Doubly nonlinear equations for the 1-Laplacian

Abstract

This paper is concerned with the Neumann problem for a class of doubly nonlinear equations for the 1-Laplacian,

$$\begin{aligned} \frac{\partial v}{\partial t} - \Delta _1 u \ni 0 \hbox { in } (0, \infty ) \times \Omega , \quad v\in \gamma (u), \end{aligned}$$

and initial data in $L^1(\Omega )$, where $\Omega$ is a bounded smooth domain in ${\mathbb {R}}^N$ and $\gamma$ is a maximal monotone graph in ${\mathbb {R}}\times {\mathbb {R}}$. We prove that, under certain assumptions on the graph $\gamma$, there is existence and uniqueness of solutions. Moreover, we proof that these solutions coincide with the ones of the Neumann problem for the total variational flow. We show that such assumptions are necessary.

1 Introduction

Consider the doubly nonlinear diffusion problem:

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{\partial v}{\partial t}(t,x)={\textrm{div}}(\alpha (\nabla u(t,x))), \quad &{} \hbox {in} \ \ (0, \infty ) \times \Omega , \\ v\in \gamma (u),\quad &{} \hbox {in} \ \ (0, \infty ) \times \Omega , \\ v(0, x) = v_0(x), \quad &{} x \in \Omega , \end{array} \right. \end{aligned}$$

(1.1)

completed with boundary conditions, being $\Omega$ a bounded domain in ${\mathbb {R}}^N$, $\gamma$ a maximal monotone graph (possibly multivalued) in ${\mathbb {R}}\times {\mathbb {R}}$ and $\alpha :{\mathbb {R}}^N\rightarrow {\mathbb {R}}^N$. Typical examples are $\alpha (\xi )=\alpha _p(\xi ):=|\xi |^{p-2}\xi$, $p>1$, and $\gamma (r)=|r|^{m-1}r$, $m>0$. In these particular cases, for $p=2$ and $m=1$ the equation reduces to the classical heat equation, while for $0<m < 1$ it is the porous medium equation (see, e.g., [26]) and the p -Laplacian diffusion equation for $p>1$ and $m=1$. In a general framework, case $0< m<p-1$ is known as a doubly nonlinear equation with slow diffusion, while the case $m>p-1$ is named a fast diffusion equation (see, e.g., [22]). Therefore, owing to the choice of $\alpha$ and the graph $\gamma$, this equation may arise a variety of different situations and it possess a wide spectrum of applications, for instance, in fluid dynamics, soil science and filtration, see [11] and [25]. Observe that, for $p =2$, other typical examples are

$$\begin{aligned} \gamma (r)= \left\{ \begin{array}{lll} r \quad &{}\hbox {if} \ \ r < 0, \\ 0 \quad &{}\hbox {if} \ \ 0\le r\le 1, \\ r-1 \quad &{}\hbox {if} \ \ r > 1, \end{array} \right. \end{aligned}$$

for a Stefan type problem, or

$$\begin{aligned} \gamma (r)= \left\{ \begin{array}{lll} 0 \quad &{}\hbox {if} \ \ r < 0, \\ \hbox {[0,1]} \quad &{}\hbox {if} \ \ r=0, \\ 1 \quad &{}\hbox {if} \ \ r > 0. \end{array} \right. \end{aligned}$$

for a Hele-Shaw-type problem.

From a mathematical point of view, there is an extensive literature related to problem (1.1). Existence, uniqueness, regularity and asymptotic behavior of solutions are treated under different restrictions on $\gamma$ and $\alpha$, and we refer some literature: [1, 2, 16, 21, 23, 24, 26] and the literature therein.

Our main aim is to deal with existence and uniqueness for the limit case $p=1$ for the function $\alpha _p$, that is, $\displaystyle \alpha _1(\xi ):=\frac{\xi }{|\xi |}$, $\gamma$ a maximal monotone graph and homogeneous Neumann boundary conditions. More precisely, by means of Crandall–Liggett’s theorem we obtain existence and uniqueness of entropy solution (see Definition 4.5) of the doubly nonlinear problem

$$\begin{aligned} \left\{ \begin{array}{lll}\frac{\partial v}{\partial t} - \Delta _1 u \ni 0 &{}\hbox {in} \ \ (0, \infty ) \times \Omega , \\ v= \gamma (u)&{}\hbox {in} \ \ (0, \infty ) \times \Omega , \\ \frac{\partial u}{\partial \eta } = 0 &{}\hbox {on} \ \ (0, \infty ) \times \partial \Omega , \\ v(0, x) = v_0(x) \quad &{} x \in \Omega ,\end{array} \right. \end{aligned}$$

(1.2)

under the condition

$$\begin{aligned} \left\{ \begin{array}{l}\gamma \hbox { is a non-decreasing continuous function such that } \gamma (0)=0\hbox { and}\\ \hbox {Rang}(\gamma )={\mathbb {R}}. \end{array}\right. \end{aligned}$$

(1.3)

For this purpose, first of all we deal with the following elliptic problem

$$\begin{aligned} \left\{ \begin{array}{ll} v - \Delta _1 u\ni f \quad \hbox {in} \ \Omega , \\ v\in \gamma (u) \quad \hbox {in}\ \Omega , \\ \frac{\partial u}{\partial \eta } =0 \quad \hbox {on} \ \partial \Omega . \end{array} \right. \end{aligned}$$

In Theorems 3.8 and 3.9, we prove the existence of solutions under the condition

$$\begin{aligned} \left\{ \begin{array}{l}\gamma \hbox { is a maximal monotone graph such that } \gamma (0)\ni 0\hbox { and}\\ \hbox {Rang}(\gamma )={\mathbb {R}}, \end{array}\right. \end{aligned}$$

(1.4)

and we prove uniqueness for continuous $\gamma$ in Theorem 3.7, that is, under assumption (1.3). Note that (1.3) implies (1.4). Moreover, we see that for non-continuous maximal monotone graphs there is non-uniqueness (Example 3.6). We also show that condition $\hbox {Rang}(\gamma )={\mathbb {R}}$ is necessary for the existence of solutions (Example 3.12).

Remark 1.1

On account of our approach to solve problem (1.2) and the above comments, condition (1.3) is natural for the study of such evolution problem. $\square$

In [5] (see also [6]), it was studied the well-posedness of the Neumann problem

$$\begin{aligned} \left\{ \begin{array}{lll}\frac{\partial v}{\partial t} - \Delta _1 v \ni 0 \quad &{}\hbox {in} \ \ (0, \infty ) \times \Omega , \\ \frac{\partial v}{\partial \eta } = 0 \quad &{}\hbox {on} \ \ (0, \infty ) \times \partial \Omega , \\ v(0, x) = v_0(x) \quad &{} x \in \Omega ,\end{array} \right. \end{aligned}$$

(1.5)

by means of the Nonlinear Semigroup Theory. For that purpose, the following operator ${\mathcal {A}}$, defined in $L^1(\Omega )\times L^1(\Omega )$, was introduced to give mathematical sense to the formal expression of $\Delta _1v:=\hbox {div}\left( \frac{\nabla v}{|\nabla v|}\right)$ (jointly with the homogenous Neumann boundary conditions).

Definition 1.2

$$\begin{aligned} \begin{aligned} (v, w) \in {\mathcal {A}} \iff {}&{} v \in L^1(\Omega ), \, T_k(v) \in BV(\Omega ) \text{ for } \text{ all } k>0, \\{}&{} \text{ and } \text{ there } \text{ exists } {\textbf {z}}\in X_1(\Omega ), \Vert {\textbf {z}}\Vert _{\infty }\le 1, \hbox { such that } \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} w=&{} - \text {div} ({\textbf {z}}) \quad \text{ in } \ {{\mathcal {D}}}^{\prime }(\Omega ), \\ \int _{\Omega } ({\textbf {z}}, DT_k(v))=&{} \int _{\Omega } \vert DT_k(v) \vert \quad \forall k >0, \end{aligned} \end{aligned}$$

and

$$\begin{aligned}{}[\textbf{z}, \nu ] = 0 \quad {\mathcal {H}}^{N-1}\hbox {-a.e. on }\partial \Omega , \end{aligned}$$

(see notation in Sects. 2.1 and 2.2).

Moreover, it was shown that ${\mathcal {A}}$ is the closure in $L^1(\Omega )\times L^1(\Omega )$ of the subdifferential of the energy functional $\Phi : L^2(\Omega ) \rightarrow (-\infty ,+\infty ]$ defined by

$$\begin{aligned} \Phi (v) = \left\{ \begin{array}{ll} \displaystyle \int _{\Omega } \vert D v \vert &{}\hbox { if }v \in BV(\Omega ) \cap L^2(\Omega ), \\ +\infty &{}\hbox { if } v \in L^2(\Omega ) \setminus BV(\Omega ). \end{array} \right. \end{aligned}$$

Since $\Phi$ is a proper convex and lower semi-continuous function, then $\partial \Phi$ is a maximal monotone operator with dense domain, generating a contraction semigroup in $L^2(\Omega )$ that solves problem (1.5) for $L^2$-data. Entropy solutions for $L^1$-data $v_0$ were introduced to characterize mild solutions of the abstract Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{l} v_t+{\mathcal {A}}(v)\ni 0,\\ v(0)=v_0, \end{array}\right. \end{aligned}$$

(1.6)

given by the Crandall–Liggett’s semigroup generation theorem ( [19]).

Remark 1.3

We show that the solutions of (1.2) are given by the solutions of (1.5) (Theorem 4.4). This is a non-trivial result; we first need to prove directly existence and uniqueness of solutions of problem (1.2). Observe that, at the level of elliptic problems, we first prove Theorem 3.9 and afterward we can prove Theorem 3.11.

