Two Generalizations of Stampacchia Lemma and Applications

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 $${\cal J} (u;\Omega) = \int_{\Omega}\ f(x,Du{(x)})dx.$$ 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 $$\begin{cases} -\mbox {div} \left( a(x, u(x)) D u(x) \right) = f(x), & x \in \Omega, u(x) = 0, & x \in {\partial \Omega}, \end{cases}$$ 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.