LITERATURE AND WORKING SEMINAR

ABSTRACT OF THE TALK
Given a bounded domain in $R^n$, and a positive integer $N>1$, divide this domain into N disjoint subdomains, $D_1,...,D_N$. Denote by $\lambda_1(D_j)$ the first Dirichlet eigenvalue of $D_j$ and look for such divisions that the $l^p$ norm of $ (\lambda_1(D_1),..., \lambda_1(D_N))$ is minimized, for a given $p$ such that $1<=p<=\infty$.
These problems were studied by many authors. I shall discuss the two most interesting cases: $p=1$ and $p=\infty$. The former was studied systematically by Caffarelli and myself and the later was studied by P.Beredo and B.Helffer among many others. I shall however, discuss the case when N becomes very large. It is related to many fundamental questions in both physics and mathematics.