Very degenerate elliptic equations under almost critical Sobolev regularity

TL;DR

This paper establishes local Lipschitz continuity and higher differentiability of minimizers for degenerate elliptic functionals with coefficients and data in an almost critical Zygmund class, advancing regularity theory under weak regularity assumptions.

Contribution

It introduces new regularity results for degenerate elliptic equations assuming coefficients and data in an almost critical Zygmund class, a setting previously unexplored.

Findings

Proved local Lipschitz continuity of minimizers.

Established higher differentiability of solutions.

Extended regularity results to weak coefficient regularity.

Abstract

We prove the local Lipschitz continuity and the higher differentiability of local minimizers of integral functionals with non autonomous integrand which is degenerate convex with respect to the gradient variable. The main novelty here is that the results are obtained assuming that the coefficients have weak derivative in an almost critical Zygmund class and the datum f is assumed to belong to the same Zygmund class.

Equations

$$\mathbb{F}(u,\Omega)=\int_{\Omega}\big{(}\mathcal{F}(x,Du)+f(x)\cdot u(x)\Big{)}\,dx$$

$$\mathbb{F}(u,\Omega)=\int_{\Omega}\big{(}\mathcal{F}(x,Du)+f(x)\cdot u(x)\Big{)}\,dx,$$

$$\ell (|\xi {|}^p-1)\le \mathcal{F}(x,\xi )\le L(|\xi {|}^p+1)$$

$$\mathcal{F}(x,\xi )=F(x,|\xi |)$$

$$⟨D_{\xi \xi }\mathcal{F}(x,\xi )\lambda ,\lambda ⟩⩾\nu |\xi {|}^{p-2}|\lambda {|}^2,$$

$$|D_{\xi \xi }\mathcal{F}(x,\xi )|⩽L_1|\xi {|}^{p-2}$$

$$|D_{x\xi }\mathcal{F}(x,\xi )|⩽k(x)|\xi {|}^{p-1},$$

$$\mathcal{F}(x,\xi )=(|\xi |-1{)}_{+}^p,p\ge 2.$$

$$|D_{x\xi }\mathcal{F}(x,\xi )|\le k(x)(1+|\xi |{)}^{p-1}(x,\xi )\in \Omega \times {\mathbb{R}}^{n\times N}$$

$$\underset{B_{\rho }}{\sup }|Du|⩽\hat{C}{\left({\int }_{B_r}(1+|Du|{)}^{p}dx,\right)}^{\frac{1}{p}}$$

$$\mathcal{G}(t):=1+{\int }_0^t(1+s{)}^{\frac{p-4}{2}}sds,$$

$$\mathcal{G}((|Du|-1{)}_{+})\in W_{\mathrm{loc}}^{1,2}(\Omega )$$

$${\int }_{B_{\rho }(x_0)}|D(\mathcal{G}((|Du|-1{)}_{+})){|}^2\le \tilde{C}{\int }_{B_r(x_0)}\left(1+|Du{|}^p\right)dx,$$

$${\int }_{B_{\rho }(x_0)}|D(\mathcal{G}((|Du|-1{)}_{+})){|}^2={\int }_{B_{\rho }(x_0)}(|Du|-1{)}_{+}^2|Du{|}^{p-4}|D^{2}u{|}^{2}dx$$

$$u_{x,r}:=\frac{1}{|B_r(x)|}{\int }_{B_r(x)}u(y)dy,$$

$$h(s)\le \vartheta h(t)+\frac{A}{(t-s{)}^{\beta }}+B,$$

$$h(r)\le \frac{cA}{(R_0-r{)}^{\beta }}+cB,$$

$$\Phi (t)={\Phi }_{\gamma }(t)=\frac{t^2}{(1+t{)}^2}(1+t{)}^{\gamma }.$$

$$t{\Phi }^{\prime }(t)$$

$${\Phi }^{\prime }(t)\le 2(1+\gamma )t(1+t^2{)}^{\gamma -2},$$

$${\Phi }^{\prime }(t)\le 2^{\gamma +1}(1+\gamma )\text{for every}t\in (0,1).$$

$$P=(|Du|-1{)}_{+}=\max \{|Du|-1,0\}$$

$$DP$$

$${\int }_{\Omega }\Psi \left(\frac{|u|}{\lambda }\right)dx<\infty ,$$

$$\Vert u{\Vert }_{L^{\Psi }(\Omega )}=\inf \left\{\lambda >0:{\int }_{\Omega }\Psi \left(\frac{|u|}{\lambda }\right)dx⩽1\right\}$$