The fact that solutions of (1.5) are solutions of (1.2) gives a kind of invariance property for the diffusion evolution problem via the 1-Laplacian, i.e., changing variables,

$$\begin{aligned} \begin{aligned}{}&{} \text{ the } \text{ solutions } \text{ of } w_t-\Delta _1 w\ni 0 \text{ and } \text{ the } \text{ solutions } \text{ of } w_t-\Delta _1 \gamma ^{-1}(w)\ni 0\\{}&{} \text{ are } \text{ the } \text{ same } \text{ provided } \text{ that }\ \gamma \text{ satisfies } ()(1.3). \end{aligned} \end{aligned}$$

Observe that, written in this way, $\gamma ^{-1}$ can be a non-continuous maximal monotone graph, hence not necessarily Lipschitz-continuous. When $\gamma ^{-1}$ is an increasing and Lipschitz-continuous function, solutions of (1.5) are solutions of (1.2), see Proposition 3.10 at the level of the elliptic problems. $\square$

2 Preliminaries

2.1 Functions of bounded variation

We will denote by ${\mathcal {M}}(\Omega )$ the set of all Lebesgue measurable functions in $\Omega$.

The natural energy space to study problem (1.2) is the space of functions of bounded variation. For further information concerning functions of bounded variation, we refer to [4] and [20]. Recall that if $\Omega$ is an open subset of ${\mathbb {R}}^N$, a function $u \in L^1(\Omega )$ whose gradient Du in the sense of distributions is a vector valued Radon measure with finite total variation in $\Omega$ is called a function of bounded variation. The class of such functions will be denoted by $BV(\Omega )$. The total variation of Du in $\Omega$ is defined by the formula

$$\begin{aligned} \begin{aligned} \vert Du \vert (\Omega ) = \sup \bigg \{ \int _{\Omega } u \ \text {div} (\phi ) \, : \, \phi \in C_0^{\infty }(\Omega , {\mathbb {R}}^N), \ \Vert \phi \Vert \le 1 \bigg \}. \end{aligned} \end{aligned}$$

The space $BV(\Omega )$ is endowed with norm

$$\begin{aligned} \Vert u \Vert _{BV(\Omega )} = \Vert u \Vert _{L^1(\Omega )} + \vert Du \vert (\Omega ). \end{aligned}$$

Recall that an ${{\mathcal {L}}}^N$-measurable subset E of ${\mathbb {R}}^N$ has finite perimeter if ${\mathrm{\chi }}_{E} \in BV({\mathbb {R}}^N)$. The perimeter of E is defined by $\textrm{Per}(E) = \vert D {\mathrm{\chi }}_E \vert ({\mathbb {R}}^N)$.

2.2 A generalized Green’s formula

Let $\Omega$ be an open bounded set in ${\mathbb {R}}^N$ with Lipschitz boundary. Following [10], for $1 \le p \le \infty$ let

$$\begin{aligned} X_p(\Omega ) = \{ \textbf{z}\in L^{\infty }(\Omega , {\mathbb {R}}^N): \textrm{div}(\textbf{z})\in L^p(\Omega ) \}. \end{aligned}$$

If $\textbf{z}\in X_p(\Omega )$ and $w \in BV(\Omega ) \cap L^{p'}(\Omega )$, we define the functional $(\textbf{z},Dw): C^{\infty }_{0}(\Omega ) \rightarrow {\mathbb {R}}$ by the formula

$$\begin{aligned} \left\langle (\textbf{z},Dw),\varphi \right\rangle = - \int _{\Omega } w \, \varphi \, \textrm{div}(\textbf{z}) \, dx - \int _{\Omega } w \, \textbf{z}\cdot \nabla \varphi \, dx. \end{aligned}$$

Then, $(\textbf{z},Dw)$ is a Radon measure in $\Omega$,

$$\begin{aligned} \int _{\Omega } (\textbf{z},Dw) = \int _{\Omega } \textbf{z}\cdot \nabla w \, dx \ \ \ \ \ \ \forall \ w \in W^{1,1}(\Omega ) \cap L^{p'}(\Omega ) \end{aligned}$$

and

$$\begin{aligned} \bigg \vert \int _{B} (\textbf{z},Dw) \bigg \vert \le \int _{B} |(\textbf{z},Dw)| \le \Vert \textbf{z}\Vert _{\infty } \int _{B} \vert Dw \vert \end{aligned}$$

(2.1)

for any Borel set $B \subseteq \Omega$.

Moreover, when $\textbf{z}\in X_p(\Omega )$ and $w \in BV(\Omega ) \cap L^{p'}(\Omega )$, we have the following integration by parts formula

$$\begin{aligned} \int _{\Omega } w \, \textrm{div} \,(\textbf{z}) \, dx + \int _{\Omega } (\textbf{z}, Dw) = \int _{\partial \Omega } [\textbf{z}, \nu ] w \, d {{\mathcal {H}}}^{N-1}, \end{aligned}$$

(2.2)

where $[\textbf{z}, \nu ]$ is the weak trace on $\partial \Omega$ of the normal component of $\textbf{z}$ (see [10]).

By (2.1), the measures $(\textbf{z},Du)$ and $\vert (\textbf{z},Du) \vert$ are absolutely continuous with respect to the measure $\vert Du\vert$ in $\Omega$.

Thus, there is a density function

$$\begin{aligned} \theta (\textbf{z}, Dw, \cdot ) =\frac{d (\textbf{z},Dw)}{d |Dw|} \in L^{\infty }(\Omega , |Dw|), \end{aligned}$$

satisfying

$$\begin{aligned} |\theta (\textbf{z}, Dw, x)|\le 1\text { for }|Dw|\text {-a.e. }x\in \Omega . \end{aligned}$$

The function $\theta (\textbf{z}, Dw, \cdot )$ is called the Radon–Nikodým derivative of $(\textbf{z},Dw)$ with respect to |Dw|. Moreover, the following results hold.

Proposition 2.1

([10], Chain rule for $(\textbf{z},D(\cdot ))$) Let $\Omega$ be a bounded domain with a Lipschitz-continuous boundary $\partial \Omega$, and, for $1\le p\le N$ and $p^{_{\prime }}$ its conjugate exponent, let $\textbf{z}\in X_p(\Omega )$ and $w \in BV(\Omega ) \cap L^{p^{\prime }}(\Omega )$. Then, for every Lipschitz-continuous, monotonically increasing function $T: {\mathbb {R}}\rightarrow {\mathbb {R}}$, one has that

$$\begin{aligned} \theta (\textbf{z}, D(T \circ w),x) = \theta (\textbf{z}, Dw, x)\qquad \text {for }\vert Dw \vert \text {-a.e. }x\in \Omega . \end{aligned}$$

We shall denote

$$\begin{aligned} \textrm{sign}_0(r):= \left\{ \begin{array}{lll} 1 \quad &{}\hbox { if} \ r> 0, \\ 0, \quad &{}\hbox { if} \ r = 0, \\ -1, \quad &{}\hbox { if} \ r< 0. \end{array} \right. \quad \quad \textrm{sign}(r):= \left\{ \begin{array}{lll} 1 \quad &{}\hbox { if} \ r > 0, \\ {[}-1,1], \quad &{}\hbox { if} \ r = 0, \\ -1, \quad &{}\hbox { if} \ r < 0, \end{array} \right. \end{aligned}$$

and $\textrm{sign}^+(r):= (\textrm{sign}(r))^+$, and $T_{k}(r):= [k - (k- \vert r \vert )^{+}]\textrm{sign}_0(r)$, $k \ge 0$.

Remark 2.2

Let us point out that although $T_k$ is only non-decreasing, we also have the following result

$$\begin{aligned} \theta (\textbf{z}, D(T_k u),x) = \theta (\textbf{z}, Du, x)\qquad \text { for }\vert DT_k \vert \text {-a.e. }x\in \Omega . \end{aligned}$$

$\square$

2.3 Accretive operators and nonlinear semigroups

An operator A on X is a possibly nonlinear and multivalued mapping $A: X\rightarrow 2^{X}$. It is standard to identify an operator A on X with its graph

$$\begin{aligned} A:=\Big \{(u,v)\in X\times X\,\Big \vert \, v\in Au\Big \}\quad \text {in }X\times X \end{aligned}$$

and so, one sees A as a subset of $X\times X$. The set $D(A):=\{u\in X\,\vert \,Au\ne \emptyset \}$ is called the domain of A, and $R(A):=\mathop {\bigcup }\limits _{u\in D(A)}Au$ the range of A.

Definition 2.3

An operator A on X is called m-accretive operator on X if A is accretive, that is, for every (u, v), $({\hat{u}},{\hat{v}})\in A$ and every $\lambda >0$,

$$\begin{aligned} \Vert u-{\hat{u}}\Vert _{X}\le \Vert u-{\hat{u}}+\lambda ( v-{\hat{v}})\Vert _{X} \end{aligned}$$

and if for all $\lambda >0$ the range condition

$$\begin{aligned} R(I+\lambda A)=X \end{aligned}$$

holds.

