PDE’s involving variable exponents

Mihai Mihăilescu
University of Craiova, Romania
3rd September 2012 2-3pm, AGR Carslaw 829

Abstract

We discuss some aspects regarding the eigenvalue problem $-{\mathrm{\Delta }}_{p(x)}u=\lambda |u{|}^{p(x)-2}u$ if $x\in \mathrm{\Omega }$, $u=0$ if $x\in \partial \mathrm{\Omega }$, where $\mathrm{\Omega }\subset {\mathbf{R}}^N$ is a bounded domain, $p:\overline{\mathrm{\Omega }}\to (1,\mathrm{\infty })$ is a continuous function and ${\mathrm{\Delta }}_{p(x)}u:=\mathrm{\nabla }\cdot (|\mathrm{\nabla }u{|}^{p(x)-2}\mathrm{\nabla }u)$ stands for the $p(x)$-Laplace operator. Let $\mathrm{\Lambda }$ be the set of eigenvalues of the above problem and ${\lambda }_{\ast }=inf\mathrm{\Lambda }$. In particular, we will emphasize, on the one hand, situations when ${\lambda }_{\ast }$ vanishes, and, on the other hand, we will advance some sufficient conditions when ${\lambda }_{\ast }$ is positive. In the case when $p\in C^1(\mathrm{\Omega })$ some extensions will be presented. In a related context some connections with a maximum principle will be pointed out.

Check also the PDE Seminar page. Enquiries to Florica C�rstea or Daniel Daners.