Remarks on a Paper by Leonetti and Siepe

TL;DR

This paper extends previous results on higher integrability of solutions to anisotropic elliptic boundary value problems by considering obstacle problems with nonhomogeneous terms under growth and monotonicity conditions.

Contribution

It generalizes Leonetti and Siepe's integrability results to obstacle problems with nonhomogeneous anisotropic elliptic equations.

Findings

Established higher integrability of solutions under new conditions.

Generalized previous boundary data integrability results.

Applicable to nonhomogeneous anisotropic elliptic equations.

Abstract

In 2012, F.Leonetti and F.Siepe [1] considered solutions to boundary value problems of some anisotropic elliptic equations of the type $$ \left\{ \begin{array}{llll} \sum\limits _{i=1}\limits^{n} D_i (a_i(x,Du(x)))=0, &x\in \Omega,\\ u(x)=\theta (x), & x\in \partial \Omega. \end{array} \right. $$ Under some suitable conditions, they obtained an integrability result, which shows that, higher integrability of the boundary datum $\theta$ forces solutions $u$ to have higher integrability as well. In the present paper, we consider ${\cal K}_{\psi,\theta}^{(p_i)}$-obstacle problems of the nonhomogeneous anisotropic elliptic equations $$ \sum_{i=1}^n D_i (a_i(x,Du(x)))=\sum_{i=1}^n D_i f^i(x). $$ Under some controllable growth and monotonicity conditions. We obtain an integrability result, which can be regarded as a generalization of the result due to Leonetti and Siepe.

Equations

$$\left\{\begin{array}[]{llll}\sum\limits_{i=1}\limits^{n}D_{i}(a_{i}(x,Du(x)))=0,&x\in\Omega,\\ u(x)=\theta(x),&x\in\partial\Omega.\end{array}\right.$$

$$\sum _{i=1}^n{D}_i(a_i(x,Du(x)))=\sum _{i=1}^n{D}_i{f}^i(x).$$

$$p_m=\underset{k=1,2,\cdots ,n}{\max }{p}_k,\overline{p}:\frac{1}{\overline{p}}=\frac{1}{n}\sum _{k=1}^n\frac{1}{p_k}.$$

$$|\{x\in \Omega :|f(x)|>\tau \}|\le \frac{k}{{\tau }^t}$$

$$W^{1,(p_k)}(\Omega )=\left\{v\in W^{1,1}(\Omega ):D_{k}v\in L^{p_k}(\Omega )\text{mbox}foreveryk=1,2,\cdots ,n\right\}$$

$$W_0^{1,(p_k)}(\Omega )=\left\{v\in W_0^{1,1}(\Omega ):D_{k}v\in L^{p_k}(\Omega )\text{mbox}foreveryk=1,2,\cdots ,n\right\},$$

$$\sum _{i=1}^n{D}_i(a_i(x,Du(x)))=\sum _{i=1}^n{D}_i{f}^i(x),$$

$$|a_i(x,z)|\le c_1{\left(h(x)+\sum _{j=1}^n|z_j{|}^{p_j}\right)}^{1-1/p_i}$$

$$\tilde{\nu }\sum _{i=1}^n|z_i-{\tilde{z}}_i{|}^{p_i}\le \sum _{i=1}^n(a_i(x,z)-a_i(x,\tilde{z}))(z_i-{\tilde{z}}_i)$$

$${\mathcal{K}}_{\psi ,\theta }^{(p_k)}(\Omega )=\left\{v\in W^{1,(p_k)}(\Omega ):v\ge \psi ,\text{mbox}a.e.andv\in \theta +W_0^{1,(p_k)}(\Omega )\right\}.$$

$$\left\{\begin{array}[]{llll}\sum\limits_{i=1}\limits^{n}D_{i}(a_{i}(x,Du(x)))=\sum\limits_{i=1}\limits^{n}D_{i}f^{i}(x),&x\in\Omega,\\ u(x)=\theta(x),&x\in\partial\Omega,\end{array}\right.$$

