one has $-1\le \lambda +g$ , then $-T_1(\lambda +g)r\le -\lambda r-gr$, that is $T_1(-\lambda -g)r\le \lambda |r|-gr$.

The previous lemma allows us to prove the following result.

Lemma 4.7 Under the assumptions of theorem 4.4 let $u_p$ be a solution of (3.1) and let ${\bf z}$ and $\beta$ be the vector field and the function found in lemma 3.5. Then it holds

\[ u([{\bf z},\nu]+ T_1(\lambda\;{\it sign}\;u-g))= 0\qquad{\mathcal{H}}^{N-1}{\unicode{x2013}a.e.\ on }\ \partial\Omega. \]

In particular it holds (4.5).

Proof. Let us take $T_{k}(u_p)$ as a test function in (3.1) obtaining that

\[ \int_\Omega |\nabla T_{k}(u_p)|^{p}\, {\rm d}x + \int_{\partial\Omega}\lambda |T_{k}(u_p)|^{p} {\rm d}{\mathcal{H}}^{N-1} = \int_\Omega f T_{k}(u_p)\, {\rm d}x + \int_{\partial\Omega}g T_{k}(u_p) {\rm d}{\mathcal{H}}^{N-1}, \]

which, applying the Young inequality, implies that

(4.7)\begin{equation} \begin{aligned} & \int_\Omega |\nabla T_{k}(u_p)|\, {\rm d}x + \int_{\partial\Omega}(\lambda|T_{k}(u_p)|-g T_{k}(u_p)) {\rm d}{\mathcal{H}}^{N-1} \\ & \le \int_\Omega f T_{k}(u_p)\, {\rm d}x + \frac{p-1}{p}|\Omega| + \frac{p-1}{p}\int_{\partial\Omega} \lambda {\rm d}{\mathcal{H}}^{N-1}. \end{aligned} \end{equation}

Owing to lemma 4.6, (4.7) becomes

(4.8)\begin{equation} \begin{aligned} & \int_\Omega |\nabla T_{k}(u_p)|\, {\rm d}x + \int_{\partial\Omega}T_1(\lambda\;{\rm sign}\;(u_p)-g) T_{k}(u_p)\, {\rm d}{\mathcal{H}}^{N-1} \\ & \le \int_\Omega f T_{k}(u_p)\, {\rm d}x + \frac{p-1}{p}\left[|\Omega| + \int_{\partial\Omega} \lambda {\rm d}{\mathcal{H}}^{N-1}\right]. \end{aligned} \end{equation}

Notice that the left-hand side of (4.8) is lower semicontinuous with respect to the $L^1$-convergence as $p\to 1^+$ thanks to proposition $1.2$ of [Reference Modica23]. Hence, taking $p\to 1^+$ in (4.8), one yields to

(4.9)\begin{equation} \int_\Omega |D T_{k}(u)| + \int_{\partial\Omega} T_1(\lambda\;{\rm sign}\; u-g) T_{k}(u)\, {\rm d}{\mathcal{H}}^{N-1} \le \int_\Omega f T_{k}(u)\, {\rm d}x. \end{equation}

Now since it follows from lemma 3.5 that $-{\rm div}\;{\bf z} = f$, from (4.9) one deduces that

\begin{align*} & \int_\Omega |D T_{k}(u)| + \int_{\partial\Omega} T_1(\lambda\;{\rm sign}\; u-g) T_{k}(u)\, {\rm d}{\mathcal{H}}^{N-1}\\ & \quad\le -\int_\Omega \;{\rm div}\;{\bf z}\, T_{k}(u) \\ & \quad= \int_\Omega ({\bf z}, DT_{k}(u)) - \int_{\partial\Omega} T_{k}(u)[{\bf z},\nu]\,{\rm d}{\mathcal{H}}^{N-1}, \end{align*}

where the last equality follows from an application of the Green formula. Now observe that $({\bf z},\,DT_{k}(u)) \le |DT_{k}(u)|$ as measures since $\|{\bf z}\|_{\infty }\le 1$; then one gets