$$W{L}^{\Psi }(\Omega )=\left\{u\in W^{1,1}(\Omega ):|\nabla u|\in L^{\Psi }(\Omega )\right\}.$$

$$\Psi (t)\simeq t^p{\log }^{\alpha }(e+t)\text{for every t≥t0≥0}.$$

$${\int }_{\Omega }|u{|}^p{\log }^{\alpha }(e+|u|)dx<\infty ,$$

$$[u]_{p,\alpha}=\biggm{[}\int|u|^{p}\log^{\alpha}\left(e+\frac{|u|}{\|u\|_{p}}\right)\,dx\biggm{]}^{\frac{1}{p}}$$

$$\frac{\Psi (t)}{t^{p-\epsilon }}\nearrow \frac{\Psi (t)}{t^{p+\epsilon }}\searrow \text{for every}\epsilon >0,$$

Full text

[TABLE]

with non autonomous integrand $\mathcal{F}(x,\xi)$ which is degenerate convex with respect to the gradient variable. The main novelty here is that the results are obtained assuming that the partial map $x\mapsto D_{\xi}\mathcal{F}(x,\xi)$ has weak derivative in the almost critical Zygmund class $L^{n}\log^{\alpha}L$ and the datum $f$ is assumed to belong to the same Zygmund class.

Mathematics Subject Classification. 35J47; 35J70; 49N60.

*Key words and phrases. Degenerate convex functionals; Lipschitz regularity; Higher differentiability. *

1 Introduction

In this paper we deal with the regularity properties of local minimizers of non autonomous and non homogeneous functionals of the form

[TABLE]

where $\Omega\subset\mathbb{R}^{n}$, $n\geq 2$ is a bounded open set, $u:\Omega\to\mathbb{R}^{N}$, $N\geq 1$ , $\mathcal{F}:\Omega\times\mathbb{R}^{n\times N}\to[0,+\infty)$ is a Carathéodory function with growth $p\geq 2$, assumed to be uniformly convex with respect to the gradient variable only at $\infty$, and $f:\Omega\subset\mathbb{R}^{n}\to\mathbb{R}^{N}$ is a given datum. To be more precise, we shall assume that $\xi\mapsto\mathcal{F}(x,\xi)$ is convex and, for an exponent $p\geq 2$, satisfies the following set of hypotheses.

1. (F0)

There exist positive constants $\ell,L$ such that

[TABLE]

for a.e. $x\in\Omega$ and every $\xi\in\mathbb{R}^{n\times N}$.

This is the unique assumption that has to hold true on the whole $\mathbb{R}^{n\times N}$. From now on we list the assumptions that need to hold only at $\infty$, i.e. outside a ball of radius $\mathcal{R}$.

1. (F1)

There exists $F:\overline{\Omega}\times[\mathcal{R},\infty)\to[0,+\infty)$ such that

[TABLE]

for a.e. $x\in\Omega$ and for every $\xi\in\mathbb{R}^{n\times N}\setminus B_{\mathcal{R}}(0)$. 2. (F2)

The partial map $\xi\mapsto\mathcal{F}(x,\xi)$ is $C^{2}(\mathbb{R}^{n\times N}\setminus B_{\mathcal{R}}(0))$ and there exists $\nu>0$ such that

[TABLE]

for a.e. $x\in\Omega$, every $\lambda\in\mathbb{R}^{n\times N}$ and every $\xi\in\mathbb{R}^{n\times N}\setminus B_{\mathcal{R}}(0)$. 3. (F3)

There exists a positive constant $L_{1}$ such that

[TABLE]

for a.e. $x\in\Omega$ and every $\xi\in\mathbb{R}^{n\times N}\setminus B_{\mathcal{R}}(0)$. 4. (F4)

There exist and exponent $\alpha>0$ and a non-negative function $k\in L^{n}\log^{\alpha}L_{{\mathop{\mathrm{missing}}{\,loc\,}}}(\Omega)$ such that

[TABLE]

for a.e. $x\in\Omega$ and for every $\xi\in\mathbb{R}^{n\times N}\setminus B_{\mathcal{R}}(0)$.

There is no loss of generality in assuming that $\mathcal{R}=1$, what we will do in what follows.

The functionals with degenerate convexity have attracted a wide interest in the last few years. Among the reasons, they arise in problems of optimal transport with congestion effects and in the construction of Wardrop equilibriums in traffic problems (see for instance Carlier, Jiménez and Santambrogio [4]). In particular, the regularity of minimizers, and more specifically their Lipschitz continuity, allows the application of the Di Perna - Lions theory to describe the equilibrium configurations. This connection was first described by Brasco, Carlier and Santambrogio [3], and has partially motivated a research line which is nowadays very active.

