Maximal $L^2$-regularity in nonlinear gradient systems and perturbations of sublinear growth

TL;DR

This paper establishes maximal $L^2$-regularity for nonlinear gradient systems with sublinear perturbations, extending classical smoothing effects and applying fixed point methods to solve perturbed evolution equations.

Contribution

It introduces a novel approach combining smoothing effects and fixed point theorems to handle sublinear perturbations in nonlinear gradient systems.

Findings

Maximal $L^2$-regularity holds for the perturbed evolution equations.

The method applies to the $p$-Laplacian and $p$-harmonic Dirichlet-to-Neumann operators.

Sublevel sets' compactness is crucial for the results.

Abstract

The nonlinear semigroup generated by the subdifferential of a convex lower semicontinuous function $\varphi$ has a smoothing effect, discovered by H. Br\'ezis, which implies maximal regularity for the evolution equation. We use this and Schaefer's fixed point theorem to solve the evolution equation perturbed by a Nemytskii-operator of sublinear growth. For this, we need that the sublevel sets of $\varphi$ are not only closed but even compact. We apply our results to the $p$-Laplacian and also to the Dirichlet-to-Neumann operator with respect to $p$-harmonic functions.

Equations

$$\{\begin{array}{ll}\dot{u}(t)+Au(t)\ni f(t)& \text{a.e. on (0,T),}\\ u(0)=u_0.& \end{array}$$

$$\Vert Gv(t){\Vert }_H\le L\Vert v(t){\Vert }_H+b(t)\text{a.e. on (0,T) and for all v∈H,}$$

$$\{\begin{array}{ll}\dot{u}(t)+Au(t)\ni Gu(t)& \text{a.e. on (0,T),}\\ u(0)=u_0.& \end{array}$$

$$\mathcal{S}:=\Big{\{}u\in X\,\Big{|}\,\text{there exists }\lambda\in[0,1]\text{ s.t. }u=\lambda\mathcal{T}{}u\Big{\}}$$

$$E_{c}:=\Big{\{}u\in D(\varphi)\,\Big{|}\;\varphi(u)\leq c\Big{\}}$$

$${\phi }_{\omega }(u):=\phi (u)+\frac{\omega }{2}\Vert u{\Vert }_H^2,(u\in H),$$

$$\partial\varphi=\Big{\{}(u,h)\in H\times H\,\Big{|}\,\liminf_{t\downarrow 0}\frac{\varphi(u+tv)-\varphi(u)}{t}\geq(h,v)_{H}\,\forall\,v\in D(\varphi)\Big{\}},$$

$$\partial\varphi=\Big{\{}(u,h)\in H\times H\,\Big{|}\,\varphi_{\omega}(u+v)-\varphi_{\omega}(u)\geq(h+\omega u,v)_{H}\,\forall\,v\in D(\varphi)\Big{\}}.$$

$$\dot{u}(t)+Au(t)\ni f(t)\text{a.e. on (0,T),}$$

$$(h+\omega u,u{)}_H\ge 0\text{for all (u,h)∈A.}$$

$$\{\begin{array}{ll}\dot{u}(t)+Au(t)\ni f(t)& \text{a.e. on (0,T),}\\ u(0)=u_0.& \end{array}$$

$$\Vert u(t){\Vert }_H$$

$${\int }_0^T\phi (u(s))\mathrm{d}s$$

$$t\phi (u(t))$$

$$\Vert \sqrt{\cdot }\dot{u}{\Vert }_{\mathcal{H}}^2$$

$$F(u)$$

$$\varphi (u)$$

$$F_1(u)$$

$${\phi }_1(u)=\{\begin{array}{ll}\frac{1}{p}{\int }_{\Omega }|\nabla u{|}^p\mathrm{d}x+{\int }_{\Omega }{F}_1(u)\mathrm{d}x& \text{if u∈V,}\\ +\infty & \text{if u∈H∖V}\end{array}$$

$$D_v\phi (u)=\underset{t\to 0+}{\lim }\frac{\phi (u+t)-\phi (u)}{t}={\int }_{\Omega }|\nabla u{|}^{p-2}\nabla u\nabla v+f_1(x,u)v\mathrm{d}x$$

$$D(\partial {\phi }_1)$$

$$\partial {\phi }_1(u)$$

$$\phi (u)={\phi }_1(u)+\varphi (u)\text{for every u∈H,}$$

$$D(A)$$

$$Au$$

$$\overline{D(A)}=\overline{D(\varphi)}=\Big{\{}u\in H\,\Big{|}\,j(u(x))\in\overline{D(\beta)}\textrm{ for a.e. $x\in\Omega$}\Big{\}}.$$

$$\{\begin{array}{ll}{\partial }_{t}u(t)-{\Delta }_{p}u(t)+\beta (u(t))+f_1(\cdot ,u(t))\ni f(t)& \text{on (0,T)×Ω,}\\ u(t)=0& \text{on (0,T)×∂Ω,}\\ u(0)=u_0& \text{on Ω.}\end{array}$$

$$E_{\omega\!,c}:=\Big{\{}u\in D(\varphi)\,|\varphi_{\omega}(u)\leq c\Big{\}}\quad\text{is compact in $H$ for every $c\in{\mathbb{R}}$.}$$

$$\phi (u):=\{\begin{array}{ll}\frac{1}{p}{\int }_{\Omega }|\nabla u{|}^p\mathrm{d}x& \text{if u∈V,}\\ +\infty & \text{if u∈H∖V}\end{array}$$

$$\Vert Gv(t){\Vert }_H\le L\Vert v(t){\Vert }_H+b(t)\text{a.e. on (0,T), for all v∈H.}$$

$$\{\begin{array}{ll}\dot{u}(t)+Au(t)\ni Gu(t)& \text{a.e. on (0,T),}\\ u(0)=u_0.& \end{array}$$

$$\phi \circ u\in W_{loc}^{1,1}((0,T])\cap L^1(0,T),$$

$$\Vert u(t){\Vert }_H\le {\left(\Vert u_0{\Vert }_H^2+\Vert b{\Vert }_{L^2(0,T)}^2\right)}^{\frac{1}{2}}{e}^{\frac{2L+1+2\omega }{2}t}\text{for all t∈[0,T].}$$

$$\bullet$$

Full text

The second author is very grateful for the warm hospitality received during his visits at the University of Ulm.

1. 1 Introduction
2. 2 Preliminaries
3. 3 Main result
4. 4 Proof of the main result
5. 5 Application to $j$-elliptic functions
6. A Brézis’ maximal $L^{2}$-regularity theorem

1. Introduction

Let $H$ be a real Hilbert space, $\varphi:H\to(-\infty,+\infty]$ a proper, convex, lower semicontinuous function and $A=\partial\varphi$ the subdifferential of $\varphi$ (see Section 2 for more details). Then $A$ is a maximal monotone (in general, multi-valued) operator on $H$, for which the following remarkable well-posedness result holds.

