**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

### PDE Seminar

# 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.