(4.10)\begin{equation} \int_{\partial\Omega} (T_1(\lambda\;{\rm sign}\; u-g) + [{\bf z},\nu]) T_{k}(u)\, {\rm d}{\mathcal{H}}^{N-1} \le 0. \end{equation}

Now observe that $(T_1(\lambda \;{\rm sign}\; u-g) + [{\bf z},\,\nu ])$ has the same sign of $u$ for $\mathcal {H}^{N-1}$–almost every point on $\partial \Omega$. Indeed, assume first that $x\in \partial \Omega$ satisfies $u(x)>0$. Then $\lambda (x)-g(x) \ge -1$ by lemma 4.6. If $\lambda (x)-g(x)\le 1$, then lemma 3.5 gives $\lambda -g +[{\bf z},\,\nu ]=\lambda -g + g-\lambda \beta =\lambda -\lambda \beta \ge 0$ since $|\beta \chi _{\{\lambda >0\}}|\le 1$. Otherwise let $x$ be such that $\lambda (x)-g(x) > 1$ then $1+[{\bf z},\,\nu ] \ge 0$ since $|[{\bf z},\,\nu ]|\le 1$. A similar argument holds when $u(x)<0$.

Thus, (4.10) implies that $(T_1(\lambda \;{\rm sign}\; u-g) + [{\bf z},\,\nu ]) u = 0$ $\mathcal {H}^{N-1}$–almost everywhere on $\partial \Omega$. Moreover since $[{\bf z},\,\nu ] = g-\lambda \beta$ it follows (4.5).

Now we focus on proving (4.3).

Lemma 4.8 Under the assumptions of theorem 4.4 let $u_p$ be a solution of (3.1) and let ${\bf z}$ be the vector field found in lemma 3.5. Then it holds

\[ ({\bf z},Du)=|Du| \quad\text{as measures in } \Omega. \]

Proof. Let us take $T_k(u_p)\varphi$ ($k> 0,\,\; 0\le \varphi \in C^1_c(\Omega )$) as a test function in (3.1) yielding to

\[ \int_\Omega |\nabla T_k(u_p)|^p\varphi \, {\rm d}x + \int_\Omega T_k(u_p) |\nabla u_p|^{p-2}\nabla u_p\cdot \nabla \varphi \, {\rm d}x = \int_\Omega f T_k(u_p)\varphi\, {\rm d}x, \]

which, from an application of the Young inequality, implies

\begin{align*} & \int_\Omega |\nabla T_k(u_p)|\varphi \, {\rm d}x + \int_\Omega T_k(u_p) |\nabla u_p|^{p-2}\nabla u_p\cdot \nabla \varphi \, {\rm d}x\\ & \quad \le \int_\Omega f T_k(u_p)\varphi\, {\rm d}x + \frac{p-1}{p}\int_\Omega \varphi \, {\rm d}x. \end{align*}

By taking $p\to 1^+$ in the previous inequality, one obtains that

\[ \int_\Omega |D T_k(u)|\varphi + \int_\Omega T_{k}(u) {\bf z}\cdot \nabla \varphi \, {\rm d}x \le \int_\Omega f T_k(u)\varphi\, {\rm d}x. \]

Hence, letting $k\to +\infty$,

\[ \int_\Omega |D u|\varphi + \int_\Omega u {\bf z}\cdot \nabla \varphi \, {\rm d}x \le \int_\Omega f u\varphi\, {\rm d}x. \]

Now, recalling that $-{\rm div}\; {\bf z} = f$ one has that

\[ \int_\Omega |D u|\varphi \le - \int_\Omega u {\bf z}\cdot \varphi \, {\rm d}x -\int_\Omega \;{\rm div}\; {\bf z} u\varphi\, {\rm d}x = \int_\Omega ({\bf z}, D u)\varphi. \]

This concludes the proof being the reverse inequality trivial since $||{\bf z}||_{\infty }\le 1$.