Theorem 1.1** **(Brézis [9]).

Let $u_{0}\in H$ such that $\varphi(u_{0})$ is finite and $f\in L^{2}(0,T;H)$. Then, there exists a unique $u\in H^{1}(0,T;H)$ such that

[TABLE]

Our aim in this article is to study a perturbed version of (1.1). Let $\mathcal{H}$ denote the space $L^{2}(0,T;H)$, ($T>0$), and $G:\mathcal{H}\to\mathcal{H}$ be a continuous mapping satisfying the sublinear growth condition

[TABLE]

for some constants $L$, $b\in L^{2}(0,T)$ satisfying $b(t)\geq 0$ for a.e. $t\in(0,T)$. Then we study the evolutionary problem

[TABLE]

For that, we will use a compactness argument in form of Schaefer’s fixed point theorem (see Theorem 2.1 in Section 2). Recall that lower semicontinuity of $\varphi$ is equivalent to saying that the sublevel sets $E_{c}:=\{u\in H\,|\,\varphi(u)\leq c\}$, ($c\in{\mathbb{R}}$), are closed. We will assume more, namely, compactness of sublevel sets $E_{c}$. In fact, we need this assumption only for the shifted function $\varphi_{\omega}$ given by $\varphi_{\omega}(u)=\varphi(u)+\frac{\omega}{2}\lVert u\rVert_{H}^{2}$ $(u\in H)$, which is important for applications. Then our main results says the following.

Theorem 1.2**.**

Let $\varphi:H\to(-\infty,+\infty]$ be a proper function such that for some $\omega\geq 0$, $\varphi_{\omega}$ is convex and has compact sublevel sets. Let $A=\partial\varphi$ and $G:\mathcal{H}\to\mathcal{H}$ be a continuous mapping satisfying (1.2). Then for every $u_{0}\in H$ with $\varphi(u_{0})$ finite, there exists $u\in H^{1}(0,T;H)$ solving (1.3).

We show in Example 3.6 that the solution is not unique in general. The proof of Theorem 1.2 is based on Brézis’ Theorem 1.1. However, we need it under the hypothesis that merely $\varphi_{\omega}$ is convex. We give a proof of this more general result (see Theorem 2.3) in the appendix of this paper. Theorem 1.2 remains also true if $u_{0}\in\overline{D(\varphi)}$ where $D(\varphi):=\{u\in H\,|\,\varphi(u)<+\infty\}$; however, the solution of (1.3) is merely in $H^{1}_{loc}((0,T];H)$ in that case.

As application, we consider $H=L^{2}(\Omega)$ and $G$ a Nemytskii operator. The operator $A$ may be the $p$-Laplacian ($1\leq p<+\infty$) with possibly lower order terms and equipped with some boundary conditions (Dirichlet, Neumann, or Robin, see [13]) or a $p$-version of the Dirichlet-to-Neumann operator considered recently in [15] and via the abstract theory of $j$-elliptic functions (see [3, 4] and [12]).

2. Preliminaries

In this section, we define the precise setting used throughout this paper and explain our mains tools: Brezis’ result for semiconvex functions and Schaefer’s fixed point theorem.

We begin by recalling that a mapping $\mathcal{T}$ defined on a Banach space $X$ is called compact if $\mathcal{T}$ maps bounded sets in into relatively compact sets.

Theorem 2.1** **([18], **Schaefer’s fixed point

theorem**).

Let $X$ be a Banach space and $\mathcal{T}:X\to X$ be continuous and compact. Assume that the “Schaefer set”

[TABLE]

is bounded in $X$. Then $\mathcal{T}$ has a fixed point.

This result is a special case of Leray-Schauder’s fixed point theorem, but Schaefer gave a most elegant proof (cf [14]), which also is valid in locally compact spaces.

Given a function $\varphi:H\to(-\infty,+\infty]$, we call the set $D(\varphi):=\{u\in H\,|\,\varphi(u)<+\infty\}$ the effective domain of $\varphi$, and $\varphi$ is said to be proper if $D(\varphi)$ is non-empty. Further, we say that $\varphi$ is lower semicontinuous if for every $c\in{\mathbb{R}}$, the sublevel set

[TABLE]

is closed in $H$, and $\varphi$ is semiconvex if there exists an $\omega\in{\mathbb{R}}$ such that the shifted function $\varphi_{\omega}:H\to(-\infty,+\infty]$ defined by

[TABLE]

is convex. Then, $\varphi_{\hat{\omega}}$ is convex for all $\hat{w}\geq\omega$, and $\varphi_{\omega}$ is lower semicontinuous if and only if $\varphi$ is lower semicontinuous.

Given a function $\varphi:H\to(-\infty,+\infty]$, its subdifferential $A=\partial\varphi$ is defined by

[TABLE]

which, if $\varphi_{\omega}$ is convex, reduces to

[TABLE]

It is standard to identify a (possibly multi-valued) operator $A$ on $H$ with its graph and for every $u\in H$, one sets $Au:=\{v\in H\,|\,(u,v)\in A\}$ and calls $D(A):=\{u\in H\,|\,Au\neq\emptyset\}$ the domain of $A$ and $\textrm{Rg}(A):=\bigcup_{u\in D(A)}\!Au$ the range of $A$.

Now, suppose $\varphi:H\to(-\infty,+\infty]$ is proper, lower semicontinuous, and semiconvex; more precisely, let’s fix $\omega\in{\mathbb{R}}$ such that $\varphi_{\omega}$ is convex. Then, under those hypotheses on $\varphi$, Brézis’ well-posedness result (Theorem 1.1) remains true.

Remark 2.2** **(Maximal $L^{2}$-regularity).

If $u_{0}\in H$ such that $\varphi(u_{0})$ is finite, then Theorem 1.1 says that for every $f\in L^{2}(0,T;H)$, the unique solution $u$ of (1.1) has its time derivative $\dot{u}\in L^{2}(0,T;H)$ and hence by the differential inclusion

[TABLE]

also $Au\in L^{2}(0,T;H)$. In other words, for $f\in L^{2}(0,T;H)$, $\dot{u}$ and $Au\in L^{2}(0,T;H)$ admit the maximal possible regularity. For this reason, we call this property maximal $L^{2}$-regularity, as it is customary for generators of holomorphic semigroups on Hilbert spaces (see [1] for a survey on this subject).

As before, we fix $T>0$, denote by $\mathcal{H}$ the space $L^{2}(0,T;H)$, and write $\lVert\cdot\rVert_{\mathcal{H}}$ for the norm $\lVert\cdot\rVert_{L^{2}(0,T;H)}$.

