Carnegie Mellon
Department of Mathematical 

Arie Leizarowitz, Technion, Israel

"On second order two-dimensional variational problems on unbounded domains"


We consider the energy functional

\begin{displaymath}J_{\Omega }=\int _{\Omega }[(\Delta u)^2-b\vert\nabla u\vert^2+\psi (u)]dxdy\end{displaymath}

where $\Omega $ is a bounded domain in $R^2$ and $\psi $ is a $C^2(R^2)$ super quadratic potential. Associated with this functional is the notion of mean energy, which is meaningful for a class of configurations $u$ on the whole plane, and we study the corresponding minimization problem. Minimizers of this problem are called equilibrium configurations. We establish apriori bounds for this unbounded-domain problem, and use it to establish existence of equilibrium configurations. We also discuss the existence of radially symmetric equilibrium configurations. Moreover, we remark on the connection between radially symmetric configurations and 1-dimensional configurations.

TUESDAY, October 10, 2000
Time: 1:30 P.M.
Location: PPB 300