4.2 The case $\partial \Omega$ Lipschitz

In the previous subsection we required that $\Omega$ has $C^1$ boundary. This fact is due to the application in lemma 4.7 of Modica's semicontinuity result that needs this hypothesis. Nevertheless, as Modica himself points out, certain functionals are lower semicontinuous with respect to the $L^1$-convergence even when the Lipschitz-continuous setting is considered.

We prove the following result.

Lemma 4.9 Let $H\>:\> BV(\Omega )\to \mathbb {R}$ be a functional defined as

\[ H(u)=\int_\Omega|Du|+\int_{\partial\Omega}\psi(x)|u|\, {\rm d}{\mathcal{H}}^{N-1} \]

where $\psi \in L^\infty (\partial \Omega )$ satisfies $0\le \psi \le 1$.

Then $H$ is lower semicontinuous with respect to the $L^1$-convergence.

Proof. We first choose an open bounded set $\Omega '$ containing $\overline \Omega$. Given $\psi \in L^\infty (\partial \Omega )$, we may find $\phi _1\in C^1(\Omega )\cap W^{1,1}(\Omega )$ such that $\phi _1\big |_{\partial \Omega }=\psi$ and $0\le \phi _1\le 1$. We may also consider $\phi _2\in C^1(\Omega '\backslash \overline \Omega )\cap W^{1,1}(\Omega '\backslash \overline \Omega )$ such that $\phi _2\big |_{\partial \Omega }=\psi$ and $0\le \phi _2\le 1$. Finally define the following continuous extension of $\psi$:

\[ \varphi(x)=\left\{\begin{array}{@{}ll} \phi_1(x) & \;{\rm if}\;x\in\Omega\\ \phi_2(x) & \;{\rm if}\;x\in\Omega'\backslash \overline\Omega\,. \end{array}\right. \]

We next claim that each $u\in BV(\Omega )$ satisfies

\begin{align*} & \int_{\Omega} \phi_1|Du|+\int_{\partial\Omega}\psi|u|\, {\rm d}{\mathcal{H}}^{N-1}\\ & \quad=\sup\left\{\int_{\Omega'} u\, \;{\rm div}\;(\varphi F)\, {\rm d}x\>:\>F\in C_0^1(\Omega')^N\ \ \|F\|_\infty\le1\right\}, \end{align*}

where $u$ is extended to $BV(\Omega ')$ by defining $u=0$ in $\Omega '\backslash \overline \Omega$.

An inequality is obvious since Green's formula implies

\begin{align*} \int_{\Omega'} u\, \;{\rm div}\;(\varphi F)\, {\rm d}x & =\int_{\Omega} u\, \;{\rm div}\;(\phi_1 F)\, {\rm d}x\\ & ={-}\int_{\Omega} \phi_1 F\cdot Du+\int_{\partial\Omega}\psi u [F, \nu]\, {\rm d}{\mathcal{H}}^{N-1}\\ & \le \int_{\Omega} \phi_1|Du|+\int_{\partial\Omega}\psi|u|\, {\rm d}{\mathcal{H}}^{N-1} \end{align*}

holds for all $F\in C_0^1(\Omega ')^N$ such that $\|F\|_\infty \le 1$.

To check the reverse inequality, we consider in $C_0^1(\Omega ')^N$ the linear map given by

\[ L(F)=\int_{\Omega'} u\, \;{\rm div}\;(\varphi F)\, {\rm d}x\,. \]

Notice that

\begin{align*} |L(F)|& =\left|\int_{\Omega'} u\, \;{\rm div}\;(\varphi F)\, {\rm d}x\right|=\left|-\int_{\Omega} \phi_1 F\cdot Du+\int_{\partial\Omega}\psi u [F, \nu]\, {\rm d}{\mathcal{H}}^{N-1}\right|\\ & \le \|F\|_\infty\left[\int_{\Omega} \phi_1|Du|+\int_{\partial\Omega}\psi|u|\, {\rm d}{\mathcal{H}}^{N-1}\right]\,. \end{align*}

From this inequality we deduce that $L$ can be extended by density to a linear and continuous map in $C_0(\Omega ')^N$ whose norm satisfies

\[ \|L\|\le \int_{\Omega} \phi_1|Du|+\int_{\partial\Omega}\psi|u|\, {\rm d}{\mathcal{H}}^{N-1}\,. \]