$${\int }_{\Omega }\sum _{i=1}^n{a}_i(x,Du(x))D_i\phi (x)dx={\int }_{\Omega }\sum _{i=1}^n{f}^i(x)D_i\phi (x)dx,$$

$${\int }_{\Omega }\sum _{i=1}^n{a}_i(x,Du(x))(D_{i}v(x)-D_{i}u(x))dx\ge {\int }_{\Omega }\sum _{i=1}^n{f}^i(x)(D_{i}v(x)-D_{i}u(x))dx$$

$$\left\{\begin{array}[]{llll}\sum\limits_{i=1}\limits^{n}D_{i}(a_{i}(x,Du(x)))=0,&x\in\Omega,\\ u(x)=\theta(x),&x\in\partial\Omega\end{array}\right.$$

$$|a_i(x,z)|\le c_2(1+|z_i|{)}^{p_i-1},i=1,2,\cdots ,n$$

$${\int }_{\Omega }(|D_1{u}_1{|}^{p_1}+|D_2{u}_2{|}^{p_2}+\cdots +|D_n{u}_n{|}^{p_n})dx.$$

$$\sum _{i=1}^n{D}_i(a_i(x,Du(x)))=0,x\in \Omega$$

$$\sum _{i=1}^n{D}_i(a_i(x,Du(x)))=\sum _{i=1}^n{D}_i{f}^i(x),x\in \Omega$$

$$u\in {\theta }_{\ast }+L_{weak}^t(\Omega )$$

$$t=\frac{{\overline{p}}^{\ast }}{1-\frac{b{\overline{p}}^{\ast }}{\overline{p}}\frac{p_m}{p_m-1}}>{\overline{p}}^{\ast },$$

$$0<b\le \underset{j=1,\cdots ,n}{\min }\left(1-\frac{p_j}{q_j}\right)\underset{i=1,\cdots ,n}{\min }\left(1-\frac{1}{p_i}\right)$$

$$b<\frac{p_m-1}{p_m}\frac{\overline{p}}{{\overline{p}}^{\ast }}.$$

$$u\in \theta +L_{weak}^t(\Omega ),$$

$$\begin{array}[]{llll}\displaystyle\min_{j=1,\cdots,n}\left(1-\frac{p_{j}}{q_{j}}\right)\min_{i=1,\cdots,n}\left(1-\frac{1}{p_{i}}\right)&=\left(1-\frac{p}{q}\right)\left(1-\frac{1}{p}\right)\\ &\displaystyle<\frac{(p-1)(n-p)}{np}=\frac{p_{m}-1}{p_{m}}\frac{\overline{p}}{\overline{p}^{*}};\end{array}$$

$$b=\underset{j=1,\cdots ,n}{\min }\left(1-\frac{p_j}{q_j}\right)\underset{i=1,\cdots ,n}{\min }\left(1-\frac{1}{p_i}\right)$$

$$t=\frac{nq}{n-q}.$$

$$u\in {\theta }_{\ast }+L_{weak}^t(\Omega )$$

$$t=\frac{nq}{n-q}.$$

$$u\in \theta +L_{weak}^t(\Omega ),$$

$$t=\frac{nq}{n-q}.$$

Full text

Remarks on a Paper by Leonetti and Siepe

GAO Hongya**  LIU Chao  TIAN Hong

College of Mathematics and Computer Science, Hebei University, Baoding, 071002, P.R.China**

††MR Subject Classification: 35J60, 35D30, 35J25.††Keywords: Integrability, anisotropic elliptic equation, anisotropic obstacle problem.††Supported by NSFC (10971224) and NSF of Hebei Province (A2011201011).

Abstract.  In 2012, F.Leonetti and F.Siepe [1] considered solutions to boundary value problems of some anisotropic elliptic equations of the type

[TABLE]

Under some suitable conditions, they obtained an integrability result, which shows that, higher integrability of the boundary datum $\theta$ forces solutions $u$ to have higher integrability as well. In the present paper, we consider ${\cal K}_{\psi,\theta}^{(p_{i})}$-obstacle problems of the nonhomogeneous anisotropic elliptic equations

[TABLE]

Under some controllable growth and monotonicity conditions. We obtain an integrability result, which can be regarded as a generalization of the result due to Leonetti and Siepe.