In this setting, the local Lipschitz regularity of the minimizers was proven in [3, Theorem 5.2] for $f\in C^{\alpha}$ and $\alpha>0$ , and in [2, Theorem 2.1] for $f\in L^{s}$, $s>n$ in case the energy density has the following special structure

[TABLE]

It is worth mentioning that the regularity of minimizers of non autonomous widely degenerate functionals is challenging also from the theoretical point of view and many contributions to its study are available.

Actually, they fit into the broader contest of asymptotically convex functionals, whose study started with the pioneering paper by Chipot and Evans ([6]) concerning the homogeneous, autonomous, quadratic growth case. Later on, still for the homogeneous, autonomous case, the Lipschitz continuity of local minimizers of asymptotically convex functionals has been established for the superquadratic growth ([25]), in the subquadratic growth case ([31]). Since then, many contributions to the regularity theory of asymptotically convex functionals have been established. Among the others we quote the results in [11, 17, 18, 19, 20, 21, 22, 23, 30, 36, 37, 38].

We’d like to recall that, in the non autonomous homogeneous case, i.e. $f\equiv 0$, Fonseca, Fusco and Marcellini proved in [16] that local minimizers of $\mathbb{F}(u,\Omega)$ are Lipschitz continuous if

[TABLE]

for some function $k\in L^{\infty}$, i.e. under a Lipschitz regularity of the partial map $x\mapsto D_{\xi}\mathcal{F}(x,\xi)$. More recently, still for $f\equiv 0$, in [13] this result was extended to the case $k\in L^{s}_{{\mathop{\mathrm{missing}}{\,loc\,}}}(\Omega)$ for some $s>n$ (see also [14, 15]).

Especially for applications, it is desirable to understand what is the situation under assumptions on the map $x\mapsto D_{\xi}\mathcal{F}(x,\xi)$ weaker than the Hölder continuity. We know, though, that if $s=n=2$ then $\nabla u\in L^{\infty}$ may fail even in the linear uniformly elliptic setting [8, 33], while in the case $n>2$ some regularity results under a Sobolev or Orlicz Sobolev assumption on the coefficients, that doesn’t imply their Hölder continuity, can be found in [24, 26, 27, 28, 32, 34].

The case $k\in L^{n}_{{\mathop{\mathrm{missing}}{\,loc\,}}}(\Omega)$ has been studied in [9] for homogeneous functionals with degenerate convexity, but obtaining only $W^{1,t}$ regularity for every $t\in[p,\infty)$.

As far as we know, no Lipschitz regularity results are available for local minimizers of this class of functionals under weaker assumptions on the coefficients and the datum and this motivated our study.

Here we will show that the gradient of the local minimizers of $\mathbb{F}(u,\Omega)$ is locally bounded in $\Omega$ under a regularity assumption on the map $x\rightarrow D_{\xi}f(x,\xi)$ and the datum $f$ that, roughly speaking, will be intermediate between $L^{n}$ and $L^{s}$, with ${s}>n$.

Actually, we are able to prove that in order to obtain the local Lipschitz regularity of the minimizers it suffices to assume that both the weak derivative with respect to $x$ of $D_{\xi}f(x,\xi)$ and the datum $f$ belong to the Orlicz Zygmund class $L^{n}\log^{\alpha}L$, with $\alpha>0$ sufficiently large with respect to the dimension $n$. More precisely, our main result is the following

Theorem 1**.**

Let $u\in W^{1,p}_{{\mathop{\mathrm{missing}}{\,loc\,}}}(\Omega)$ be a local minimizer of the functional $\mathbb{F}(u,\Omega)$ in (1.1), and assume that the energy density $\mathcal{F}(x,\xi)$ satisfies assumptions (F0)–(F4). If $f\in L^{n}\log^{\alpha}L_{{\mathop{\mathrm{missing}}{\,loc\,}}}(\Omega)$, with $\alpha>3n$ and where $\alpha$ is the exponent appearing in (F4), then $Du\in L^{\infty}_{\mathrm{loc}}(\Omega)$ and the following estimate

[TABLE]

holds for every balls $B_{\rho}\subset B_{r}\Subset\Omega$ and for a constant $\hat{C}=\hat{C}(n,p,\alpha,\nu,L_{1},\rho,r,||k||_{L^{n}\log^{\alpha}L(B_{r})},||f||_{L^{n}\log^{\alpha}L(B_{r})})$.

