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Two Generalizations of Stampacchia Lemma and Applications

Han Yingxiao, Fang Mi, Xia Liuye, Gao Hongya

Abstract

We present two generalizations of the classical Stampacchia Lemma which contain a non-decreasing non-negative function $g$, and give applications. As a first application, we deal with variational integrals of the form $$ {\cal J} (u;Ω) = \int_Ω\ f(x,Du{(x)})dx. $$ We consider a minimizer $u: Ω\subset \mathbb R^n \to \mathbb R $ among all functions with a fixed boundary value $u_{\ast }$ on $\partial Ω$. Under some nonstandard growth conditions of the integrand $f(x,ξ)$ we derive some regularity results; as a second application, we consider elliptic equations of the form $$ \begin{cases} -\mbox {div} \left( a(x, u(x)) D u(x) \right) = f(x), & x \in Ω, u(x) = 0, & x \in {\partial Ω}, \end{cases} $$ under the conditions $$ \frac {α}{(1+|s|) ^θ\ln ^θ(e+|s|)} \le a (x,s) \le β, \ \ \ 0<α\le β<\infty, \ θ\ge 0, $$ we obtain some regularity properties of its weak solutions.

Two Generalizations of Stampacchia Lemma and Applications

Abstract

We present two generalizations of the classical Stampacchia Lemma which contain a non-decreasing non-negative function , and give applications. As a first application, we deal with variational integrals of the form We consider a minimizer among all functions with a fixed boundary value on . Under some nonstandard growth conditions of the integrand we derive some regularity results; as a second application, we consider elliptic equations of the form under the conditions we obtain some regularity properties of its weak solutions.
Paper Structure (3 sections, 17 theorems, 218 equations)

This paper contains 3 sections, 17 theorems, 218 equations.

Table of Contents

  1. Introduction and preliminaries.
  2. The first generalization and applications.
  3. The second generalization and applications.

Key Result

Lemma 1.1

Let $c, \alpha, \beta$ be positive constants and $k_0\in \mathbb{R}$. Let $\varphi: [k_0,+\infty)$$\rightarrow [0, +\infty)$ be non-increasing and such that for every $h,k$ with $h>k\ge k_0$. It results that: (i) if $\beta > 1$ then where (ii) if $\beta = 1$ then for any $k\ge k_0$, (iii) if $0< \beta < 1$ and $k_0 > 0$ then for any $k\ge k_0$,

Theorems & Definitions (28)

  • Lemma 1.1
  • Lemma 1.2
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • Lemma 1.5
  • Lemma 1.6
  • Theorem 2.1
  • Remark 2.1
  • ...and 18 more