Further, after possibly replacing $\varphi$ by a translation, we may always assume without loss of generality that $0\in D(\partial\varphi_{\omega})$ and $\varphi_{\omega}$ attains a minimum at [math] with $\varphi_{\omega}(0)=0$ (for further details see [5, p159] or the appendix of this paper). By the convexity of $\varphi_{\omega}$, this implies that $(0,0)\in\omega I_{H}+A$, that is,

[TABLE]

With this assumption in mind, we now state Brézis’ $L^{2}$-maximal regularity theorem for semiconvex functions.

Theorem 2.3** **(**Brézis’ $L^{2}$-maximal regularity for

semiconvex $\varphi$).**

Let $u_{0}\in\overline{D(\varphi)}$ and $f\in\mathcal{H}$. Then, there exists a unique $u\in H^{1}_{loc}((0,T];H)\cap C([0,T];H)$ satisfying

[TABLE]

Moreover, one has that $\varphi\circ u\in W^{1,1}_{loc}((0,T])\cap L^{1}(0,T)$,

[TABLE]

Finally, if $u_{0}\in D(\varphi)$, then $u\in H^{1}(0,T;H)$.

To keep this paper self-contained, we provide a proof of this result in the appendix of this paper.

Definition 2.4**.**

Given $f\in\mathcal{H}$ and $u_{0}\in H$, we call a function $u:[0,T]\to H$ a (strong) solution of (2.3) (respectively, of (1.1)) if $u\in H^{1}_{loc}((0,T];H)\cap C([0,T];H)$, $u(0)=u_{0}$, and for a.e. $t\in(0,T)$, one has that $u(t)\in D(A)$ and $f(t)-\dot{u}(t)\in Au(t)$.

For illustrating the theory developed in this paper, we consider the following standard example: the Dirichlet $p$-Laplacian perturbed by a lower order term.

Example 2.5**.**

Let $\Omega$ be an open subset of ${\mathbb{R}}^{d}$, ($d\geq 1$), $H=L^{2}(\Omega)$, and for $\frac{2d}{d+2}\leq p<\infty$, let $V=W^{1,p}_{0}(\Omega)$ be the closure of $C_{c}^{1}(\Omega)$ with respect to the norm $\lVert u\rVert_{V}:=\lVert\nabla u\rVert_{L^{p}(\Omega;{\mathbb{R}}^{d})}$. Then, one has that $V$ is continuously embedded into $H$ (cf [11, Theorem 9.16]); we write for this $V\hookrightarrow H$.

Further, let $f=\beta+f_{1}$ be the sum of a maximal monotone graph $\beta$ of ${\mathbb{R}}$ satisfying $(0,0)\in\beta$ and a Lipschitz-Carathéodory function $f_{1}:\Omega\times{\mathbb{R}}\to{\mathbb{R}}$ satisfying $f(x,0)=0$; that is, for a.e. $x\in\Omega$, $f_{1}(x,\cdot)$ be Lipschitz continuous (with constant $\omega>0$) uniformly for a.e. $x\in\Omega$, and $f_{1}(\cdot,u)$ is measurable on $\Omega$ for every $u\in{\mathbb{R}}$. Then, there is a proper, convex and lower semicontinuous function $j:{\mathbb{R}}\to(-\infty,+\infty]$ satisfying $j(0)=0$ and $\partial j=\beta$ in ${\mathbb{R}}$ (see [5, Example 1., p53]). We set

[TABLE]

for every $u\in L^{2}(\Omega)$. Now, let $\varphi_{1}:H\to(-\infty,+\infty]$ be given by

[TABLE]

for every $u\in H$. Then the domain $D(\varphi_{1})$ of $\varphi_{1}$ is $V$. The function $\varphi_{1}$ is lower semicontinuous on $H$, proper, $\varphi_{1\!,\omega}$ is convex, and for every $u\in V$, $\varphi_{1}$ is Gâteaux-differentiable with

[TABLE]

for every $v\in V$. Since $V$ is dense in $H$, the operator $\partial\varphi_{1}$ is a single-valued operator on $H$ with domain

[TABLE]

The operator $\partial\varphi_{1}$ is the negative Dirichlet $p$-Laplacian $-\Delta_{p}^{\!D}$ on $\Omega$ with a Lipschitz continuous lower order term $f_{1}$. Next, we add the function $\phi$ given by (2.8) to the $\varphi_{1}$. For this, note that $\phi$ is proper (since for $u_{0}\equiv 0$, $\phi(u_{0})=0$) with $\textrm{int}(D(\phi))\neq\emptyset$, convex (since $j$ is convex), and lower semicontinuous on $H$. Thus, the function $\varphi:H\to(-\infty,+\infty]$ given by

[TABLE]

is convex, lower semicontinuous, and proper with domain $D(\varphi)=\{u\in V\,|\,j(u)\in L^{1}(\Omega)\}$ and the operator $A=\partial\varphi$ is given by

[TABLE]

and $A$ is single-valued provided $D(\varphi)$ is dense in $L^{2}(\Omega)$. Here, we note that

[TABLE]

Due to Theorem 2.3, for every $u_{0}\in\overline{D(\varphi)}$ and $f\in\mathcal{H}$, there is a unique solution $u\in H^{1}_{loc}((0,T];H)\cap C([0,T];H)$ of the parabolic boundary-value problem

[TABLE]

Here, we write $\partial_{t}u(t)$ instead of $\dot{u}(t)$ since we rewrote the abstract Cauchy problem (2.3) as an explicit parabolic partial differential equation.

3. Main result

Throughout this section, let $\varphi:H\to(-\infty,+\infty]$ be a proper function. We assume that there is an $\omega\in{\mathbb{R}}$ such that $\varphi_{\omega}$ is convex and the sublevel set

[TABLE]

$\mbox{}_{\Box}$

Remark 3.1**.**

We emphasize that condition (3.1) does not imply that $\varphi$ has compact sublevel sets. This becomes more clear if one considers as $\varphi$ the function associated with the negative Neumann $p$-Laplacian $-\Delta_{p}^{\!\!N}$ on a bounded, open subset $\Omega$ of ${\mathbb{R}}^{d}$ with a Lipschitz boundary $\partial\Omega$. For $\max\{1,\frac{2d}{d+2}\}<p<\infty$, ($d\geq 1$), let $V=W^{1,p}(\Omega)$, $H=L^{2}(\Omega)$, $\mathcal{H}=L^{2}(0,T;L^{2}(\Omega))=L^{2}((0,T)\times\Omega)$, and $\varphi:H\to(-\infty,+\infty]$ be given by

[TABLE]

for every $u\in H$. Then, for every $c>0$, the sublevel set $E_{0,c}$ of $\varphi$ contains the sequence $(u_{n})_{n\geq 0}$ of constant functions $u_{n}\equiv n$, which does not admit any convergent subsequence in $H$. On the other hand, for every $\omega>0$ and $c>0$, the sublevel set $E_{\omega\!,c}$ is a bounded set in $V$ and by Rellich-Kandrachov’s compactness, one has that $V\hookrightarrow H$ by a compact embedding. Thus, for every $\omega>0$ and $c>0$, the sublevel set $E_{\omega\!,c}$ is compact in $L^{2}(\Omega)$.