Applying the Riesz representation theorem, there exists a Radon measure $\mu$ on $\Omega '$ such that $L(F)=\int _{\Omega '} F\cdot \mu$ for every $F \in C_0(\Omega ')^N$ and its total variation is $\int _{\Omega '}|\mu |=\|L\|$. Thus,

\[ \int_{\Omega'} F\cdot \mu=L(F)=\int_{\Omega'} u\, \;{\rm div}\;(\varphi F)\, {\rm d}x={-}\int_\Omega \phi_1 F\cdot Du+\int_{\partial\Omega}u\psi [F, \nu]\, {\rm d}{\mathcal{H}}^{N-1} \]

for all $F\in C_0^1(\Omega ')^N$. We deduce that

\[ \int_{\Omega} \phi_1|Du|+\int_{\partial\Omega}\psi|u|\, {\rm d}{\mathcal{H}}^{N-1}=\int_{\Omega'} |\mu|=\sup\left\{L( F)\>:\> F\in C_0(\Omega')^N\ \ \|F\|_\infty\le1\right\}\,. \]

By density, we conclude that

\[ \int_{\Omega} \phi_1|Du|+\int_{\partial\Omega}\psi|u|\, {\rm d}{\mathcal{H}}^{N-1}=\sup\left\{L( F)\>:\> F\in C_0^1(\Omega')^N\ \ \|F\|_\infty\le1\right\} \]

and the claim is proven.

As a straightforward consequence the functional

\[ u\mapsto \int_{\Omega} \phi_1|Du|+\int_{\partial\Omega}\psi|u|\, {\rm d}{\mathcal{H}}^{N-1} \]

is lower semicontinuous with respect to the $L^1$–convergence. Therefore,

\[ H(u)=\int_{\Omega} (1-\phi_1)|Du|+\int_{\Omega} \phi_1|Du|+\int_{\partial\Omega}\psi|u|\, {\rm d}{\mathcal{H}}^{N-1} \]

is the sum of two lower semicontinuous functionals, so that lemma is proven.

The previous lemma can be applied to the functional

(4.11)\begin{equation} I(u)=\int_\Omega |D u| + \int_{\partial\Omega}T_1(\lambda\;{\rm sign}\;(u)-g) u\, {\rm d}{\mathcal{H}}^{N-1}, \quad u\in BV(\Omega)\,, \end{equation}

as shown in proposition 4.10 below. Here we only have to take into account the inequalities $|a^+-b^+|\le |a-b|$ and $|a^--b^-|\le |a-b|$, which hold for all real numbers.

Proof. First write $I=I_1+I_2$, where

\[ I_1(u)=I(u^+)=\int_\Omega |D u^+|\, {\rm d}x + \int_{\partial\Omega}T_1(\lambda-g) u^+\, {\rm d}{\mathcal{H}}^{N-1} \]

and

\[ I_2(u)=I({-}u^-)=\int_\Omega |D u^-|\, {\rm d}x + \int_{\partial\Omega}T_1(\lambda+g) u^-\, {\rm d}{\mathcal{H}}^{N-1} \]

Take a sequence $u_n$ in $BV(\Omega )$ that converges to $u$ strongly in $L^1(\Omega )$. Then $u_n^+$ converges to $u^+$ and $u_n^-$ converges to $u^-$ as $n\to \infty$, so that lemma 4.9 implies that

\[ I_1(u)\le \liminf_{n\to\infty}I_1(u_n) \]

and

\[ I_2(u)\le \liminf_{n\to\infty}I_2(u_n)\,. \]

Therefore, its sum $I$ is lower semicontinuous.

5.1 The case with $\lambda \in L^1(\partial \Omega )$

Here we briefly spend a few words for the case of a nonnegative $\lambda \in L^1(\partial \Omega )$.