Note that A is accretive if the resolvent $J_\lambda := (I + \lambda A)^{-1}$ are contractions for all $\lambda >0$. The Yosida approximation of A is defined as

$$\begin{aligned} A_\lambda := \frac{1}{\lambda } (I - J_\lambda ), \quad \hbox {for} \ \lambda >0. \end{aligned}$$

We have that

$$\begin{aligned} A_\lambda : D(J_\lambda ) \rightarrow X \ \hbox {is Lipschitz-continuous with Lipschitz constant} \ \frac{2}{\lambda }. \end{aligned}$$

Moreover,

$$\begin{aligned} \begin{aligned} A_\lambda u \in A J_\lambda u, \quad u = J_\lambda u + \lambda A_\lambda u \quad \text{ and } \quad \Vert A_\lambda u \Vert \le \inf \{ \Vert v \Vert \, : \, v \in Au \}. \end{aligned} \end{aligned}$$

In the case that the Banach space is $L^1(\Omega )$, with $\Omega \subset {\mathbb {R}}^N$ an open set and norm

$$\begin{aligned} \Vert u \Vert _1:= \int _\Omega \vert u(x) \vert dx, \quad u \in L^1(\Omega ), \end{aligned}$$

it is well known (see [14]) that

$$\begin{aligned} \begin{aligned} \begin{array}{ll}\text{ an } \text{ operator } A \text{ on } L^1(\Omega ) \text{ is } \text{ accretive } \iff \text{ for } \text{ every } (u,v), ({\hat{u}},{\hat{v}})\in A \\ \text{ there } \text{ exists } \xi \in \text {sign}(u - {\hat{u}}) \text{ such } \text{ that } \displaystyle \int _\Omega (v - {\hat{v}}) \xi dx \ge 0. \end{array} \end{aligned} \end{aligned}$$

Definition 2.4

We say that an operator A on $L^1(\Omega )$ is T-accretive if for every (u, v), $({\hat{u}},{\hat{v}})\in A$ and every $\lambda >0$,

$$\begin{aligned} \Vert (u-{\hat{u}})^+\Vert _{1}\le \Vert (u-{\hat{u}}+\lambda ( v-{\hat{v}}))^+\Vert _{1}. \end{aligned}$$

If A is an m-accretive operator on a Banach space X, then by the classical existence theory (see, e.g., [14, Theorem 6.5], or [12, Corollary 4.2]), the first-order Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \frac{d u}{dt}+A(u(t))\ni g(t) &{}\text {on }(0,T),\\ u(0)=u_{0}, \end{array} \right. \end{aligned}$$

(2.3)

is well-posed for every $u_{0}\in \overline{D(A)}^{_{X}}$, and $g\in L^{1}(0,T;X)$ in the following mild sense.

Definition 2.5

For given $u_{0}\in \overline{D(A)}^{_{X}}$ and $g\in L^{1}(0,T;X)$, a function $u\in C([0,T];X)$ is called a mild solution of Cauchy problem (2.3) if $u(0)=u_{0}$ and for every $\varepsilon >0$, there is a partition $0=t_{0}<t_{1}<\cdots < t_{N}=T$ and a step function

$$\begin{aligned} u_{\varepsilon ,N}(t)=u_{0}\,{\mathrm{\chi }}_{\{t=0\}}(t)+\sum _{i=1}^{N}u_{i}\,{\mathrm{\chi }}_{(t_{i-1},t_{i}]}(t), \quad t\in [0,T], \end{aligned}$$

satisfying

$$\begin{aligned} \bullet \quad&t_{i}-t_{i-1}<\varepsilon \quad \text { for all }i=1,\dots ,N,\\ \bullet \quad&\sum _{N=1}^{N}\int _{t_{i-1}}^{t_{i}}\Vert g(t)-{\overline{g}}_{i} \Vert _X\,dt<\varepsilon ,\quad \text {where }{\overline{g}}_{i}:=\frac{1}{t_{i}-t_{i-1}}\int _{t_{i-1}}^{t_{i}}g(t)\, dt,\\ \bullet \quad&\frac{u_{i}-u_{i-1}}{t_{i}-t_{i-1}}+A u_{i}\ni {\overline{g}}_{i} \quad \text { for all }i=1,\dots ,N, \end{aligned}$$

and

$$\begin{aligned} \sup _{t\in [0,T]}\Vert u(t)-u_{\varepsilon ,N}(t)\Vert _{X}<\varepsilon . \end{aligned}$$

In the case $g =0$, the unique mild solution is given by the Crandall–Liggett’s exponential formula

$$\begin{aligned} u(t) = e^{-tA} u_0:= \lim _{n \rightarrow \infty } \left( I + \frac{t}{n} A \right) ^{-n} u_0. \end{aligned}$$

Mild solutions are limits of step functions which are not necessarily differentiable in time. This leads to the notion of strong solution of the Cauchy problem (2.3).

Definition 2.6

For given $u_{0}\in \overline{D(A)}^{_{X}}$ and $g\in L^{1}(0,T;X)$, a function $u\in C([0,T];X)\cap W^{1,1}_{\textrm{loc}}((0,T);X)$ is called a strong solution of the Cauchy problem (2.3) if $u(0)=u_{0}$ and, for a.e. $t\in (0,T)$, $u(t)\in D(A)$ and $Au(t)\ni g(t)-\frac{d u}{dt}(t)$.

We now recall a Bénilan–Crandall relation between functions $u, v\in L^1(\Omega ,\nu )$. Denote by $J_0$ and $P_0$ the following sets of functions:

$$\begin{aligned} J_0:= & {} \{ j: {\mathbb {R}}\rightarrow [0, +\infty ]: \ j \text{ is } \text{ convex, } \text{ lower } \text{ semi-continuous } \text{ and } \ j(0) = 0 \},\\ P_0:= & {} \left\{ \rho \in C^\infty ({\mathbb {R}}): \ 0\le \rho '\le 1, \hbox { supp}(\rho ') \hbox { is compact and } 0\notin \hbox {supp}(\rho ) \right\} . \end{aligned}$$

Assume that $\nu (\Omega ) < +\infty$ and let $u,v\in L^1(\Omega ,\nu )$. The following relation between u and v is defined in [15]:

$$\begin{aligned} u\ll v \ \hbox { if} \ \int _{\Omega } j(u)\, d\nu \le \int _{\Omega } j(v) \, d\nu \ \ \hbox {for every} \ j \in J_0. \end{aligned}$$

Moreover, the following equivalences are proved in [15, Proposition 2.2]:

$$ \begin{aligned} \begin{aligned} \int _\Omega v\rho (u)d\nu \ge 0\quad \forall \rho \in P_0\ \Longleftrightarrow&{} \ u\ll u+\lambda v\quad \forall \lambda>0, \\ \int _\Omega v\rho (u)d\nu \ge 0\quad \forall \rho \in P_0\ \Longleftrightarrow&{} \ \int _{\{u<-h\}}vd\nu \le 0 \quad \& \\ {}&{} 0 \le \int _{\{u>h\}}vd\nu \quad \forall h>0. \end{aligned} \end{aligned}$$

The following result is given in [15]

Proposition 2.7

Let $\Omega \subset {\mathbb {R}}^N$ be an open bounded set.

• (i) For any $u,v \in L^1(\Omega )$, if $\int _\Omega u p(u) dx \le \int _\Omega v p(u) dx$ for all $p \in P_0$, then $u \ll v$.

• (ii) If $u,v \in L^1(\Omega )$ and $u \ll v$, then $\Vert u \Vert _p \le \Vert v \Vert _p$ for all $1 \le p \le \infty$.

• (iii) If $v \in L^1(\Omega )$, then $\{ u\in L^1(\Omega ) \, : \, u \ll v \}$ is a weakly compact subset of $L^1(\Omega )$.

• (iv) If $u_n, u \in L^1(\Omega )$ satisfy $u_n \ll u$ and $u_n \rightarrow u$ weakly in $L^1(\Omega )$, then $u_n \rightarrow u$ in $L^1(\Omega )$.

Let $\gamma \subset {\mathbb {R}}\times {\mathbb {R}}$ be a maximal monotone graph. We denote by $\gamma ^0(r)$ the element of $\gamma (r)$ of minimal absolute value. Then, for the Yosida approximations of $\gamma$ we have that ([18, Proposition 2.6])

$$\begin{aligned} \hbox {for }r \in D(\gamma ): \quad \lim _{\lambda \downarrow 0} \gamma _\lambda (r) = \gamma ^0(r) \quad \hbox {and} \quad \vert \gamma _\lambda (r) \vert \uparrow \vert \gamma ^0 (r) \vert \ \ \hbox {as} \ \lambda \downarrow 0. \end{aligned}$$

3 The elliptic problem

From [4, Theorem 2], given $f \in L^1(\Omega )$ there exists a unique entropy solution v of the elliptic problem

$$\begin{aligned} \left\{ \begin{array}{ll} v - \Delta _1 v\ni f \quad \hbox {in} \ \Omega , \\ \frac{\partial v}{\partial \eta } =0 \quad \hbox {on} \ \partial \Omega , \end{array} \right. \end{aligned}$$

defined as follows: $v \in L^1(\Omega )$ with $T_k(v) \in BV(\Omega )$ for all $k >0$ and such that there exists $\textbf{z}\in X_1(\Omega )$ with $\Vert \textbf{z}\Vert _\infty \le 1$,

$$\begin{aligned} - \textrm{div} (\textbf{z})= & {} f - v \quad \hbox {in} \ {\mathcal {D}}^\prime (\Omega ), \\ (\textbf{z},DT_k(v))= & {} \vert DT_k(v) \vert \quad \hbox {as measures for all} \ k>0 \end{aligned}$$

and

$$\begin{aligned}{}[\textbf{z}, \nu ] = 0 \quad {\mathcal {H}}^{N-1}\hbox {-a.e. on }\partial \Omega . \end{aligned}$$

Let $\gamma$ be a maximal monotone graph in ${\mathbb {R}}\times {\mathbb {R}}$ with $0\in \gamma (0)$. Following such concept, we give the following concept of entropy solution of the following elliptic problem

$$\begin{aligned} (S^\gamma _f)\ \left\{ \begin{array}{ll} v - \Delta _1 u\ni f \quad \hbox {in} \ \Omega , \\ v \in \gamma (u) \quad \hbox {in}\ \Omega , \\ \frac{\partial u}{\partial \eta } =0 \quad \hbox {on} \ \partial \Omega . \end{array} \right. \end{aligned}$$

Definition 3.1

For $f \in L^1(\Omega )$, we say that v is an entropy solution of problem $(S^\gamma _f)$ if $v \in L^1(\Omega )$ and there exist $u \in {\mathcal {M}}(\Omega )$ with $T_k(u) \in BV(\Omega )$ for all $k>0$ and $\textbf{z}\in X_1(\Omega )$ with $\Vert \textbf{z}\Vert _\infty \le 1$ such that

$$\begin{aligned} v\in & {} \gamma (u)\quad \hbox {a.e. in }\Omega , \end{aligned}$$

(3.1)

$$\begin{aligned} - \textrm{div} (\textbf{z})= & {} f - v \quad \hbox {in} \ {\mathcal {D}}^\prime (\Omega ), \nonumber \\ (\textbf{z},DT_k(u))= & {} \vert DT_k(u) \vert \quad \hbox {as measures for all} \ k>0, \nonumber \\ {[}\textbf{z}, \nu ]= & {} 0 \quad {\mathcal {H}}^{N-1}\hbox {-a.e. on }\partial \Omega . \end{aligned}$$

(3.2)

For data in $f \in L^\infty (\Omega )$, we also define the following concept of weak solution of problem $(S^\gamma _f)$.

