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 $g$, and give applications. As a first application, we deal with variational integrals of the form $$\mathcal{J}(u;\Omega )={\int }_{\Omega }f(x,Du(x))dx.$$ We consider a minimizer $u:\Omega \subset {\mathbb{R}}^n\to \mathbb{R}$ among all functions with a fixed boundary value $u_{\ast }$ on $\partial \Omega$. Under some nonstandard growth conditions of the integrand $f(x,\xi )$ 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 $$\frac{\alpha }{(1+|s|{)}^{\theta }{\ln }^{\theta }(e+|s|)}\le a(x,s)\le \beta ,0<\alpha \le \beta <\infty ,\theta \ge 0,$$ we obtain some regularity properties of its weak solutions.

arXiv comment26 pages