Michal Kowalczyk, Universidad de Chile"Critical points of the regular part of the harmonic Green's function with Robin boundary condition" AbstractIn this talk I will consider the Green's function for the Laplacian in a smooth bounded domain $\Omega \subset R^N$ with Robin boundary condition $$ \pd{G_{\lambda}{\nu} + \lambda b(x) G_{\lambda} = 0, \quad \mbox{on} \ \partial \Omega, $$ and its regular part $S_{\lambdd}(x,y)$, where $b>0$ is smooth. I will show that in general, as $\lambda \to \infty$, the Robin function $R_{\lambda}(x) = S_{\lambda}(x,x)$ has at least 3 critical points. Moreover, in the case $b\equiv const$, $R_{\lambda}$ has critical points near non-degenerate critical points of the mean curvature of the boundary, and when $b \not\equiv const$ there are critical points of $b$. I will discuss applications of these results in the context of concentration phenomena for elliptic singular perturbation problems. This is a joint work with J. D. Davila and M. Montenegro.
TUESDAY, SEPTEMBER 11, 2007 |