Definition 3.2

For $f \in L^\infty (\Omega )$, we say that v is a weak solution of problem $(S^\gamma _f)$ if $v\in L^\infty (\Omega )$ and there exist $u \in BV(\Omega )\cap L^\infty (\Omega )$ and $\textbf{z}\in X_1(\Omega )$ with $\Vert \textbf{z}\Vert _\infty \le 1$ such that

$$\begin{aligned} v\in & {} \gamma (u)\quad \hbox {a.e. in }\Omega , \nonumber \\ - \textrm{div} (\textbf{z})= & {} f - v \quad \hbox {in} \ {\mathcal {D}}'(\Omega ), \nonumber \\ (\textbf{z},D u)= & {} \vert Du \vert \quad \hbox {as measures}, \nonumber \\ {[}\textbf{z}, \nu ]= & {} 0 \quad {\mathcal {H}}^{N-1}\hbox {-a.e. on }\partial \Omega . \end{aligned}$$

(3.3)

We have that every weak solution is an entropy solution.

Working as in [6, Lemma 2.4], it is easy to see the two following results.

Lemma 3.3

For $f \in L^\infty (\Omega )$, the following assertions are equivalent:

• (a) v is weak solution of problem $(S^\gamma _f)$;

• (b) there exist $u \in BV(\Omega )\cap L^\infty (\Omega )$ and ${\textbf {z}}\in X_\infty (\Omega )$ with $\Vert \textbf{z}\Vert _\infty \le 1$ satisfying (3.1), (3.2) and

  $$\begin{aligned} \begin{aligned} \int _{\Omega } (\varphi - u)(f- v) \, dx\le&{} \int _{\Omega } {\textbf {z}}\cdot \nabla \varphi \, dx - \int _{\Omega } \vert Du \vert , \\{}&{} \qquad \forall \varphi \in W^{1,1}(\Omega ) \cap L^\infty (\Omega ); \end{aligned} \end{aligned}$$

  (3.4)

• (c) there exist $u \in BV(\Omega )\cap L^\infty (\Omega )$ and ${\textbf {z}}\in X_\infty (\Omega )$ with $\Vert \textbf{z}\Vert _\infty \le 1$ satisfying (3.1), (3.2) and

  $$\begin{aligned} \begin{aligned} \int _{\Omega } (\varphi - u)(f- v) \, dx\le&{} \int _{\Omega } ({\textbf {z}}, D\varphi ) - \int _{\Omega } \vert Du \vert , \\{}&{} \qquad \forall \varphi \in BV(\Omega )\cap L^\infty (\Omega ); \end{aligned} \end{aligned}$$

  (3.5)

• (d) there exist $u \in BV(\Omega )\cap L^\infty (\Omega )$ and ${\textbf {z}}\in X_\infty (\Omega )$ with $\Vert \textbf{z}\Vert _\infty \le 1$ satisfying (3.1), (3.2) and

  $$\begin{aligned} \int _{\Omega } \varphi (f- v) \, dx = \int _{\Omega } (\textbf{z}, D\varphi ), \ \forall \varphi \in BV(\Omega )\cap L^\infty (\Omega ). \end{aligned}$$

Lemma 3.4

For $f \in L^1(\Omega )$, the following assertions are equivalent:

• (a) v is an entropy solution of problem $(S^\gamma _f)$;

• (b) there exist $u \in {\mathcal {M}}(\Omega )$ with $T_k(u) \in BV(\Omega )$ for all $k>0$ and $\textbf{z}\in X_1(\Omega )$ with $\Vert \textbf{z}\Vert _\infty \le 1$ satisfying (3.1), (3.2) and

  $$\begin{aligned} \begin{aligned} \int _{\Omega } (\varphi - T_k(u))(f- v) \, dx\le&{} \int _{\Omega } {\textbf {z}}\cdot \nabla \varphi \, dx - \int _{\Omega } \vert DT_k(u) \vert , \\{}&{} \qquad \forall \varphi \in W^{1,1}(\Omega ) \cap L^\infty (\Omega ); \end{aligned} \end{aligned}$$

• (c) there exist $u \in {\mathcal {M}}(\Omega )$ with $T_k(u) \in BV(\Omega )$ for all $k>0$ and $\textbf{z}\in X_1(\Omega )$ with $\Vert \textbf{z}\Vert _\infty \le 1$ satisfying (3.1), (3.2) and

  $$\begin{aligned} \begin{aligned} \int _{\Omega } (\varphi - T_k(u))(f- v) \, dx\le&{} \int _{\Omega } ({\textbf {z}}, D\varphi ) - \int _{\Omega } \vert DT_k(u) \vert , \\{}&{} \qquad \forall \varphi \in L^\infty (\Omega ) \cap BV(\Omega ); \end{aligned} \end{aligned}$$

  (3.6)

As can be verified in the above lemma, the notion of entropy solution for the 1-Laplacian defined here is analogous to the concept of entropy solution for the p-Laplacian ($1<p<N$) introduced in the pioneering paper [13].

Remark 3.5

Let v be an entropy solution of problem $(S^\gamma _f)$. Then, there exist $u \in {\mathcal {M}}(\Omega )$ with $T_k(u) \in BV(\Omega )$ for all $k>0$ and $\textbf{z}\in X_1(\Omega )$ with $\Vert \textbf{z}\Vert _\infty \le 1$ such that $v\in \gamma (u)$ and (3.6) holds true. Then, if we take $T_k(u) \pm 1$ as test function in (3.6), it follows that

$$\begin{aligned} \int _\Omega f(x) dx = \int _\Omega v(x) dx. \end{aligned}$$

Therefore, denoting

$$\begin{aligned} \gamma ^-:= \inf \textrm{Ran}(\gamma ) \quad \hbox {and} \quad \gamma ^+:= \sup \textrm{Ran}(\gamma ), \end{aligned}$$

the following condition must be satisfied

$$\begin{aligned} \gamma ^- {\mathcal {L}}^N (\Omega ) \le \int _\Omega f(x) dx \le \gamma ^+ {\mathcal {L}}^N (\Omega ). \end{aligned}$$

Therefore, in the case $\hbox {Rang}(\gamma )={\mathbb {R}}$ this is always true for any $f\in L^1(\Omega )$. $\square$

It is worthy to mention that if $\gamma$ is a multivalued maximal monotone graph, the corresponding problem $(S^\gamma _f)$ has more than one weak solution, as we show in the next example.

Example 3.6

Let $\gamma$ be a multivalued graph such that

$$\begin{aligned} \gamma (0)=[0,1]. \end{aligned}$$

Consider $\Omega := ]-1,1[$ and $f(x):= \frac{1}{2}$ for all $x \in ]-1,1[$. We define $\textbf{z}: ]-1,1[ \rightarrow {\mathbb {R}}$ as

$$\begin{aligned} \textbf{z}(x):= \left\{ \begin{array}{lll} 0, \quad &{}\hbox {if} \ x \in \left]-1, - \frac{1}{2} \right] \cup \left[ \frac{1}{2}, 1 \right[, \\ \frac{1}{2} x + \frac{1}{4},\quad &{}\hbox {if} \ x \in \left[ -\frac{1}{2}, - \frac{1}{4} \right] \\ -\frac{1}{2} x,\quad &{}\hbox {if} \ x \in \left[ -\frac{1}{4}, \frac{1}{4} \right] , \\ \frac{1}{2} x - \frac{1}{4},\quad &{}\hbox {if} \ x \in \left[ \frac{1}{4},\frac{1}{2} \right] . \end{array} \right. \end{aligned}$$

Then, $\Vert \textbf{z}\Vert _\infty \le 1$, $[\textbf{z}, \nu ]=0$ and

$$\begin{aligned} v(x):= \textrm{div} \, \textbf{z}(x) + f(x) = \left\{ \begin{array}{lll} \frac{1}{2}, \quad &{}\hbox {if} \ x \in \left]-1, - \frac{1}{2} \right] \cup \left[ \frac{1}{2}, 1 \right[, \\ 1,\quad &{}\hbox {if} \ x \in \left[ -\frac{1}{2}, - \frac{1}{4} \right] \\ 0,\quad &{}\hbox {if} \ x \in \left[ -\frac{1}{4}, \frac{1}{4} \right] , \\ 1,\quad &{}\hbox {if} \ x \in \left[ \frac{1}{4},\frac{1}{2} \right] . \end{array} \right. \end{aligned}$$

Clearly, $v \in \gamma (0)$. Therefore, v is a weak solution of problem $(S^\gamma _f)$. Now, taking $\textbf{z}=0$, it follows that f is also a weak solution of problem $(S^\gamma _f)$. In particular, for the Stefan type problem, there is not uniqueness of weak solution of problem $(S^\gamma _f)$.

Due to the above example, we need to impose some restriction to the maximal monotone graph $\gamma$ in order to get uniqueness of entropy solution of problem $(S^\gamma _{f_i})$. In this direction, we have the following result for graphs satisfying (1.3) without the range condition.

