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

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.