Let $G:\mathcal{H}\to\mathcal{H}$ be a continuous function with sublinear growth; that is, there are $L\geq 0$ and $b\in L^{2}(0,T)$ satisfying $b(t)\geq 0$ a.e. on $(0,T)$ such that

[TABLE]

Here we let $Gv(t):=(Gv)(t)$ to use less heavy notation. Then, our main result of this paper reads as follows.

Theorem 3.2**.**

Let $u_{0}\in\overline{D(\varphi)}$ and $f\in\mathcal{H}$. Then, there exists a solution $u\in H^{1}_{loc}((0,T];H)\cap C([0,T];H)$ of

[TABLE]

In particular, if $u_{0}\in D(\varphi)$, then problem (3.3) has a solution $u\in H^{1}(0,T;H)$.

Note that $Gu\in\mathcal{H}$. Thus, the inclusion in (3.3) means that $Gu(t)-\dot{u}(t)\in Au(t)$ a.e. on $(0,T)$. In particular, the following regularity estimates hold for strong solutions of (3.3).

Remark 3.3**.**

For given $u_{0}\in\overline{D(\varphi)}$ and $f\in\mathcal{H}$, the solution $u$ of (3.3) satisfies

[TABLE]

[TABLE]

$\mbox{}_{\Box}$

The main example of perturbations $G$ allowed in Theorem 3.2 are Nemytskii operators on $\mathcal{H}=L^{2}(0,T;L^{2}(\Omega))$. Let $\Omega\subseteq{\mathbb{R}}^{d}$ be open and $g:(0,T)\times\Omega\times{\mathbb{R}}\to{\mathbb{R}}$ be a Carathéodory function, that is,

[TABLE]

Assume furthermore that $g$ has sublinear growth, that is, there exist $L\geq 0$ and $f\in L^{2}(0,T;L^{2}(\Omega))$ such that

[TABLE]

Proposition 3.4**.**

Let $\mathcal{H}=L^{2}(0,T;L^{2}(\Omega))$. Then, the relation

[TABLE]

defines a continuous operator $G:\mathcal{H}\to\mathcal{H}$ of sublinear growth (1.2).

The proof is routine (cf [19, Proposition 26.7]) if one uses that $f_{n}\to f$ in $\mathcal{H}$ if and only if each subsequence of $(f_{n})_{n\geq 1}$ has a dominated subsequence converging to $f$ a.e. (which is well known from the completeness proof of $L^{2}$).

We illustrate our result by reconsidering Example 2.5 adding a perturbation of Nemytskii type.

Example 3.5** **(Example 2.5 revisited).

For $\max\{1,\frac{2d}{d+2}\}<p<\infty$, let $V=W^{1,p}_{0}(\Omega)$, $H=L^{2}(\Omega)$, $\mathcal{H}=L^{2}((0,T)\times\Omega)$ and let $\varphi$ be given by (2.9). Then, there is an $\omega>0$ such that $\varphi_{\omega}$ is convex and for every $c>0$, the sublevel set $E_{\omega\!,c}$ is compact in $L^{2}(\Omega)$. Furthermore, let $g:(0,T)\times\Omega\times{\mathbb{R}}\to{\mathbb{R}}$ be a Carathédory function with sublinear growth and $u_{0}\in\overline{D(\varphi)}$. Then, there is at least one solution $u\in H^{1}_{loc}((0,T];H)\cap C([0,T];H)$ of the parabolic boundary-value problem

[TABLE]

In general, the solutions in Example 3.5 are not unique. We give an example.

Example 3.6** **(Non-uniqueness).

Let $g(u)=\sqrt{\lvert u\rvert}$, $u\in{\mathbb{R}}$, and $\Omega$ be an open and bounded subset of ${\mathbb{R}}^{d}$, $d\geq 1$, with a Lipschitz boundary $\partial\Omega$. Then, there are $L$, $b>0$ such that $\hat{g}$ satisfies

[TABLE]

Thus, for $H=L^{2}(\Omega)$, one has that $\mathcal{H}=L^{2}((0,T)\times\Omega)$ and the associated Nemytskii operator $G:\mathcal{H}\to\mathcal{H}$ defined by (3.6) satisfies the sublinear growth condition (1.2). For $\max\{1,\frac{2d}{d+2}\}<p<+\infty$, let $\varphi:L^{2}(\Omega)\to(-\infty,+\infty]$ be the energy function (3.2) associated with the negative Neumann $p$-Laplacian $-\Delta_{p}^{\!N}$ on $\Omega$. Then, by Theorem 3.2, for every $u_{0}\in L^{2}(\Omega)$ and every $T>0$, there is a solution $u\in H^{1}_{loc}((0,T];L^{2}(\Omega))\cap C([0,T];L^{2}(\Omega))$ of

[TABLE]

Here, $\lvert\nabla u\rvert^{p-2}D_{\nu}u$ denotes the (weak) co-normal derivative of $u$ on $\partial\Omega$ (cf [13]). Now, for the initial value $u_{0}\equiv 0$ on $\Omega$, the constant zero function $u\equiv 0$ is certainly a solution of (3.7). For constructing a non-trivial solution of (3.7) with initial value $u_{0}\equiv 0$, let $w\in C^{1}[0,T]$ be a non-trivial solution of the following classical ordinary differential equation

[TABLE]

For instance, one non-trivial solution is $w(t)=t^{2}/4$. Since for every constant $c\in{\mathbb{R}}$, $-\Delta_{p}^{\!N}(c\mathds{1}_{\Omega})=0$, the function $u(t):=w(t)$ is another non-trivial solution of (3.7) with initial value $u_{0}\equiv 0$.

4. Proof of the main result

For the proof of Theorem 3.2, we need some auxiliary results. The first concerns continuity and is standard (see Bénilan [8, (6.5), p87] or Barbu [5, (4.2), p128]).

Lemma 4.1**.**

Let $f_{1}$, $f_{2}\in\mathcal{H}$, $u_{1}$, $u_{2}\in H^{1}(0,T;H)$ such that

[TABLE]

Then,

[TABLE]

for every $t\in[0,T]$.

Next, we establish the compactness of the solution operator $P$ associated with evolution problem (2.3). Note, for convenience, we write here $\mathcal{H}$ to denote $\mathcal{H}$, ($T>0$), and recall that the closure $\overline{D(\varphi)}$ in $H$ of the effective domain of a semiconvex function $\varphi$ is a convex subset of $H$.

Lemma 4.2**.**

Let $P:\overline{D(\varphi)}\times\mathcal{H}\to\mathcal{H}$ be the mapping defined dy

[TABLE]

Then, $P$ is continuous and compact.

Proof.

(a) By Lemma 4.1, the map $P$ is continuous from $\overline{D(\varphi)}\times\mathcal{H}$ to $\mathcal{H}$.