Theorem 3.7

Assume that $\gamma : D(\gamma ) \subset {\mathbb {R}}\rightarrow {\mathbb {R}}$ is a non-decreasing continuous function with $\gamma (0)=0$. Given $f_i \in L^1(\Omega )$ and $v_i$ entropy solutions of $(S^\gamma _{f_i})$, for $i=1,2$, then

$$\begin{aligned} \Vert (v_1 - v_2)^+ \Vert _1 \le \Vert (f_1 - f_2)^+ \Vert _1. \end{aligned}$$

(3.7)

In particular,

$$\begin{aligned} \Vert v_1 - v_2 \Vert _1 \le \Vert f_1 - f_2 \Vert _1. \end{aligned}$$

(3.8)

Proof

For $i=1,2$, we have that there exists $u_i \in L^1(\Omega )$ with $T_k(u_i) \in BV(\Omega )$ for all $k>0$ and there exists $\textbf{z}_i \in X_1(\Omega )$ with $\Vert \textbf{z}_i \Vert _\infty \le 1$ such that $v_i=\gamma (u_i)$ and

$$\begin{aligned} - \textrm{div} (\textbf{z}_i)= & {} f_i - v_i \quad \hbox {in} \ {\mathcal {D}}^\prime (\Omega ), \nonumber \\ (\textbf{z}_i,DT_k(u_i))= & {} \vert DT_k(u_i) \vert \quad \hbox {as measures for all} \ k>0, \nonumber \\ {[}\textbf{z}_i, \nu ]= & {} 0 \quad {\mathcal {H}}^{N-1}\hbox {-a.e. on }\partial \Omega . \end{aligned}$$

(3.9)

Let $p_\epsilon$ be a smooth strictly monotone approximation of the sign function. Then, applying integration by parts formula (2.2), we have

$$\begin{aligned} \int _\Omega (f_i - v_i) p_\epsilon (T_k(u_1) - T_k(u_2)) dx= & {} - \int _\Omega \textrm{div} (\textbf{z}_i) p_\epsilon (T_k(u_1) - T_k(u_2)) dx \\= & {} \int _\Omega (\textbf{z}_i,Dp_\epsilon (T_k(u_1) - T_k(u_2))). \end{aligned}$$

Thus,

$$\begin{aligned} \int _\Omega (v_1 - v_2)p_\epsilon (T_k(u_1) - T_k(u_2)) dx= & {} - \int _\Omega (\textbf{z}_1 - \textbf{z}_2,Dp_\epsilon (T_k(u_1) - T_k(u_2))) \\{} & {} + \int _\Omega (f_1 - f_2)p_\epsilon (T_k(u_1) - T_k(u_2)) dx \\\le & {} - \int _\Omega (\textbf{z}_1 - \textbf{z}_2,Dp_\epsilon (T_k(u_1) - T_k(u_2)))\\{} & {} + \Vert f_1 - f_2 \Vert _1 \end{aligned}$$

Now, from (3.9) and (2.1), we have

$$\begin{aligned} \int _B (\textbf{z}_1 - \textbf{z}_2, D(T_k(u_1) - T_k(u_2))) \ge 0, \quad \hbox {for all Borel set} \ B \subset \Omega . \end{aligned}$$

This implies that

$$\begin{aligned} \theta (\textbf{z}_1 - \textbf{z}_2, D(T_k(u_1) - T_k(u_2)), x) \ge 0 \quad \vert D(T_k(u_1) - T_k(u_2))\vert \hbox {-a.e.} \end{aligned}$$

Since, according to Proposition 2.1, we have

$$\begin{aligned} \theta (\textbf{z}_1 - \textbf{z}_2, Dp_\epsilon (T_k(u_1) - T_k(u_2)), x) = \theta (\textbf{z}_1 - \textbf{z}_2, D(T_k(u_1) - T_k(u_2)), x) \end{aligned}$$

$\vert D(T_k(u_1) - T_k(u_2))\vert \hbox {-a.e.}$ and $\vert Dp_\epsilon (T_k(u_1) - T_k(u_2)) \vert \hbox {-a.e.}$, we get

$$\begin{aligned}{} & {} \int _\Omega (\textbf{z}_1 - \textbf{z}_2,Dp_\epsilon (T_k(u_1) - T_k(u_2))) \\{} & {} \quad = \int _\Omega \theta (\textbf{z}_1 - \textbf{z}_2, Dp_\epsilon (T_k(u_1) - T_k(u_2)), x) d \vert Dp_\epsilon (T_k(u_1) - T_k(u_2)) \vert \ge 0. \end{aligned}$$

Therefore,

$$\begin{aligned} \int _\Omega (v_1 - v_2)p_\epsilon (T_k(u_1) - T_k(u_2)) dx \le \Vert f_1 - f_2 \Vert _1. \end{aligned}$$

Taking limit as $k \rightarrow \infty$, we get

$$\begin{aligned} \int _\Omega (v_1 - v_2)p_\epsilon (u_1 - u_2) dx \le \Vert f_1 - f_2 \Vert _1. \end{aligned}$$

Taking now limit as $\epsilon \rightarrow 0^+$, we have that there exists $\xi (x) \in \textrm{sign}((u_1(x) - u_2(x))$ ${\mathcal {L}}^N$-a.e. $x \in \Omega$ such that

$$\begin{aligned} \int _\Omega (v_1 - v_2)\xi (x) dx \le \Vert f_1 - f_2 \Vert _1. \end{aligned}$$

Now, since $v_i = \gamma (u_i)$, $i=1,2$, and $\gamma$ is non-decreasing and $\gamma (0)=0$, we have $\xi (x) \in \textrm{sign}((v_1(x) - v_2(x))$, if $v_1(x) \not = v_2(x)$. Hence, since $\gamma$ is continuous, which for a maximal monotone graph is equivalent to say that $\gamma (r)$ is always univalued for any $r\in D(\gamma )$,

$$\begin{aligned} \Vert v_1 - v_2 \Vert _1 = \int _{\{ v_1 \not = v_2 \}} (v_1 - v_2)\xi (x) dx \le \Vert f_1 - f_2 \Vert _1, \end{aligned}$$

and (3.8) holds.

The proof of (3.7) is similar but using a smooth monotone approximation of the $\textrm{sign}^+$. $\square$

Let us now prove existence for problem $(S^\gamma _f)$ for graphs satisfying condition (1.4).

Theorem 3.8

Let $\gamma$ be a graph satisfying (1.4) and $f \in C^\infty _c(\Omega )$. Then, there exists $v_f$ weak solution of problem $(S^\gamma _f)$ with $v_f\ll f.$

Moreover, for any ${{\tilde{f}}}\in C^\infty _c(\Omega )$, it follows

$$\begin{aligned} \Vert (v_f-v_{{{\tilde{f}}}})^+\Vert _1\le \Vert (f-{{\tilde{f}}})^+ \Vert _1, \end{aligned}$$

for the weak solutions constructed here.

Proof

Given $f \in C^\infty _c(\Omega )$, we must find $v \in L^\infty (\Omega )$ and $u \in BV(\Omega )\cap L^\infty (\Omega )$ with

$$\begin{aligned} v \in \gamma (u)\quad \hbox {a.e. in }\Omega , \end{aligned}$$

and $\textbf{z}\in X_1(\Omega ),$ with $\Vert \textbf{z}\Vert _\infty \le 1$, satisfying (3.2) and (3.4).

By [8, Theorem 3.9 (i)], for any $p>1$, there exist $u_p \in W^{1,p}(\Omega )$ and $v_p \in \gamma (u_p) \in L^1(\Omega )$ such that

$$\begin{aligned} \int _\Omega \vert \nabla u_{p} \vert ^{p-2} \nabla u_{p} \cdot \nabla \varphi dx + \int _\Omega v_p \varphi dx = \int _\Omega f \varphi dx, \end{aligned}$$

(3.10)

for all $\varphi \in W^{1,p}(\Omega ) \cap L^\infty (\Omega )$. Moreover,

$$\begin{aligned} v_p \ll f,\quad \hbox {for all} \ p >1, \end{aligned}$$

(3.11)

and, since $\hbox {Rang}(\gamma )={\mathbb {R}}$,

$$\begin{aligned} \Vert u_p \Vert _\infty \le M_1,\quad \hbox {for all} \ p >1. \end{aligned}$$

Taking $\varphi =u_p\in W^{1,p}(\Omega )\cap L^\infty (\Omega )$ as a test function and taking into account that $u_pv_p\ge 0$ it follows that

$$\begin{aligned} \int _\Omega \vert \nabla u_p \vert ^p=\int _\Omega u_p(f-v_p)\le \int _\Omega fu_p \le M_2. \end{aligned}$$

(3.12)

Therefore, by Hölder inequality we get

$$\begin{aligned} \int _\Omega \vert \nabla u_p \vert \le M_3. \end{aligned}$$

Thus, there exists $u\in BV(\Omega )\cap L^\infty (\Omega )$ such that, up to a subsequence (no relabeled),

$$\begin{aligned} u_p \rightarrow u \, \hbox { in } L^q(\Omega ), \, \hbox { for }\, 1\le q < 1^*:=\frac{N}{N-1}. \end{aligned}$$

(3.13)

Moreover, inequality (3.12) allows to establish the following statements (see [5]): There exists a bounded vector field ${\textbf{z}}\in L^\infty (\Omega ;{\mathbb {R}}^N)$ with $|\vert {\textbf{z}}|\vert _\infty \le 1$ such that

$$\begin{aligned} \vert \nabla u_p \vert ^{p-2}\nabla u_p \rightharpoonup \textbf{z}, \, \hbox { in } L^r(\Omega ;{{\mathbb {R}}}^N), \, \hbox { for all } 1\le r <\infty , \end{aligned}$$

(3.14)

as $p\rightarrow 1^+$. In particular,

$$\begin{aligned} -\hbox {div}(\textbf{z})= f-v, \quad \hbox {in }\, {\mathcal {D}}^\prime (\Omega ). \end{aligned}$$

On the other hand, by (3.11) we obtain that

$$\begin{aligned} v_p \rightharpoonup v, \quad \hbox {in } L^q(\Omega ), \, 1\le q < \infty , \end{aligned}$$

