SMS scnews item created by Daniel Daners at Wed 29 Aug 2012 1053
Type: Seminar
Distribution: World
Expiry: 3 Sep 2012
Calendar1: 3 Sep 2012 1400-1500
CalLoc1: AGR Carslaw 829
Auth: daners@bari.maths.usyd.edu.au

# PDE’s involving variable exponents

### Mihailescu

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

## Abstract

We discuss some aspects regarding the eigenvalue problem $$-\Delta_{p(x)}u=\lambda|u|^{p(x)-2}u$$ if $$x\in\Omega$$, $$u=0$$ if $$x\in\partial\Omega$$, where $$\Omega\subset\mathbf R^N$$ is a bounded domain, $$p\colon\bar\Omega\rightarrow(1,\infty)$$ is a continuous function and $$\Delta_{p(x)}u:=\nabla\cdot\bigl(|\nabla u|^{p(x)-2}\nabla u\bigr)$$ stands for the $$p(x)$$-Laplace operator. Let $$\Lambda$$ be the set of eigenvalues of the above problem and $$\lambda_*=\inf\Lambda$$. In particular, we will emphasize, on the one hand, situations when $$\lambda_*$$ vanishes, and, on the other hand, we will advance some sufficient conditions when $$\lambda_*$$ is positive. In the case when $$p\in C^1(\Omega)$$ 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.

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