(b) We show that $P$ is compact. Let $(u_{n}^{(0)})_{n\geq 1}\subseteq\overline{D(\varphi)}$ and $(f_{n})_{n\geq 1}\subseteq\mathcal{H}$ such that $\lVert u_{n}^{(0)}\rVert_{H}+\lVert f_{n}\rVert_{\mathcal{H}}\leq c$ and $u_{n}=P(u_{n}^{(0)},f_{n})$ for every $n\geq 1$. Then, by (2.4), (2.5) and by (2.7), for every $\delta\in(0,T)$, there is a $c_{\delta}>0$ such that

[TABLE]

Since $H^{1}(\delta,T;H)\hookrightarrow C^{\gamma}([\delta,T];H)$ for some $\gamma\in(0,1)$, it follows that the sequence $(u_{n})_{n\geq 1}$ is equicontinuous on $[\delta,T]$ for each $0<\delta<T$. Choose a countable dense subset $D:=\{t_{m}|\,m\in\mathbb{N}\}$ of $(0,T]$. Let $m\geq 1$. Then by (2.6),

[TABLE]

and since by (2.4), $(u_{n}(t_{m}))_{n\geq 1}$ is bounded in $H$, there is a $c^{\prime}>0$ such that $(u_{n}(t_{m}))_{n\geq 1}$ is in the sublevel set $E_{\!\omega,c^{\prime}}$. Thus and by the assumption (3.1), $(u_{n}(t_{m}))_{n\geq 1}$ has a convergent subsequence in $H$. By Cantor’s diagonalization argument, we find a subsequence $(u_{n_{k}})_{k\geq 1}$ of $(u_{n})_{n\geq 1}$ such that

[TABLE]

It follows from the equicontinuity of $(u_{n_{k}})_{k\geq 1}$ that $u_{n_{k}}$ converges in $C([\delta,T];H)$ for all $\delta\in(0,T]$. In particular, $(u_{n_{k}}(t))_{k\geq 1}$ converges in $H$ for every $t\in(0,T)$ and by (2.4), $(u_{n_{k}})_{k\geq 1}$ is uniformly bounded in $L^{\infty}(0,T;H)$. Thus, it follows from Lebesgue’s dominated convergence theorem that $u_{n_{k}}=P(u_{n_{k}}^{(0)},f_{n_{k}})$ converges in $\mathcal{H}$. ∎

Remark 4.3**.**

In the previous proof, we have actually shown that $P$ is compact from $\overline{D(\varphi)}\times\mathcal{H}$ into the Fréchet space $C((0,T];H)$.

With these preliminaries, we can now give the proof of our main result. Here, we got inspired from the linear case (cf [2]).

Proof of Theorem 3.2.

First, let $u_{0}\in\overline{D(\varphi)}$.

Let $v\in\mathcal{H}$. Then $Gv\in\mathcal{H}$ and so, by Brézis’ maximal $L^{2}$-regularity result (Theorem 2.3), there is a unique solution $u\in H^{1}_{loc}((0,T];H)\cap C([0,T];H)$ of the evolution problem

[TABLE]

Let $\mathcal{T}{}v:=P(u_{0},Gv)$. Then by the continuity of $G$ and since $P(u_{0},\cdot):\mathcal{H}\to\mathcal{H}$ is continuous and compact (Lemma 4.2), the mapping $\mathcal{T}:\mathcal{H}\to\mathcal{H}$ is continuous and compact.

a) We consider the Schaefer set

[TABLE]

We show that $\mathcal{S}$ is bounded in $\mathcal{H}$. Let $u\in\mathcal{S}$. We may assume that $\lambda\in(0,1]$, otherwise, $u\equiv 0$. Then, one has that $u\in H^{1}_{loc}((0,T];H)\cap C([0,T];H)$ and

[TABLE]

It follows from (2.2) that

[TABLE]

Thus and by (1.2),

[TABLE]

for a.e. $t\in(0,T)$. It follows from Gronwall’s lemma that (3.4) holds for every $t\in[0,T]$. Thus, $\mathcal{S}$ is bounded in $\mathcal{H}$. Now, Schaefer’s fixed point theorem implies that there exists $u\in\mathcal{H}$ such that $u=\mathcal{T}{}u$; that is, $u\in H^{1}_{loc}((0,T];H)\cap C([0,T];H)$ is a solution of the evolution problem (3.3).

b) Let $u_{0}\in D(\varphi)$. Then, by the first part of this proof, there is a solution solution $u\in H^{1}_{loc}((0,T];H)\cap C([0,T];H)$ of the evolution problem (3.3). However, by Brézis’ maximal regularity result applied to $f=Gu\in\mathcal{H}$, it follows that $u\in H^{1}(0,T;H)$. This completes the proof of this theorem. ∎

5. Application to $j$-elliptic functions

In the previous examples (cf Examples 2.5 and Example 3.6), $V$ is a Banach space injected in $H$. Recently, in [12], Chill, Hauer and Kennedy extended results of [3], [4] by Arendt and Ter Elst to a nonlinear framework of $j$-elliptic functions $\varphi:V\to(-\infty,+\infty]$ generating a quasi maximal monotone operator $\partial_{j}\varphi$ on $H$, where $j:V\to H$ is just a linear operator which is not necessarily injective. This enabled the authors of [12] to show that several coupled parabolic-elliptic systems can be realized as a gradient system in a Hilbert space $H$ and to extend the linear variational theory of the Dirichlet-to-Neumann operator to the nonlinear $p$-Laplace operator (see also [6, 7, 16] for further applications and extensions of this theory).

The aim of this section is to illustrate that the main Theorem 3.2 of Section 3 can also be applied to the framework of $j$-elliptic functions.

Let us briefly recall some basic notions and facts about $j$-elliptic functions from [12]. Let $V$ be a real locally convex topological vector space and $j:V\to H$ be a linear operator which is merely weak-to-weak continuous (and, in general, not injective). Given a function $\varphi:V\to(-\infty,+\infty]$, then the $j$-subdifferential is the operator

[TABLE]

The function $\varphi$ is called $j$-semiconvex if there exists $\omega\in{\mathbb{R}}$ such that the “shifted” function $\varphi_{\omega}:V\to(-\infty,+\infty]$ given by

[TABLE]

is convex. If $V=H$ and $j=I_{H}$, then $j$-semiconvex functions $\varphi$ are the semiconvex ones (see Section 1). The function $\varphi$ is called $j$-elliptic if there exists $\omega\geq 0$ such that $\varphi_{\omega}$ is convex and for every $c\in{\mathbb{R}}$, the sublevel sets $\{\hat{u}\in V\,|\,\varphi_{\omega}(u)\leq c\}$ are relatively weakly compact. Finally, we say that the function $\varphi$ is lower semicontinuous if the sublevel sets $\{\varphi\leq c\}$ are closed in the topology of $V$ for every $c\in{\mathbb{R}}$. It was highlighted in [12, Lemma 2.2] that

1. (a)

If $\varphi$ is $j$-semiconvex, then there is an $\omega\in{\mathbb{R}}$ such that

[TABLE] 2. (b)

If $\varphi$ is Gâteaux differentiable with directional derivative $D_{\hat{v}}\varphi$, ($\hat{v}\in V$), then

[TABLE]

The main result in [12] is that the $j$-subdifferential $\partial_{j}\varphi$ of a $j$-elliptic function $\varphi$ is already a classical subdifferential. More precisely, the following holds.