Indeed, let us stress that, even for $\lambda \in L^1(\partial \Omega )$, the quotient which appears in $M$ is well defined. Nevertheless, now the supremum is taken over all $u\in W^{1,1}(\Omega )\cap L^1(\partial \Omega,\, \lambda )\backslash \{0\}$

Then if one considers the following approximation scheme

(5.1)\begin{equation} \begin{cases} \displaystyle -\Delta_p u_p = f & \text{ in }\Omega,\\ \displaystyle |\nabla u_p|^{p-2}\nabla u_p\cdot \nu +\lambda |u_p|^{p-2}u_p = g & \text{ on } \partial\Omega, \end{cases} \end{equation}

where $f\in L^{N,\infty }(\Omega )$, $g\in L^\infty (\partial \Omega )$ and $0\le \lambda \in L^1(\partial \Omega )$ but not identically null, the existence of $u_p\in W^{1,p}(\Omega ) \cap L^p(\partial \Omega,\, \lambda )$ satisfying (5.1) follows from the minimization of the following functional

\[ Q(u)=\frac{1}{p}\int_{\Omega}\left|\nabla u\right|^{p}{\rm d}x +\int_{\partial\Omega}\frac{\lambda}{p}\left|u\right|^{p} \ {\rm d}{\mathcal{H}}^{N-1} - \int_{\partial\Omega}gu \ {\rm d}{\mathcal{H}}^{N-1} - \int_{\Omega}fu \ {\rm d}x. \]

Indeed, we consider the space $W^{1,p}(\Omega ) \cap L^p(\partial \Omega,\, \lambda )$ endowed with the norm defined as $\displaystyle \|u\|^p_{p,\lambda }=\int _\Omega |\nabla u|^p{\rm d}x+\int _{\partial \Omega }\lambda |u|^p\, {\rm d}{\mathcal {H}}^{N-1}$. We remark that $\|u\|_{p,\lambda }$ is not anymore an equivalent norm to the $W^{1,p}$–norm. By the way one can convince himself that it always holds the inequality

\[ \|u\|_{p,\lambda} \ge C\|u\|_{W^{1,p}(\Omega)}, \]

which allows to deduce all continuous and compact embeddings which holds for $W^{1,p}(\Omega )$. Since $Q$ can be written as

\[ Q(u)=\frac1p\|u\|_{p,\lambda}^p - \int_{\partial\Omega}gu \ {\rm d}{\mathcal{H}}^{N-1} - \int_{\Omega}fu \ {\rm d}x\,, \]

it follows that these embeddings lead to coercivity. We also deduce from these embeddings that $Q$ is weakly lower semicontinuous. Standard results then yield the desired minimizer.

Any minimizer $u_p$ of the previous functional satisfies that

\[ \displaystyle \int_{\Omega}\left|\nabla u_p\right|^{p-2}\nabla u_p\cdot \nabla \varphi\ {\rm d}x +\int_{\partial\Omega}\lambda\left|u_p\right|^{p-2}u_p\varphi\ {\rm d}{\mathcal{H}}^{N-1} = \int_{\Omega}f\varphi\ {\rm d}x + \int_{\partial\Omega}g\varphi\ {\rm d}{\mathcal{H}}^{N-1}, \]

where $\varphi \in W^{1,p}(\Omega ) \cap L^p(\partial \Omega,\, \lambda )$. Therefore $u_p$ itself can be taken as a test function. Now similar estimates for $\|u_p\|_{\lambda }$ can be obtained. Notice that $\|u_p\|_{\lambda } \ge C\|u_p\|_{BV(\Omega )}$ but they are not equivalent. With this approach in mind one can show that the results of both § 3 and 4 still hold if $0\le \lambda \in L^1(\Omega )$ (but not identically null) with natural modifications.

5.2 The radial case

Here we deal with the case $\Omega$ as a ball of radius $R$ centered at the origin, namely:

\[ \Omega = B_R := \{x\in \mathbb{R}^N: |x|< R\}. \]

Hence let us consider the following problem

\[ \begin{cases} \displaystyle -\Delta_p u_p = \frac{A}{\left|x\right|} & \text{in} \ \Omega,\\ |\nabla u_p|^{p-2}\nabla u_p \cdot \nu + \lambda u^{p-1}_p=\gamma & \text{on}\ \partial\Omega\,, \end{cases} \]

where $A,\,\lambda$ and $\gamma$ are positive constants, and we are firstly interested in the asymptotic behaviour of $u_p$ as $p\to 1^+$. We explicitly observe that the datum $A/\left |x\right |\in L^{N,\infty }(\Omega )$.