(3.15)

being $v\ll f$. This result, in addition to (3.13), implies that

$$\begin{aligned} v\in \gamma (u)\quad \hbox {a.e. in }\Omega . \end{aligned}$$

In order to show that v is a weak solution of problem $(S^\gamma _f)$, for each $\varphi \in W^{1,1}(\Omega )\cap L^\infty (\Omega )$ we consider the sequence $\{ \varphi _n \}\subset {\mathcal {C}}^\infty ({\bar{\Omega }})$ such that $\varphi _n \rightarrow \varphi$ in $W^{1,1}(\Omega )$. Now, taking $\varphi _n-u_p$ as a test function in (3.10) and taking limits it follows

$$\begin{aligned} \lim _{p\rightarrow 1^+} \int _\Omega \vert \nabla u_p \vert ^p= & {} \lim _{p\rightarrow 1^+} \int _\Omega \vert \nabla u_p \vert ^{p-2}\nabla u_p \nabla \varphi _n \nonumber \\{} & {} - \lim _{p\rightarrow 1^+} \int _\Omega (f-v_p)(\varphi _n-u_p). \end{aligned}$$

(3.16)

By (3.13) and (3.15), we get

$$\begin{aligned} \int _\Omega (f-v_p)(\varphi _n-u_p)\rightarrow \int _\Omega (f-v)(\varphi _n-u), \end{aligned}$$

and by (3.14)

$$\begin{aligned} \int _\Omega \vert \nabla u_p \vert ^{p-2}\nabla u_p \nabla \varphi _n \rightarrow \int _\Omega \textbf{z}\,\nabla \varphi _n. \end{aligned}$$

In addition, using Young’s inequality and the weak lower semi-continuity of the total variation, we obtain

$$\begin{aligned} \lim _{p\rightarrow 1^+} \int _\Omega |\nabla u_p|^p&\ge \lim _{p\rightarrow 1^+} \left( p\int _\Omega |\nabla u_p|-(p-1)\vert \Omega \vert \right) \\&\ge \lim _{p\rightarrow 1^+} \inf \left( \int _\Omega |\nabla u_p|-(p-1)\vert \Omega \vert \right) \\&=\int _\Omega |Du| . \end{aligned}$$

Therefore, expression (3.16) yields

$$\begin{aligned} \int _\Omega |Du| \le \int _\Omega \textbf{z}\,\nabla \varphi _n-\int _\Omega (f-v)(\varphi _n-u). \end{aligned}$$

Finally, taking limits as $n\rightarrow \infty$ we obtain inequality (b) from Lemma 3.3, which means that v is a weak solution of problem $(S^\gamma _f)$.

The second part is a consequence of [8, Theorem 3.9 (ii)] and the above construction. $\square$

Theorem 3.9

Assume that $\gamma$ satisfies condition (1.4). Then, for any $f \in L^1(\Omega )$ there exists an entropy solution of problem $(S^\gamma _f)$.

Proof

Given $f \in L^1(\Omega )$, let $f_n\in {\mathcal {C}}_c^\infty (\Omega )$ be such that $f_n\rightarrow f$ in $L^1(\Omega )$. For any $n \in {{\mathbb {N}}}$, by Theorem 3.8 there exists a weak solution $v_n$ of problem $(S^\gamma _{f_n})$ such that $v_n \ll f_n$. Thus, there exists $u_n \in BV(\Omega )\cap L^\infty (\Omega )$ and there exists $\textbf{z}_n \in X_1$ with $\Vert \textbf{z}_n \Vert _\infty \le 1$ such that

$$\begin{aligned} v_n\in & {} \gamma (u_n)\quad \hbox {a.e. in }\Omega , \nonumber \\ -\hbox {div}(\mathbf{z_n})= & {} f_n-v_n, \quad \hbox {in }\, {\mathcal {D}}^\prime (\Omega ), \end{aligned}$$

(3.17)

and

$$\begin{aligned} \int _\Omega (f_n-v_n)(\varphi -u_n)+\Vert Du_n\Vert \le \int _\Omega ({\mathbf{z_n}},D\varphi ), \quad \forall \, \varphi \in BV(\Omega )\cap L^\infty (\Omega ).\qquad \quad \end{aligned}$$

(3.18)

Taking $\varphi =u_n-T_k(u_n)$ in (3.18), we have

$$\begin{aligned} -\int _\Omega (f_n-v_n)T_k(u_n)+ \int _\Omega \vert Du_n\vert \le \int _\Omega (\textbf{z}_n,Du_n) - \int _\Omega \vert DT_k(u_n)\vert . \end{aligned}$$

Then, by (2.1) and since $v_nT_k(u_n)\ge 0$, we get

$$\begin{aligned} \int _\Omega \vert DT_k(u_n)\vert \le \int _\Omega f_nT_k(u_n)\le k \Vert f\Vert _1. \end{aligned}$$

Then, by the compact embedding, taking subsequences and using a diagonal process, we have

$$\begin{aligned} T_k(u_n)\rightarrow \sigma _k,\quad \, n\rightarrow \infty , \hbox { in }L^q(\Omega ) \hbox { and a.e. for }1 \le q < 1^*, \end{aligned}$$

with

$$\begin{aligned} \vert \sigma _k \vert \le k. \end{aligned}$$

Let us see now that (remark that this argument is not needed if $[0,+\infty [\subset D(\gamma )$, similarly for the argument with the negative part)

$$\begin{aligned} {\mathcal {L}}^N(\{x\in \Omega : \sigma _k^{+}(x)=k \}) \rightarrow 0,\quad \hbox {as } k\rightarrow \infty . \end{aligned}$$

(3.19)

In fact, since $\gamma ^{0}$ is lower semi-continuous and $\hbox {Rang}(\gamma ^{0})={\mathbb {R}}$, by applying Fatou’s lemma it follows that

$$\begin{aligned} {\mathcal {L}}^N(\{x\in \Omega : \sigma _k^{+}(x)=k \})= & {} \int _{\{x\in \Omega : \sigma _k^{+}(x)=k \}}\frac{\gamma ^0(\sigma _k^{+}(x))}{\gamma ^0(k)}\le \frac{1}{\gamma ^0(k)}\liminf _{n \rightarrow \infty } \int _\Omega \gamma ^0(T_k(u_n)^+) \\\le & {} \frac{1}{\gamma ^0(k)} \liminf _{n \rightarrow \infty } \int _\Omega v_n^+ \le \frac{1}{\gamma ^0(k)} \Vert f\Vert _1 \rightarrow 0, \quad k\rightarrow \infty . \end{aligned}$$

Similarly, it is shown that

$$\begin{aligned} {\mathcal {L}}^N(\{x\in \Omega : \sigma _k^{-}(x)=k \}) \rightarrow 0,\quad k\rightarrow \infty . \end{aligned}$$

(3.20)

By (3.19) and (3.20), if we define

$$\begin{aligned} u(x):= \sigma _k(x) \quad \hbox {on} \ \{ x \in \Omega : \ \vert \sigma _k(x) \vert = k \}, \end{aligned}$$

we have that u is measurable and

$$\begin{aligned} u_n \ \ \hbox { converges to }u\hbox { a.e. in }\Omega . \end{aligned}$$

Now, by using the second part in Theorem 3.8, we get

$$\begin{aligned} \Vert v_n - v_m \Vert _1 \le \Vert f_n - f_m \Vert _1 \quad \hbox {for all} \ n,m \in {{\mathbb {N}}}. \end{aligned}$$

Therefore,

$$\begin{aligned} v_n \rightarrow v \quad \hbox {in} \ \ L^1(\Omega ), \end{aligned}$$

(3.21)

and

$$\begin{aligned} v\in \gamma (u) \quad \hbox {a.e. in }\Omega . \end{aligned}$$

On the other hand, since $\textbf{z}_n \in X_1(\Omega )$ with $\Vert \textbf{z}_n \Vert _\infty \le 1$, we may assume that

$$\begin{aligned} \textbf{z}_n \rightarrow \textbf{z}\quad \hbox {in the weak}^*\hbox { topology of } \ \ L^\infty (\Omega ,{\mathbb {R}}^N). \end{aligned}$$

(3.22)

Then, from (3.17),

$$\begin{aligned} -\hbox {div}(\textbf{z})= f-v, \quad \hbox {in }\, {\mathcal {D}}^\prime (\Omega ). \end{aligned}$$

Given now $\varphi \in W^{1,1}(\Omega ) \cap L^\infty (\Omega )$ and taking $\varphi + u_n -T_k(u_n)$ as test function in (3.18), we obtain

$$\begin{aligned} \int _\Omega (f_n-v_n)(\varphi -T_k(u_n))+\Vert Du_n\Vert\le & {} \int _\Omega (\textbf{z}_n,D(\varphi + u_n -T_k(u_n)) \\\le & {} \int _\Omega \textbf{z}_n \cdot \nabla \varphi \, dx + \int _\Omega (\textbf{z}_n,D(u_n -T_k(u_n)) \\\le & {} \int _\Omega \textbf{z}_n \cdot \nabla \varphi \, dx + \int _\Omega \vert D(u_n -T_k(u_n)) \vert . \end{aligned}$$

Thus, applying [5, Lemma 3], we arrive to

$$\begin{aligned} \int _\Omega (f_n-v_n)(\varphi -T_k(u_n))+\Vert DT_k(u_n)\Vert \le \int _\Omega \textbf{z}_n \cdot \nabla \varphi \, dx. \end{aligned}$$

Then, taking limit as $n \rightarrow \infty$ and having in mind (3.21), (3.22) and the lower semi-continuity of the total variation, we get

$$\begin{aligned} \int _\Omega (f-v)(\varphi -T_k(u))+\Vert DT_k(u)\Vert \le \int _\Omega \textbf{z}\cdot \nabla \varphi \, dx. \end{aligned}$$

$\square$

Let us now prove that, under assumption (1.3), the unique solution of problem $(S^\gamma _f)$ coincides by the unique solution of $(S^{Id}_f)$. Let us first see an easy situation.

Proposition 3.10

Let $\gamma ^{-1}$ be an increasing and Lipschitz-continuous function with $\gamma (0)=0$ and $\hbox {Rang}(\gamma )={\mathbb {R}}$. Let $v\in BV(\Omega )\cap L^\infty (\Omega )$ be the unique weak solution of $(S^{Id}_f)$ for $f\in L^\infty (\Omega )$. Then, v is also a weak solution of problem $(S^{\gamma }_f)$.