Theorem 5.1** ([12, Corollary 2.7]).**

*Let $\varphi:V\to(-\infty,+\infty]$ be proper, lower semicontinuous, and $j$-elliptic. Then there is a proper, lower semicontinuous, semiconvex function $\varphi^{H}:H\to(-\infty,+\infty]$ such that $\partial_{j}\varphi=\partial\varphi^{H}$. The function $\varphi^{H}$ is unique up to an additive constant. *

Thus the operator $A=\partial_{j}\varphi$ has the properties of maximal regularity we used before. The following result gives a description of $\varphi^{H}$ in the convex case and will be important for our intentions in this paper.

Theorem 5.2** ([12, Theorem 2.9]).**

Assume that $\varphi:V\to(-\infty,+\infty]$ is convex, proper, lower semicontinuous and $j$-elliptic, and let $\varphi^{H}:H\to(-\infty,+\infty]$ be the function from Corollary 5.1. Then, there is a constant $c\in{\mathbb{R}}$ such that

[TABLE]

with effective domain $D(\varphi^{H})=j(D(\varphi))$.

For our perturbation result, we need the compactness of the sublevel sets of $\varphi^{H}$. With the help of Theorem 5.2 we can establish a criterion in terms of the given $\varphi$ for this property.

Lemma 5.3**.**

Let $\varphi:V\to(-\infty,+\infty]$ be proper, lower semicontinuous $j$-semiconvex, and $j$-elliptic. Assume that

[TABLE]

then there is an $\omega\geq 0$ such that for every $c\in{\mathbb{R}}$, the sublevel set

[TABLE]

Remark 5.4**.**

If $V$ is a normed space, then by the Eberlein-Šmulian Theorem hypothesis (5.1) is equivalent to $j$ maps weakly convergent sequences in $V$ to norm convergent sequences in $H$. This in turn is equivalent to $j$ being compact if $V$ is reflexive.

Proof of Lemma 5.3.

By hypothesis, there is an $\omega\geq 0$ such that $\varphi_{\omega}$ is convex, lower semicontinuous, and for every $c\in{\mathbb{R}}$, the sublevel sets $\{\hat{u}\in V\,|\,\varphi_{\omega}(u)\leq c\}$ are weakly relatively compact and closed. By Corollary 5.1, there is a lower semicontinuous, proper function $\varphi^{H}:H\to(-\infty,+\infty]$ such that $\varphi^{H}_{\omega}$ is convex and $\partial\varphi^{H}_{\omega}=\partial_{j}\varphi_{\omega}$. Applying Theorem 5.2 to $\varphi_{\omega}$ and $\varphi^{H}_{\omega}$, we have that

[TABLE]

and some constant $d\in{\mathbb{R}}$. For $c\in{\mathbb{R}}$, let $(u_{n})_{n\geq 1}$ be an arbitrary sequence in $E_{\omega\!,c}$. By (5.2), for every $n\in\mathbb{N}$, there is a $\hat{u}_{n}\in j^{-1}(\{u_{n}\})$ such that

[TABLE]

By hypothesis, all sublevel sets of $\varphi_{\omega}$ are weakly relatively compact in $V$. Thus, by our hypothesis, the image under $j$ is relatively compact in $H$. Consequently, there are a subsequence $(u_{n_{l}})_{l\geq 1}$ of $(u_{n})_{n\geq 1}$ and a $u\in H$ such that $u_{n_{l}}=j(\hat{u}_{n_{l}})\to u$ in $H$ as $l\to+\infty$. Since $\varphi^{H}_{\omega}(u_{n_{l}})\leq c$ and since $\varphi^{H}$ is lower semicontinuous, it follows that $\varphi^{H}(u)\leq c$. This shows that $E_{\omega\!,c}$ is compact. ∎

Now, applying Lemma 5.3 to Theorem 3.2, we can state the following existence theorem.

Theorem 5.5**.**

Let $\varphi:V\to(-\infty,+\infty]$ be proper, lower semicontinuous $j$-semiconvex, and $j$-elliptic. Assume that the mapping $j$ satisfies (5.1) and let $G:\mathcal{H}\to\mathcal{H}$ be a continuous mapping of sublinear growth (1.2). Then, for $A=\partial_{j}\varphi$ the nonlinear evolution problem (3.3) admits for every $u_{0}\in\overline{j(D(\varphi))}$ and $f\in\mathcal{H}$ at least one solution $u\in H^{1}_{loc}((0,T];H)\cap C([0,T];H)$. In particular, one has that $\varphi\circ u$ belongs to $W^{1,1}_{loc}((0,T])\cap L^{1}(0,T)$ and inequality (3.4) holds. If $u_{0}\in D(\varphi)$, then problem (3.3) has a solution $u\in H^{1}(0,T;H)$.

We complete this section by considering the following evolution problem involving the Dirichlet-to-Neumann operator associated with the $p$-Laplacian (cf [15, 12]).

Example 5.6**.**

Let $\Omega$ be a bounded domain with a Lipschitz continuous boundary $\partial\Omega$. Then, for $\frac{2d}{d+1}<p<+\infty$, the trace operator $\textrm{Tr}:W^{1,p}(\Omega)\to L^{2}(\partial\Omega)$ is a completely continuous operator (cf [17, Théorème 6.2] for the case $p<d$, the other cases $p=d$ and $p>d$ can be deduced from [17, Conséquence 6.2 & 6.3]). Now, we take

$V=W^{1,p}(\Omega)$, $H=L^{2}(\partial\Omega)$, and $j=\textrm{Tr}$.

Then, $j$ is a linear bounded mapping satisfying hypothesis (5.1). In fact, $j$ is a prototype of a non-injective mapping. Furthermore, let $\varphi:V\to{\mathbb{R}}$ be the function given by

[TABLE]

Then, $\varphi$ is continuously differentiable on $V$ and convex. Thus, the Tr-subdifferential operator $\partial_{\textrm{Tr}}\varphi$ is given by

[TABLE]

Moreover, by inequality [15, (20)], for any $\omega>0$, the shifted function $\varphi_{\omega}$ has bounded level sets in $V$. Since $V$ is reflexive, every level set of $\varphi_{\omega}$ is weakly compact in $V$. In addition, by [15, Lemma 2.1], $j(D(\varphi))$ is dense in $H$.