Hence we look for a function $u_p(r)$ ($r=|x|$) satisfying

\[ -\frac{1}{r^{N-1}}\left(r^{N-1} |u_p'(r)|^{p-2} u_p'(r)\right)^{'} = \frac{A}{r}, \]

which gives

\[ [r^{N-1}({-}u_p'(r))^{p-1}]' = \frac{A}{r^{2-N}} \]

and

(5.2)\begin{equation} -u_{p}'(r)=\left(\frac{A}{N-1}\right)^{\frac{1}{p-1}}. \end{equation}

Now integrating between $r$ and $R$ (with an abuse of notation) one has

\[ u_p(r) = u_p(R) + \left(\frac{A}{N-1}\right)^{\frac{1}{p-1}}\left(R- r\right), \]

and since it follows from the boundary condition and from (5.2) that

\[ u_p(R)=\left\{\frac{1}{\lambda}\left[\frac{A}{N-1}+\gamma\right]\right\}^{\frac{1}{p-1}} \]

then one also has

\[ u_p(r) = \left\{\frac{1}{\lambda}\left[\frac{A}{N-1}+\gamma\right]\right\}^{\frac{1}{p-1}} + \left(\frac{A}{N-1}\right)^{\frac{1}{p-1}}\left(R- r\right). \]

Let us underline that:

1. (1) if $A>N-1$, then $u_{p}\to +\infty$ in $\Omega$;

2. (2) if $A=N-1$, then

   1. (a) if $\lambda <1+\gamma$, then $u_{p}\to +\infty$;

   2. (b) if $\lambda =1+\gamma$, then $u_{p}\to 1+(R-r)$;

   3. (c) if $\lambda >1+\gamma$, then $u_{p}\to R-r$;

3. (3) if $A< N-1$, then

   1. (a) if $\lambda < \frac {A}{N-1}+\gamma$, then $u_{p}\to +\infty$ in $\bar \Omega$;

   2. (b) if $\lambda =\frac {A}{N-1}+\gamma$, then $u_{p}\to 1$;

   3. (c) if $\lambda >\frac {A}{N-1}+\gamma$, then $u_{p}\to 0$.

Remark 5.1 Let $\Omega =B_{R}$ and $A,\,\gamma \ge 0$. A posteriori from lemma 3.4 and 3.7, last example assures what follows.

• • In the cases: $A>N-1$; $A=N-1$ and $\lambda <1+\gamma$; $(N-1)(\lambda -\gamma ) < A< N-1$, then

  \[ M(A/\left|x\right|,\gamma,\lambda) >1. \]

• • In the cases: $A=N-1$ and $\lambda \ge 1+\gamma$; $A=(\lambda -\gamma )(N-1)< N-1$, then

  \[ M(A/\left|x\right|,\gamma,\lambda) =1. \]

• • In the case $A<\min \{N-1,\,(\lambda -\gamma )(N-1)\}$ then

  \[ M(A/\left|x\right|,\gamma,\lambda) <1. \]

We conclude that

\[ M(A/\left|x\right|,\gamma,\lambda)=\max\left\{\frac{A}{N-1}, \frac1\lambda\left[\frac{A}{N-1}+\gamma\right]\right\} \]

5.3 A variational approach

In the case $\lambda (x)=\lambda$ positive constant, and $g\equiv 0$, the argument used in [Reference Della Pietra, Nitsch, Oliva and Trombetti12] allows to prove that the functional

\[ J_{p}(u)=\frac{1}{p}\int_{\Omega}\left|\nabla u\right|^{p} \ {\rm d}x +\frac{\lambda}{p}\int_{\partial\Omega}\left|u\right|^{p} \ {\rm d}{\mathcal{H}}^{N-1} -\int_{\Omega}fu \ {\rm d}x \]