§1   Introduction and statement of results

Let $\Omega$ be a bounded open subset of $\mbox{R}^{n}$, $n\geq 2$. For $p_{k}>1$, $k=1,2,\cdots,n$, we denote by $p_{m}$ and $\overline{p}$ the maximum value and the harmonic mean of $p_{k}$, respectively, i.e.,

[TABLE]

For $t>0$, the weak $L^{t}$-spaces, or Marcinkiewicz spaces, $L_{weak}^{t}(\Omega)$, is defined (see [2, Chapter 1, Section 2] or [3, Chapter 2, Section 5]) by all measurable functions $f$ such that

[TABLE]

for some positive constant $k=k(f)$ and every $\tau>0$, where $|E|$ is the $n$-dimensional Lebesgue measure of $E\subset\mbox{R}^{n}$. We recall that if $f\in L_{weak}^{t}(\Omega)$ for some $t>1$, then $f\in L^{s}(\Omega)$ for every $1\leq s<t$.

The anisotropic Sobolev spaces $W^{1,(p_{k})}(\Omega)$ and $W_{0}^{1,(p_{k})}(\Omega)$ are defined, respectively, by

[TABLE]

and

[TABLE]

where $D_{k}v=\frac{\partial v}{\partial x_{k}}$, $k=1,2,\cdots,n$.

Let us consider the following divergence elliptic equation

[TABLE]

and suppose that the Carathéodory functions $a_{i}(x,\xi):\Omega\times\mbox{R}^{n}\rightarrow\mbox{R}$, $i=1,2,\cdots,n$, satisfy

[TABLE]

for almost every $x\in\Omega$, every $z=(z_{1},z_{2},\cdots,z_{n})\in\mbox{R}^{n}$ and any $i=1,2,\cdots,n$. Furthermore, there exists $\tilde{\nu}\in(0,+\infty)$ such that

[TABLE]

for almost every $x\in\Omega$ and any $z,\tilde{z}\in\mbox{R}^{n}$. The integrability conditions for $f^{i}(x)$, $i=1,2,\cdots,n$, in (1.1) and $h(x)\geq 0$ in (1.2) will be given later.

Let $\psi$ be any function in $\Omega$ with values in $\mbox{R}\cup\{\pm\infty\}$ and $\theta\in W^{1,(p_{k})}(\Omega)$. We introduce

[TABLE]

The function $\psi$ is an obstacle and $\theta$ determines the boundary values.

Definition 1.1**.**

A function $u\in\theta+W_{0}^{1,(p_{k})}(\Omega)$ is called a solution to the boundary value problem

[TABLE]

if

[TABLE]

holds true for any $\varphi\in W_{0}^{1,(p_{k})}(\Omega)$.

Definition 1.2**.**

A solution to the ${\cal K}_{\psi,\theta}^{(p_{k})}$-obstacle problem is a function $u\in{\cal K}_{\psi,\theta}^{(p_{k})}(\Omega)$ such that

[TABLE]

whenever $v\in{\cal K}_{\psi,\theta}^{(p_{k})}(\Omega)$.

In a recent paper [1], F.Leonetti and F.Siepe considered solutions $u\in\theta+W_{0}^{1,(p_{k})}(\Omega)$ to the boundary value problem

[TABLE]

under the conditions

[TABLE]

and (1.3), and obtained an integrability result, which shows that, higher integrability of the boundary datum $\theta$ forces solutions $u$ to have higher integrability as well.

Note that the assumptions (1.2)′ and (1.3) are suggested by the Euler equation of the anisotropic functional

[TABLE]

Later, Gao, Zhang and Li [4] considered ${\cal K}_{\psi,\theta}^{(p_{i})}$-obstacle problems for the homogeneous elliptic equations

[TABLE]

under the conditions (1.2)′ and (1.3). A similar result was obtained, which shows that, higher integrability of the datum $\theta_{*}=\max\{\theta,\psi\}$ forces solutions $u$ to have higher integrability as well.