Now, let $g:(0,T)\times\Omega\times{\mathbb{R}}\to{\mathbb{R}}$ be a Carathédory function with sublinear growth. Then by Theorem 5.5, for every $u_{0}\in L^{2}(\partial\Omega)$, there is at least one solution $u\in H^{1}_{loc}((0,T];L^{2}(\partial\Omega))\cap C([0,T];L^{2}(\partial\Omega))$ of the elliptic-parabolic boundary-value problem

[TABLE]

Appendix A Brézis’ maximal $L^{2}$-regularity theorem

To keep this paper self-contained, we show in this appendix that Brézis’ maximal $L^{2}$-regularity result (Theorem 2.3) remains true for proper, lower semicontinuous functions $\varphi:H\to(-\infty,+\infty]$, which are semiconvex.

Under the above hypotheses on $\varphi$, the subdifferential operator $A=\partial\varphi$ is quasi maximal monotone. Note that an operator $A$ on $H$ is called maximal monotone if firstly, $A$ is monotone, that is,

[TABLE]

and secondly, $A$ satisfies the range condition

[TABLE]

Now, an operator $A$ is called quasi maximal monotone if there is and $\omega\in{\mathbb{R}}$ such that $\omega I_{H}+A$ is maximal monotone.

One important property of the class of maximal monotone operators in Hilbert spaces is that their graph is closed in $H\times H_{w}$, where $H_{w}$ means that $H$ is equipped with the weak topology $\sigma(H^{\ast},H)$.

Proposition A.1** ([10, Proposition 2.5]).**

Let $A$ be an maximal monotone operator, $((u_{n},v_{n}))_{n\geq 1}\subseteq A$, $u$, $v\in H$ such that $u_{n}\rightharpoonup u$ and $v_{n}\rightharpoonup v$ weakly in $H$ as $n\to+\infty$ and $\limsup_{n\to+\infty}(u_{n},v_{n})_{H}\leq(u,v)_{H}$. Then $(u,v)\in A$ and $(u_{n},v_{n})_{H}\to(u,v)_{H}$ as $n\to+\infty$.

For the class of $\omega$-quasi maximal monotone operators in Hilbert spaces the following existence and regularity result holds. Here, we recall [5, Theorem 4.5] in the Hilbert spaces framework and note that in Hilbert spaces monotone operators are accretive and vice versa.

Theorem A.2** **(Existence & regularity for smooth $f$).

Let $A$ be an $\omega$-quasi maximal monotone operator for some $\omega\in{\mathbb{R}}$, $f\in W^{1,1}(0,T;H)$, $u_{0}\in D(A)$. Then there is a unique solution $u\in W^{1,\infty}(0,T;H)$ of problem (2.3).

Further, since $\partial\varphi_{\omega}=A+\omega I_{H}$ has dense domain in $\overline{D(\varphi)}$ by [10, Proposition 2.11] (or [5, p.48]), the domain $D(A)$ of the subdifferential operator $A=\partial\varphi$ is dense in $\overline{D(\varphi)}$. For later use, we fix this observation in the next proposition.

Proposition A.3**.**

Let $\varphi:H\to(-\infty,+\infty]$ be proper, semiconvex, and lower semicontinuous. Then the domain $D(A)$ of $A=\partial\varphi$ is dense in $\overline{D(\varphi)}$.

We also need the following chain rule for convex functions $\varphi$.

Lemma A.4** ([10, Lemma 3.3]).**

Let $\varphi:H\to(-\infty,+\infty]$ be proper, convex, and lower semicontinuous, and $u\in H^{1}(0,T;H)$. Assume, there is a $g\in\mathcal{H}$ such that $(u(t),g(t))\in\partial\varphi$ for a.e. $t\in(0,T)$. Then $\varphi\circ u$ is absolutely continuous on $[0,T]$ and

[TABLE]

Note, we may always assume without loss of generality that $0\in D(\partial\varphi_{\omega})$, $\varphi_{\omega}$ attains a minimum at [math] (that is, (2.2) holds), and $\varphi_{\omega}(0)=0$. Otherwise, one chooses any $(u_{0},v_{0})\in\partial\varphi_{\omega}$ and replaces $\varphi$ by

[TABLE]

Then,

[TABLE]

$\varphi_{\omega}\geq 0$, $0\in D(\tilde{\varphi})$, and $\tilde{\varphi}_{\omega}(0)=0$. Moreover, for each solution $y$ of inclusion

[TABLE]

the function $u(t):=y(t)+u_{0}$ is a solution of (2.1). This shows that there is no loss of generality by assuming that for $\varphi_{\omega}$, inequality (2.2) holds and $\varphi_{\omega}\geq 0$.

With this, we can now outline the proof of Brézis’ $L^{2}$-maximal regularity result.

Proof of Theorem 2.3.

Let $f\in\mathcal{H}$, $u_{0}\in D(\varphi)$, $f_{n}\in H^{1}(0,T;H)$ such that $f_{n}\to f$ in $\mathcal{H}$. Moreover, for every $n\geq 1$, there are $u_{n}^{(0)}\in D(A)$ such that

[TABLE]

and $u_{n}^{(0)}\to u_{0}$ in $H$ (see the last paragraph on [5, p.161]). By Theorem A.2, there is a unique solution $u_{n}\in W^{1,\infty}(0,T;H)$ of problem

[TABLE]

Then, by Lemma 4.1, $(u_{n})_{n\geq 1}$ is a Cauchy sequence in $C([0,T];H)$. Hence there is a $u\in C([0,T];H)$ such that $u_{n}\to u$ in $C([0,T];H)$. In particular, $u(0)=u_{0}$.

(a) We show that $u$ satisfies (3.4). Adding $\omega u_{n}$ on both sides of

[TABLE]

and then multiplying the resulting inclusion by $u_{n}$ yields

[TABLE]

for every $h\in Au_{n}(t)$ for a.e. $t\in(0,T)$. Applying (2.2), and then integrating over $(0,t)$, for $t\in(0,T]$ leads to

[TABLE]

Now, the Gronwall inequality gives that $u_{n}$ satisfies the uniform bound (2.4) and by letting $n\to+\infty$ using that $u_{n}\to u$ in $C([0,T];H)$, we have that $u$ satisfies (2.4).