$\Gamma$-converges in $BV(\Omega )$ to

\[ J(u)=\int_\Omega|D u|+\min\{\lambda,1\}\int_{\partial\Omega}|u| \ {\rm d}{\mathcal{H}}^{N-1}-\int_{\Omega}fu \ {\rm d}x. \]

Let us observe that the minimizers of $J_{p}$ in $W^{1,p}(\Omega )$ are solutions to (3.1). Formally, (4.1) is the Euler–Lagrange equation related to $J$. Then, if $M(f,\,0,\,\lambda )\le 1$ it follows that the minimizers of $J_{p}$ converge (in $BV(\Omega )$) to a minimizer of $J$.

Actually, the truncation appearing in the boundary datum seems to be natural. If one considers $\lambda >1$ and the functional

\[ \tilde J(u)=\int_\Omega|D u|+\lambda\int_{\partial\Omega}|u| \ {\rm d}{\mathcal{H}}^{N-1}-\int_{\Omega}fu \ {\rm d}x, \]

it is easy to convince that

(5.3)\begin{equation} \min_{u\in BV(\Omega)} \tilde J(u)= \min_{u\in BV(\Omega)} J(u). \end{equation}

Indeed, if $v$ is a minimum for $J$, theorem $3.1$ of [Reference Littig and Schuricht18] assures the existence of a sequence $v_k\in C^\infty _c(\Omega )$ which converges to $v$ in $L^q(\Omega )$ for any $q\le \frac {N}{N-1}$ and such that $\int _\Omega |\nabla v_k|\, {\rm d}x$ converges to $\int _{\mathbb {R}^N} |Dv|$ as $k\to \infty$. Hence $\tilde J(v_k)=J(v_k)$ for all $k>0$ and $\min _{u\in BV(\Omega )} \tilde J(u)\le \lim _{k\to +\infty } J(v_{k})= \min _{u\in BV(\Omega )} J(u)$. Being the reverse inequality trivial, it holds (5.3).

5.4 A sharp estimate on $M(f,\,g,\,\lambda )$

Let $f\equiv 1$, $g\equiv 0$, $\lambda \ge 0$ and let $\Omega$ be a Lipschitz bounded domain. Then, by pointing out the dependence of $M$ by $\Omega$,

\[ M(1,0,\lambda)= M(1,0,\lambda,\Omega)= \sup_{u\in W^{1,1}(\Omega)\setminus\{0\}}\left\{\frac{\displaystyle\int_{\Omega}\left|u\right|{\rm d}x}{\displaystyle\int_{\Omega}\left|\nabla u\right|{\rm d}x+\lambda\int_{\partial\Omega}\left|u\right|{\rm d}{\mathcal{H}}^{N-1}}\right\}. \]

In this case we denote by $\Lambda (\Omega,\,\lambda )=\frac {1}{M(1,\,0,\,\lambda,\,\Omega )}$, and the value $\Lambda (\Omega,\,\lambda )$ is the limit, as $p\to 1$, of the first Robin $p$-Laplace eigenvalue (see [Reference Della Pietra, Nitsch, Oliva and Trombetti12]). It has been proved in [Reference Della Pietra, Nitsch, Oliva and Trombetti12] that when $\lambda >0$ and $\Omega$ is a Lipschitz bounded domain, then $\Lambda (\Omega,\,\lambda )\in ]0,\,+\infty [$ and

(5.4)\begin{equation} \Lambda(\Omega,\lambda)\ge \min\{\lambda,1\}\frac{N}{R}, \end{equation}

where $R$ is the radius of the ball having the same volume than $\Omega$. Moreover, for any $\lambda \ge 0$, inequality (5.4) is an equality when $\Omega$ is a ball. Then (5.4) gives an explicit upper bound for $M(1,\,0,\,\lambda,\,\Omega )$, and then an explicit condition in order to obtain that the solutions $u_{p}$ of (3.1), for this particular choice of the coefficients, go to zero in $\Omega$ as $p\to 1$.