Integrability property is important among the regularity theories of nonlinear elliptic PDEs and systems, see [5-12]. In the present paper, we consider ${\cal K}_{\psi,\theta}^{(p_{i})}$-obstacle problems of the nonhomogeneous anisotropic elliptic equations

[TABLE]

under the conditions (1.2) and (1.3) with suitable functions $h$ and $f^{i}$, $i=1,2,\cdots,n$. The main result of this paper is the following theorem.

Theorem 1.1**.**

Let $\theta\in W^{1,(q_{k})}(\Omega)$, $q_{k}\in(p_{k},+\infty)$, $k=1,2,\cdots,n$, $0\leq h\in L^{\tau}(\Omega)$ with $\tau=\max\limits_{k=1,\cdots,n}\frac{q_{k}}{p_{k}}$, $\psi\in[-\infty,+\infty]$ be such that $\theta_{*}=\max\{\psi,\theta\}\in\theta+W_{0}^{1,(q_{k})}(\Omega)$. Moreover $\overline{p}<n$. Then for any solution $u\in{\cal K}_{\psi,\theta}^{(p_{k})}(\Omega)$ to the ${\cal K}_{\psi,\theta}^{(p_{k})}$-obstacle problem, we have

[TABLE]

provided that $f^{i}\in L^{p_{i}/(p_{i}-1-bp_{i})}(\Omega)$, $i=1,\cdots,n$, where

[TABLE]

$b$* is any number verifying*

[TABLE]

and

[TABLE]

The idea of the proof of Theorem 1.1 comes from [1]. Theorem 1.1 can be regarded as a generalization of [1, Theorem 2.1]. The difficulty in the proof of Theorem 1.1 is that, under the condition (1.2), we need to derive that the constant $M$ in [1] is finite. To this aim, we need to restrict the constant $b$ in Theorem 1.1 to satisfy (1.10) instead of [1,(2.9)].

For solutions to boundary value problems (1.4), we have

Theorem 1.2**.**

Let $\theta\in W^{1,(q_{k})}(\Omega)$, $q_{k}\in(p_{k},+\infty)$, $k=1,2,\cdots,n$, $0\leq h\in L^{\tau}(\Omega)$ with $\tau$ be as in Theorem 1.1. Moreover $\overline{p}<n$. Then for any solution $u\in\theta+W_{0}^{1,(p_{k})}(\Omega)$ to the boundary value problem (1.4), we have

[TABLE]

provided that $f^{i}\in L^{p_{i}/(p_{i}-1-bp_{i})}(\Omega)$, $i=1,\cdots,n$, where $t$ verifies (1.9)-(1.11).

Proof.

Take the obstacle function $\psi$ to be minus infinity in Theorem 1.1 we arrive at the desired result. ∎

When we are in the isotropic case, that is, $q_{i}=q>p=p_{i}$ for every $i=1,2,\cdots,n$, we denote $W^{1,(p_{i})}(\Omega)=W^{1,p}(\Omega)$, $W_{0}^{1,(p_{i})}(\Omega)=W_{0}^{1,p}(\Omega)$ and ${\cal K}_{\psi,\theta}^{(p_{i})}(\Omega)={\cal K}_{\psi,\theta}^{p}(\Omega)$.

When $n>q_{i}=q>p=p_{i}$ for every $i=1,\cdots,n$, then

[TABLE]

then we take

[TABLE]

and we get

[TABLE]

Thus we have the following two corollaries.

Corollary 1.1**.**

Let $\theta\in W^{1,q}(\Omega)$, $q\in(p,n)$, $0\leq h\in L^{q/p}(\Omega)$, $\psi\in[-\infty,+\infty]$ be such that $\theta_{*}=\max\{\psi,\theta\}\in\theta+W_{0}^{1,q}(\Omega)$. Then for any solution $u\in{\cal K}_{\psi,\theta}^{p}(\Omega)$ to the ${\cal K}_{\psi,\theta}^{p}$-obstacle problem, we have

[TABLE]

provided that $f^{i}\in L^{q/(p-1)}(\Omega)$, $i=1,\cdots,n$, where

[TABLE]

Corollary 1.2**.**

Let $\theta\in W^{1,q}(\Omega)$, $q\in(p,n)$, $0\leq h\in L^{q/p}(\Omega)$. Then for any solution $u\in\theta+W_{0}^{1,p}(\Omega)$ to the boundary value problem (1.4), we have

[TABLE]

provided that $f^{i}\in L^{q/(p-1)}(\Omega)$, $i=1,\cdots,n$, where

[TABLE]

§2   Proof of Theorem 1.1

Proof.