(b) Next, we show that $u\in H^{1}(0,T;H)$. First, we add $\omega u_{n}$ on both sides of (A.2), and then multiply the resulting inclusion by $\dot{u}_{n}$. Now, by Lemma A.4,

[TABLE]

for a.e. $t\in(0,T)$. From this and by (A.1), one deduces that

[TABLE]

Note that $\varphi_{\omega}$ is bounded from below by an affine function. Thus and by part (a), $(\dot{u}_{n})_{n\geq 1}$ is bounded in $\mathcal{H}$. Since $\mathcal{H}$ is reflexive, $(\dot{u}_{n})_{n\geq 1}$ admits a weakly convergent subsequence in $\mathcal{H}$. From this, by the limit $u_{n}\to u$ in $C([0,T];H)$, we can conclude that $u\in H^{1}(0,T;H)$. Moreover, by the lower semicontinuity of $\varphi_{\omega}$, one see that $u$ satisfies

[TABLE]

for every $t\in(0,T]$, which is equivalent to

[TABLE]

(c) We conclude showing that $u$ is a solution of the evolution problem (2.1). For this, we use the lifted operator $\mathcal{A}$ in $\mathcal{H}$ given by

[TABLE]

Since $\omega I_{H}+A=\partial\varphi_{\omega}$ is maximal monotone on $H$, we have that $\mathcal{A}_{\omega}:=\omega\mathcal{I}_{\mathcal{H}}+\mathcal{A}$ is maximal monotone on $\mathcal{H}$ (see [10, Exemple 2.3.3]). Moreover, $u_{n}\to u$ in $\mathcal{H}$, and after having chosen a subsequence, $v_{n}:=f_{n}+\omega u_{n}-\dot{u}_{n}\rightharpoonup v:=f+\omega u-\dot{u}$ weakly in $\mathcal{H}$. Thus, by Proposition A.1, $u\in D(\mathcal{A})$ and $v\in\mathcal{A}_{\omega}u$, this is equivalent to $u(t)\in D(A)$ and $f(t)-\dot{u}(t)\in Au(t)$ for a.e. $t\in(0,T)$.

(d) Next, let $f\in\mathcal{H}$, $u_{0}\in\overline{D(\varphi)}$, and $u_{n}^{(0)}\in D(\varphi)$ such that $u_{n}^{(0)}\to u_{0}$ in $H$. By the previous part, for every $n\geq 1$, there are solutions $u_{n}\in H^{1}(0,T;H)$ of problem

[TABLE]

By Lemma 4.1, $(u_{n})_{n\geq 1}$ is a Cauchy sequence in $C([0,T];H)$ and so, there is a $u\in C([0,T];H)$ such that $u_{n}\to u$ in $C([0,T];H)$ as $n\to+\infty$. Moreover, by the same argument as in part (a), one sees that each $u_{n}$ and $u$ satisfies (2.4).

(e) Next, we show that

[TABLE]

Since $f(t)-\dot{u}_{n}(t)\in\partial\varphi(u_{n}(t))$, it follows from the definition of $A=\partial\varphi$ that

[TABLE]

for every $v\in H$ and a.e. $t\in(0,T)$. Thus taking $v=0$ and using that $\varphi_{\omega}\geq 0$, one sees that

[TABLE]

for a.e. $t\in(0,T)$. Integrating over $(0,T)$, one sees that

[TABLE]

From this, it follows that (A.3) holds. Then, since $u_{n}\to u$ in $C([0,T];H)$ and $\varphi_{\omega}(u_{n})\geq 0$, it follows from the lower semicontinuity of $\varphi_{\omega}$ and by Fatou’s lemma that (A.4) holds for $u$ and hence, $\varphi\circ u\in L^{1}(0,T)$ satisfying (2.5).

(f) We show that $u\in H^{1}_{loc}((0,T];H)$ with $\sqrt{\cdot}\dot{u}\in\mathcal{H}$, and there is a subsequence $(u_{n_{k}})_{k\geq 1}$ of $(u_{n})_{n\geq 1}$ such that $\dot{u}_{n_{k}}\rightharpoonup\dot{u}$ weakly in $L^{2}_{loc}((0,T];H)$. We first add $\omega u_{n}$ on both sides of

[TABLE]

and then multiply the resulting inclusion by $t\cdot\dot{u}_{n}(t)$. Then by Lemma A.4,

[TABLE]

for a.e. $t\in(0,T)$. Applying Cauchy-Schwarz’s and Young’s inequality on the right hand side of this equation, and subsequently integrating over $(0,t)$ for $t\in(0,T]$ gives

[TABLE]

Further, by (A.4) applied to $T=t$, one has

[TABLE]

Recall that $u_{n}\to u$ in $C([0,T];H)$. Thus, $(\sqrt{\cdot}\dot{u}_{n})_{n\geq 1}$ is bounded in $\mathcal{H}$ and so by the reflexivity of $\mathcal{H}$, one has that $u\in H^{1}_{loc}((0,T];H)$ with $\sqrt{\cdot}\dot{u}\in\mathcal{H}$. In particular, $(\dot{u}_{n})_{n\geq 1}$ is bounded in $L^{2}(\delta,T;H)$ for every $\delta\in(0,T]$. Thus, a diagonal sequence arguments shows that there is a subsequence $(u_{n_{k}})_{k\geq 1}$ of $(u_{n})_{n\geq 1}$ such that $\dot{u}_{n_{k}}\rightharpoonup\dot{u}$ weakly in $L^{2}_{loc}((0,T];H)$.

(g) Next, we show that $u$ is a solutions of (2.3) and $\varphi\circ u\in W^{1,1}_{loc}((0,T])$. To see that $u$ is a solution of (2.3) recall that $u_{n}^{(0)}\to u_{0}$ in $H$ and the solutions $u_{n}$ of (A.2) converge to $u$ in $C([0,T];H)$. Thus, $u(0)=u_{0}$ and since for every $\delta\in(0,T]$, $\dot{u}_{n}\rightharpoonup\dot{u}$ weakly in $L^{2}(\delta,T;H)$, it follows by the same argument as in part (c) from the maximal monotonicity of the operator $\omega I_{\mathcal{H}}+\mathcal{A}_{\delta}$ in $L^{2}(\delta,T;H)$ with

[TABLE]

that $u(t)\in D(A)$ for a.e. $t\in(0,T)$ and $f(t)-\dot{u}(t)\in A(u(t))$. Moreover, since now, $g(t):=f(t)-\dot{u}(t)+\omega u(t)\in\partial\varphi_{\omega}(u(t))$ for a.e. $t\in(0,T)$ and $g\in L^{2}(\delta,T;H)$ for every $\delta\in(0,T]$, it follows from Lemma A.4 that $\varphi\circ u\in W^{1,1}_{loc}((0,T])$. This completes the proof of Brézis’ $L^{2}$-maximal regularity result for semiconvex $\varphi$.

(h) Finally, we show that $u$ satisfies (2.6) and (2.7). Since $u$ is a solution of (2.3), we can add $\omega u$ on both side of

[TABLE]

and then multiply the resulting inclusion by $t\cdot\dot{u}(t)$. Recall, $\sqrt{\cdot}\dot{u}\in\mathcal{H}$. Thus by Lemma A.4,

[TABLE]

for a.e. $t\in(0,T)$. Next, by Cauchy-Schwarz’s and Young’s inequality, and subsequently integrating over $(0,t)$ for $t\in(0,T]$ gives

[TABLE]

from which we can conclude (2.6) and (2.7). ∎