Note that, by virtue of the embedding Theorems in the Orlicz-Sobolev setting ([7]), our assumption on the partial map $x\mapsto D_{\xi}\mathcal{F}(x,\xi)$ implies the continuity of the coefficients but not their Hölder continuity.

As usual, the proof of this Theorem is divided in two parts: an a priori $L^{\infty}$ estimate, and an approximation argument.

The a priori estimate is achieved by the use of test functions that vanish on the set where the functional in not uniformly convex. The novelty here is the use of a suitable Young’s inequality in Orlicz space and an appropriate modification of the Moser iteration argument.

The size of the exponent $\alpha$ in assumption (F4) is used in order to let this iteration argument work.

Once the a priori estimate is established, we perform an approximation procedure with a sequence of functionals regular enough to apply the a priori estimate to their minimizers.

The lack of the uniqueness of the minimizers, due to the lack of uniform convexity of the integrand, could effect the approximation argument. We overcome this difficulty, as done in [5], by adding to the approximating functionals a penalization term that forces the approximating sequence to converge to an arbitrarily fixed local minimizers.

It is worth mentioning that in order to start the Moser iteration procedure, we need to estimate quantities involving the second derivatives of the local minimizers. Therefore, another result that we are able to establish here consists in showing that minimizers turn out to gain one degree of differentiability, in the weak sense, away from the degeneracy set. This can be quantitatively stated as follows. Denoting by

[TABLE]

we have the following higher differentiability result.

Theorem 2**.**

Let $u\in W^{1,p}(\Omega;\mathbb{R}^{n\times N})$ be a local minimizer of the functional $\mathbb{F}(\cdot,\Omega)$ in (1.1), and assume that the energy density $\mathcal{F}(x,\xi)$ satisfies assumptions (F0)-(F4). If $f\in L^{n}\log^{\alpha}L_{{\mathop{\mathrm{missing}}{\,loc\,}}}(\Omega)$ with $\alpha>3n$, then

[TABLE]

and the following Caccioppoli type inequality holds,

[TABLE]

for every balls $B_{\rho}\subset B_{r}(x_{0})\Subset\Omega$, for some $\widetilde{C}=\widetilde{C}(n,p,\alpha,\nu,L_{1},\rho,r,||k||_{L^{n}\log^{\alpha}L(B_{r})},||f||_{L^{n}\log^{\alpha}L(B_{r})})$.

The proof of Theorem 2 is established as before combining an a priori estimate with an approximation argument.

This time the regularity of the approximating minimizers is transferred to the limit by the use of a measure theory result ( see Proposition 8 below) that is an improved version of Proposition 6 in [10].

Let us mention that in equation (1.3), the term on the left hand side is equivalent to

[TABLE]

so that the above result is, in fact, a weighted bound for $D^{2}u$ with the weight $(|Du|-1)_{+}^{2}\,|Du|^{p-4}$.

The paper is structured as follows. In Section 2 we recall some preliminaries. In Section 3 we prove both the interior a priori Lipschitz and the a priori second order Sobolev estimates. In Section 4 we prove the Lipschitz regularity of the minimizers of $\mathbb{F}(u,\Omega)$, i.e. Theorem 1. In Section 5 we prove the interior second order Sobolev regularity of the minimizers of $\mathbb{F}(u,\Omega)$, i.e. Theorem 2.

2 Preliminary results

We will follow the usual convention and denote by $c$ or $C$ a general constant that may vary on different occasions, even within the same line of estimates. Relevant dependencies on parameters and special constants will be suitably emphasized using parentheses or subscripts. All the norms we use will be the standard Euclidean ones and denoted by $|\cdot|$ in all cases. In particular, for matrices $\xi$, $\eta\in\mathbb{R}^{n\times m}$ we write $\langle\xi,\eta\rangle:=\text{trace}(\xi^{T}\eta)$ for the usual inner product of $\xi$ and $\eta$, and $|\xi|:=\langle\xi,\xi\rangle^{\frac{1}{2}}$ for the corresponding euclidean norm. By $B_{r}(x)$ we will denote the ball in $\mathbb{R}^{n}$ centered at $x$ of radius $r$. The integral mean of a function $u$ over a ball $B_{r}(x)$ will be denoted by $u_{x,r}$, that is

[TABLE]

where $|B_{r}(x)|$ is the Lebesgue measure of the ball in $\mathbb{R}^{n}$. If no confusion arises, we shall omit the dependence on the center.

The following lemma is an important application in the so called hole-filling method. Its proof can be found for example in [29, Lemma 6.1] .

Lemma 3**.**

Let $h:[r,R_{0}]\to\mathbb{R}$