SMS scnews item created by Daniel Daners at Wed 18 May 2011 1026
Type: Seminar
Distribution: World
Expiry: 26 May 2011
Calendar1: 26 May 2011 1400-1500
CalLoc1: AGR Carslaw 829
Auth: daners@bari.maths.usyd.edu.au

# Nonnegative solutions of elliptic equations on symmetric domains and their nodal structure

### Polacik

Peter Poláčik
University of Minnesota, USA
Thursday 26 May 2010, 2-3pm, Access Grid Room (note unusual time and location)

## Abstract

We consider the Dirichlet problem for a class of fully nonlinear elliptic equations on a bounded domain $$\Omega$$. We assume that $$\Omega$$ is symmetric about a hyperplane $$H$$ and convex in the direction perpendicular to $$H$$. By a well-known result of Gidas, Ni and Nirenberg and its generalizations, all positive solutions are reflectionally symmetric about $$H$$ and decreasing away from the hyperplane in the direction orthogonal $$H$$. For nonnegative solutions, this result is not always true. We show that, nonetheless, the symmetry part of the result remains valid for nonnegative solutions: any nonnegative solution $$u$$ is symmetric about $$H$$. Moreover, we prove that if $$u\not\equiv 0$$, then the nodal set of $$u$$ divides the domain $$\Omega$$ into a finite number of reflectionally symmetric subdomains in which $$u$$ has the usual Gidas-Ni-Nirenberg symmetry and monotonicity properties. Examples of nonnegative solutions with nontrivial nodal structure will also be given.

Check also the PDE Seminar page. Enquiries to Florica Cîrstea or Daniel Daners.

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