For $L>0$ and a function $w$, let $T_{L}(w)$ be the truncation of $w$ at level $L$, that is,

[TABLE]

Let $u\in{\cal K}_{\psi,\theta}^{(p_{k})}(\Omega)$ be a solution to the ${\cal K}_{\psi,\theta}^{(p_{k})}$-obstacle problem. If we take

[TABLE]

then $v\in{\cal K}_{\psi,\theta}^{(p_{k})}(\Omega)$. Indeed, it is obvious that $v\in W^{1,(p_{k})}(\Omega)$; for the second and the third cases of the above definition for $v$, we obviously have $v\geq\psi$, and for the first case, $u-\theta_{*}<-L$, we have $\theta_{*}>u+L\geq\psi+L$, this implies $v=\theta_{*}-L\geq\psi$; and since $u\in\theta+W_{0}^{1,(p_{k})}(\Omega)$ and $u\geq\psi$, then $\theta_{*}=\max\{\psi,\theta\}=\theta=u$ on $\partial\Omega$, thus $v=u=\theta$ on $\partial\Omega$, this implies $v\in\theta+W_{0}^{1,(p_{k})}(\Omega)$.

Definition 1.2 together with the definition of $v$ yields

[TABLE]

Monotonicity (1.3) allows us to write

[TABLE]

which together with (2.1) implies

[TABLE]

We now use anisotropic growth (1.2) and the Hölder inequality in (2.2), obtaining that

[TABLE]

[TABLE]

Let $t_{i}$ be such that

[TABLE]

for every $i=1,\cdots,n$; $t_{i}$ will be chosen later. We use Hölder inequality as follows

[TABLE]

We would like to choose $t_{i}$ such that the exponent

[TABLE]

does not depend on $i$, and simultaneously

[TABLE]

is finite. To the first aim, we solve (2.5) with respect to $t_{i}$, obtaining that

[TABLE]

Since we need $t_{i}>0$, we require that $p_{i}-1-bp_{i}>0$, that is $b<1-\frac{1}{p_{i}}<1$; moreover, the limitation $p_{i}<t_{i}$ is equivalent to $b>0$. Finally, since we required $t_{i}\leq q_{i}$, we need

[TABLE]

To the second aim, since we need

[TABLE]

we then require

[TABLE]

Note that (2.8) is equivalent to

[TABLE]

We now show (2.8) occurs if

[TABLE]

holds true. In fact, from (2.9), for every $i=1,\cdots,n$, one has

[TABLE]

This implies

[TABLE]

which is equivalent to (2.8).

Thus, for every $b$ such that (1.10) holds true, we can define $t_{i}$ as in (2.7) obtaining that $p_{i}<t_{i}\leq q_{i}$, $b$ in (2.5) is independent of $i$, and $M_{1}$ in (2.6) is finite.

Under the above assumptions on exponents, (2.4) becomes

[TABLE]

where $M_{1}$ be as in (2.6).

We use Hölder inequality again, obtaining that

[TABLE]

where

[TABLE]

From the assumption $f^{i}\in L^{p_{i}/(p_{i}-1-bp_{i})}(\Omega)$, $i=1,2,\cdots,n$ and (2.5), one has

[TABLE]

for every $i=1,2,\cdots,n$. This implies $M_{2}<\infty$.

If we insert (2.10) and (2.11) into (2.3), we easily get

[TABLE]

which can be considered as an inequality of the following type

[TABLE]

with

[TABLE]

and

[TABLE]

Following the idea of [1], one can derive that

[TABLE]

with $t$ verifies (1.9)-(1.11). This completes the proof of Theorem 1.1. ∎