Workshops## On a structure of stationary solutions to a nonlinear nonlocal elliptic PDE arising in the Ohmic heating problem

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Kazuhiro Kurata

2011-09-01

11:30:00 - 12:20:00

101 , Mathematics Research Center Building (ori. New Math. Bldg.)

In this talk, I present recent results on the structure of stationary solutions to a nonlocal nonlinear ellptic boundary value problem: \[ -\Delta u=\delta \frac{f(u)}{(\int_{\Omega}f(u)\, dx)^2} in \Omega, B u=0, x\in\partial\Omega, \] where $\Omega$ is a bounded smooth domain in ${\bf R}^n$, $\delta>0$, and $f(t)$ is a positive, decreasing $C^1$ function on $[0, +\infty)$ with $\int_0^{+\infty}f(t)\, dt <+\infty$. Here, $B$ represents the boundary condition, e.g. Dirichlet boundary condition $B u=u$ or Robin boundary condition $B u=\partial u/\partial \nu +\alpha u, \alpha>0$, where $\nu$ is an outward unit normal vector on $\partial \Omega$. This problem arises in the Ohming heating problem and has been studied by many authors, e.g. Lacey 1995. We consider this problem under various boundary conditions, especially two cases: the Robin boundary condition case and the mixed boundary condition case. We report different structure of the set of stationary solutions between two cases.This is a joint work with my student Tatsuyuki Kakiuchi.

For material related to this talk, click here.