Proof

By setting $u:= \gamma ^{-1}(v)$ (which is well defined since $\hbox {Rang}(\gamma )={\mathbb {R}}$), we have

$$\begin{aligned} u \in BV(\Omega )\cap L^\infty (\Omega ) \end{aligned}$$

and

$$\begin{aligned} |Du|\ll |Dv|\quad \end{aligned}$$

(see [4, Theorems 3.101 and 3.99]). Now, by Proposition 2.1 and (3.3) it follows

$$\begin{aligned} \theta (z,Du,.)=\theta (z,D\gamma ^{-1}(v),.)= \theta (z,Dv,.)=1\quad |Dv|\hbox {-a.e., hence }|Du|\hbox {-a.e.}; \end{aligned}$$

consequently,

$$\begin{aligned} (\textbf{z},Du) = \vert Du \vert \quad \hbox {as measures}. \end{aligned}$$

Therefore, v is a weak solution of problem $(S^\gamma _f)$. $\square$

Theorem 3.11

Under condition (1.3), the entropy solution of problem $(S^\gamma _f)$ is given by the entropy solution of problem $(S^{Id}_f)$, i.e., of problem

$$\begin{aligned} \left\{ \begin{array}{ll} v - \Delta _1 v\ni f \quad \hbox {in} \ \Omega , \\ \frac{\partial v}{\partial \eta } =0 \quad \hbox {on} \ \partial \Omega . \end{array} \right. \end{aligned}$$

Proof

By Theorem 3.7, it is enough to prove it for data $f\in C_c^\infty (\Omega )$. So, our aim is to see that the (weak) solution v to

$$\begin{aligned} \left\{ \begin{array}{ll} v - \Delta _1 v\ni f \quad \hbox {in} \ \Omega , \\ \frac{\partial v}{\partial \eta } =0 \quad \hbox {on} \ \partial \Omega , \end{array} \right. \end{aligned}$$

(3.23)

is the (weak) solution of

$$\begin{aligned} \left\{ \begin{array}{ll} v - \Delta _1 u\ni f \quad \hbox {in} \ \Omega , \\ v=\gamma (u) \quad \hbox {in} \ \Omega , \\ \frac{\partial u}{\partial \eta } =0 \quad \hbox {on} \ \partial \Omega . \end{array} \right. \end{aligned}$$

(3.24)

Let ${\tilde{\gamma }}_n(r)=\gamma _{1/n}(r)+\frac{1}{n}r$, where $\gamma _{1/n}$ is the Yosida approximation of $\gamma$ (and where $\frac{1}{n}r$ can be deleted if the graph of $\gamma$ has not flat zones). Then, ${\tilde{\gamma }}_n(r)$ is a Lipschitz-continuous and increasing function also satisfying $\hbox {Rang}({\tilde{\gamma }}_n)={\mathbb {R}}$. Therefore, from Theorem 3.8, there exist $v_n \in L^\infty (\Omega )$, $u_n \in BV(\Omega )\cap L^\infty (\Omega )$ and $\textbf{z}_n \in X_1(\Omega )$ with $\Vert \textbf{z}_n \Vert _\infty \le 1$ such that

$$\begin{aligned} v_n= & {} {\tilde{\gamma }}_n(u_n)\quad \hbox {in }\Omega , \nonumber \\ v_n\ll & {} f, \nonumber \\ - \textrm{div} (\textbf{z}_n)= & {} f - v_n \quad \hbox {in} \ {\mathcal {D}}^\prime (\Omega ), \end{aligned}$$

(3.25)

$$\begin{aligned} ({\textbf {z}}_n,Du_n)= & {} {} \vert Du_n \vert \quad \text{ as } \text{ measures }, \nonumber \\ {[}{} {\textbf {z}}_n, \nu ]= & {} {} 0 \quad {\mathcal {H}}^{N-1}\text{-a.e. } \text{ on } \partial \Omega . \end{aligned}$$

(3.26)

Moreover by (3.5), we have

$$\begin{aligned} \begin{aligned} \int _{\Omega } (\varphi - u_n)(f- v_n) \, dx\le&{} \int _{\Omega } ({\textbf {z}}_n, D\varphi ) - \int _{\Omega } \vert Du_n \vert , \\{}&{} \qquad \forall \varphi \in BV(\Omega )\cap L^\infty (\Omega ). \end{aligned} \end{aligned}$$

(3.27)

Since $v_n:= {\tilde{\gamma }}_n(u_n)$, we have

$$\begin{aligned} v_n \in BV(\Omega ) \end{aligned}$$

and

$$\begin{aligned} |Dv_n|\ll |Du_n|. \end{aligned}$$

By (3.26) and Proposition 2.1, we have

$$\begin{aligned} \theta (z_n,Dv_n,.)=\theta (z_n,D{\tilde{\gamma }}_n(u_n),.)= \theta (z_n,Du_n,.)=1\quad |Du_n|\hbox {-a.e., hence }|Dv_n|\hbox {-a.e.}, \end{aligned}$$

consequently,

$$\begin{aligned} (\textbf{z}_n,Dv_n) = \vert Dv_n \vert \quad \hbox {as measures}, \end{aligned}$$

and, therefore, we get that $v_n$ is a (weak) solution of problem (3.23) with vector field $\textbf{z}_n$. Therefore, by uniqueness of problem (3.23), we have

$$\begin{aligned} v_n=v. \end{aligned}$$

And, by (3.27), we have

$$\begin{aligned} \begin{aligned} \int _{\Omega } (\varphi - u_n)(f- v) \, dx\le&{} \int _{\Omega } ({\textbf {z}}_n, D\varphi ) - \int _{\Omega } \vert Du_n \vert , \\{}&{} \qquad \forall \varphi \in BV(\Omega )\cap L^\infty (\Omega ). \end{aligned} \end{aligned}$$

(3.28)

Now, since ${\tilde{\gamma }}_n(u_n)=v \ll f$, we get

$$\begin{aligned} \Vert \gamma _1(u_n)\Vert _q^q \le \Vert \gamma _{1/n}(u_n)\Vert _q^q \le \Vert {\tilde{\gamma }}_n(u_n)\Vert _q^q \le \Vert f\Vert _q^q, \end{aligned}$$

for $q\in [1,\infty ]$. In particular, $\Vert u_n\Vert _\infty \le C_1:= \max \{-\gamma _1^{-1}(-\Vert f\Vert _\infty ), \gamma _1^{-1}(\Vert f\Vert _\infty ) \}$, for all $n\in {\mathbb {N}}$.

Finally, taking $\varphi =0$ as a test function in (3.28) we obtain

$$\begin{aligned} \int |Du_n| \le \int u_n(f-v)\le C_2, \quad \hbox {for all } n\in {\mathbb {N}}, \end{aligned}$$

so that $\{u_n\}$ is bounded in $BV(\Omega )$. It follows that there exists $u\in BV(\Omega )$ such that up to a subsequence (no relabeled)

$$\begin{aligned} u_n \rightarrow u \hbox { in } L^m(\Omega ), \hbox { for } 1\le m <\frac{N}{N-1}, \end{aligned}$$

and

$$\begin{aligned} u_n(x) \rightarrow u(x) \hbox { for almost every } x\in \Omega . \end{aligned}$$

This implies that $v=\gamma (u)$.

On the other hand, since $\textbf{z}_n \in X_1(\Omega )$ with $\Vert \textbf{z}_n \Vert _\infty \le 1$, we may assume that

$$\begin{aligned} \textbf{z}_n \rightarrow \textbf{z}\quad \hbox {in the weak}^*\hbox { topology of } \ \ L^\infty (\Omega ,{\mathbb {R}}^N). \end{aligned}$$

In particular, from (3.25)

$$\begin{aligned} -\hbox {div}(\textbf{z})= f-v, \quad \hbox {in }\, {\mathcal {D}}^\prime (\Omega ). \end{aligned}$$

Then, by [6, Proposition C.12] and having in mind the lower semi-continuity of the total variation, taking limits in (3.28) as $n \rightarrow \infty$, we get

$$\begin{aligned} \int _{\Omega } (\varphi - u)(f- v) \, dx \le \int _{\Omega } (\textbf{z}, D\varphi ) - \int _{\Omega } \vert Du \vert , \ \forall \varphi \in BV(\Omega )\cap L^\infty (\Omega ). \end{aligned}$$

Therefore, by Lemma 3.3, we have that v is a solution of (3.24). $\square$

In the next example will be see that the condition $\text{ Rang }(\gamma ) = {\mathbb {R}}$ is necessary in the above theorem.

Example 3.12

Let, for $n\in {\mathbb {N}}$, ${\beta _n}(r)=\frac{1}{n}\arctan (r)$, and let $f\in L^1(\Omega )$, $\int _\Omega f=0$. If $(S^{{\beta _n}}_f)$ has a solution for all $n\in {\mathbb {N}}$, then there exists $u_n \in {\mathcal {M}}(\Omega )$, $T_k(u_n) \in BV(\Omega )$ for all $k>0$, $\beta _n(u_n)\in L^1(\Omega )$, and there exists $\textbf{z}_n \in X_1(\Omega )$ with $\Vert \textbf{z}_n \Vert _\infty \le 1$, such that

$$\begin{aligned} - \textrm{div} (\textbf{z}_n) = f - \beta _n(u_n) \quad \hbox {in} \ {\mathcal {D}}^\prime (\Omega ). \end{aligned}$$

(3.29)

Now, since

$$\begin{aligned} -\frac{\pi }{2n}\le \beta _n(u_n)\le \frac{\pi }{2n}, \end{aligned}$$

taking limits in (3.29) we find $\textbf{z}\in X_1(\Omega ),$ $\Vert \textbf{z}\Vert _{\infty }\le 1,$ such that

$$\begin{aligned} - \textrm{div} (\textbf{z})=f \quad \hbox {{ in} }{{\mathcal {D}}}^{\prime }(\Omega ), \end{aligned}$$

(3.30)

and we get a contradiction with the well-known fact that there exist $f\in L^1(\Omega )$, $\int _\Omega f=0$, such that the above equation has not solution in $X_1(\Omega )$ (see, for instance, [17]). Nevertheless, let us see, with an easy example, that there are $L^\infty$-functions for which (3.30) has no solution in $X_1(\Omega )$ with $\Vert \textbf{z}\Vert _{\infty }\le 1$. In fact, (3.30) implies that

$$\begin{aligned} \left| \int _\Omega f(x)\varphi (x)dx\right| \le \sum _{i=1}^N\int _\Omega \left| \frac{\partial \varphi }{\partial x_i}(x)\right| dx\quad \forall \varphi \in W^{1,1}_0(\Omega ). \end{aligned}$$

(3.31)

Take $\Omega =B_1(0)$ the ball in ${\mathbb {R}}^N$ centered at 0 of radius 1, and, for $k>0$,

$$\begin{aligned}f(x)=\left\{ \begin{array}{ll} -k, &{}|x|\le 1/2,\\ \frac{k}{2^N-1}, &{}1/2<|x|<1, \end{array} \right. \end{aligned}$$

which satisfies $\int _{\Omega } f=0$. Take now $\varphi (x)=1-|x|$, which belongs to $W^{1,1}_0(\Omega )$. Then, on the one hand,

$$\begin{aligned} \sum _{i=1}^N\int _\Omega \left| \frac{\partial \varphi }{\partial x_i}(x)\right| dx\le N |\Omega |, \end{aligned}$$

and, on the other hand, since $\int _{\Omega } f=0$,

$$\begin{aligned} \left| \int _\Omega f(x)\varphi (x)dx\right| =\left| \int _\Omega |x| f(x)dx\right| =\frac{N|\Omega |k}{2(N+1)(2^N-1)}, \end{aligned}$$

which, for $k>2(N+1)(2^N-1)$, contradicts (3.31). $\square$

4 The evolution problem

In this section, we study the evolution problem (1.2).

We do this through the Nonlinear Semigroup Theory, and therefore, we introduce an operator ${\mathcal {B}}$ in $L^1(\Omega )$ that allows to rewrite problem (1.2) as the abstract Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \frac{d v}{dt}+{\mathcal {B}}(v(t))\ni 0 &{}\text {on }(0,T),\\ v(0)=v_{0}. \end{array} \right. \end{aligned}$$

(4.1)

Definition 4.1

$(v,w) \in {\mathcal {B}}$ if and only if $v,w \in L^1(\Omega )$ and there exist $u \in {\mathcal {M}}(\Omega )$ such that $T_k(u) \in BV(\Omega )$ for all $k >0$ and $v \in \gamma (u)$ a.e. in $\Omega$, and there exists $\textbf{z}\in X_1(\Omega )$ with $\Vert \textbf{z}\Vert _\infty \le 1$, satisfying:

$$\begin{aligned} - \textrm{div}(\textbf{z}) = w \quad \hbox {in} \ \ {\mathcal {D}}^{\prime }(\Omega ) \end{aligned}$$

and

$$\begin{aligned} \int _\Omega (\varphi - T_k(u)) w dx \le \int _\Omega \textbf{z}\cdot \nabla \varphi dx - \int _\Omega \vert DT_k(u) \vert , \ \ \forall \varphi \in BV(\Omega )\cap L^\infty (\Omega ). \end{aligned}$$

Note that for $f \in L^1(\Omega )$, we have that

$$\begin{aligned} (I + {\mathcal {B}})^{-1}f= v \iff v \ \hbox {is an entropy solution of problem} \ (S^\gamma _f). \end{aligned}$$

By Theorems 3.7 and 3.9, we have the following result.

Theorem 4.2

Under condition (1.3), ${\mathcal {B}}$ is a T-accretive and m-accretive operator on $L^1(\Omega )$.

As a consequence of the above result, by Crandall–Liggett’s theorem, it follows that, for every initial data $v_0 \in \overline{D({\mathcal {B}})}^{L^1(\Omega )}$, the abstract Cauchy problem (4.1) has a unique mild solution v(t) given by the exponential formula

$$\begin{aligned} v(t) = e^{-t{\mathcal {B}}} v_0 = \lim _{n \rightarrow \infty } \left( I + \frac{t}{n} {\mathcal {B}} \right) ^{-n}v_0. \end{aligned}$$

In [5] (see also [6]), it is shown that the operator ${\mathcal {A}}$ given in Definition 1.2 is an m-completely accretive operator in $L^1(\Omega )$ and that for every initial data $v_0 \in L^1(\Omega )$ the mild solution

$$\begin{aligned} v(t) = e^{-t{\mathcal {A}}} v_0 = \lim _{n \rightarrow \infty } \left( I + \frac{t}{n} {\mathcal {A}} \right) ^{-n}v_0 \end{aligned}$$

is a strong solution.

Theorem 4.3

The following equality holds

$$\begin{aligned} \overline{D({\mathcal {B}})}^{L^1(\Omega )}= L^1(\Omega ). \end{aligned}$$

Proof

It is enough to prove that $C_c^\infty (\Omega )\subset \overline{D({\mathcal {B}})}^{L^1(\Omega )}$. So, take $f\in C_c^\infty (\Omega )$ and take $f_n=\left( I+\frac{1}{n}{\mathcal {B}}\right) ^{-1}f$. Observe that $f_n\in D({\mathcal {B}})$ and it is a (weak) solution of

$$\begin{aligned} (S^\gamma _f)\ \left\{ \begin{array}{ll} v - \frac{1}{n}\Delta _1 u\ni f \quad \hbox {in} \ \Omega , \\ v \in \gamma (u) \quad \hbox {in}\ \Omega , \\ \frac{\partial u}{\partial \eta } =0 \quad \hbox {on} \ \partial \Omega . \end{array} \right. \end{aligned}$$

Therefore, by Theorem 3.8 and Lemma 3.3 (d), $f_n\in L^\infty (\Omega )$, $f_n\ll f$, and there exist $u_n \in BV(\Omega )\cap L^\infty (\Omega )$ and $\textbf{z}_n \in X_1(\Omega )$ with $\Vert \textbf{z}\Vert _\infty \le 1$ satisfying (3.1), (3.2) and

$$\begin{aligned} \int _{\Omega } \varphi (f- f_n) \, dx = \frac{1}{n}\int _{\Omega } (\textbf{z}_n, D\varphi ), \ \forall \varphi \in BV(\Omega )\cap L^\infty (\Omega ). \end{aligned}$$

Now, for $\varphi \in BV(\Omega )\cap L^\infty (\Omega )$, since

$$\begin{aligned} \lim _n\frac{1}{n}\int _{\Omega } (\textbf{z}_n, D\varphi )=0, \end{aligned}$$

we have that

$$\begin{aligned} \lim _n\int _{\Omega } \varphi (f- f_n) \, dx =0. \end{aligned}$$

Now, since $f_n\ll f$, by (iv) in Proposition 2.7 we get

$$\begin{aligned} \lim _nf_n=f\quad \hbox {in }L^1(\Omega ). \end{aligned}$$

$\square$

Then, as a consequence of Theorem  3.11, we obtain the following result:

Theorem 4.4

Under condition (1.3) and for every initial data $v_0 \in L^1(\Omega )$, the abstract Cauchy problem (4.1) has a unique strong solution v(t). Moreover, this solution coincides with the unique strong solution of problem (1.6).

We introduce the following concept of solution of problem (1.2).

Definition 4.5

A measurable function $v: (0,T)\times \Omega \rightarrow {\mathbb {R}}$ is an entropy solution of (1.2) in $(0,T)\times \Omega$ if $v \in C([0,T],L^1(\Omega )) \cap W^{1,1}_{loc}(0,T; L^1(\Omega ))$, $v(0)=v_0$, and, for almost all $t \in (0, T)$, there exists $u(t)\in {\mathcal {M}}(\Omega )$ with $T_{k}(u(t)) \in BV(\Omega )$ for all $k > 0$, and there exists $\textbf{z}(t) \in L^{\infty }(\Omega )$ with $\Vert \textbf{z}(t) \Vert _{\infty } \le 1$, such that

$$\begin{aligned} v(t,x)\in & {} \gamma (u(t,x)) \quad \hbox {a.e.} \ x \in \Omega , \\ v_{t}(t)= & {} \textrm{div}(\textbf{z}(t))\quad \hbox {in }\mathcal D^{\prime }(\Omega ) \end{aligned}$$

and

$$\begin{aligned} \int _{\Omega } \left( T_{k}(u(t)) - w\right) v_{t}(t) \, dx \le \int _{\Omega } \textbf{z}(t) \cdot \nabla w \, dx- \int _{\Omega } \vert DT_{k}(u(t)) \vert \end{aligned}$$

for every $w \in W^{1,1}(\Omega ) \cap L^{\infty }(\Omega )$.

As a consequence of the above result, we have the following existence and uniqueness result.

Theorem 4.6

Under condition (1.3) and for every initial data $v_0 \in L^1(\Omega )$, there exists a unique entropy solution of (1.2) in $(0,T)\times \Omega$ for every $T >0$ such that $v(0) = v_0$. Moreover, if v(t) and ${\hat{v}}(t)$ are entropy solutions corresponding to initial data $v_0$ and ${\hat{v}}_0$, respectively, then

$$\begin{aligned} \Vert (v(t) - {\hat{v}}(t))^+ \Vert _1 \le \Vert (v_0 - {\hat{v}}_0)^+\Vert _1 \quad \hbox {for all }t \ge 0. \end{aligned}$$

In particular,

$$\begin{aligned} \Vert v(t) - {\hat{v}}(t) \Vert _1 \le \Vert v_0 - {\hat{v}}_0\Vert _1 \quad \hbox {for all }t \ge 0. \end{aligned}$$