# Shape optimization problems on metric measure spaces

## Bozhidar Velichkov

### (Scuola Normale Superiore, Pisa)

Date: Oct 11, 2011, time: 17:50

Place: Centro De Giorgi, Scuola Normale Superiore

Abstract. We consider shape optimization problems of the form $\min\{\lambda_1(\Omega):\Omega \subset X, \vert O \vert = m\} \ \ \ \ \ \ (1)$ where $X$ is a metric measure space of nite measure and $\lambda_1$ is a generalization, through a variational formulation, of the first eigenvalue of the Dirichlet laplacian.

We work in a purely abstract setting, de ning the Sobolev space over a general
measure space $(X, \mu)$, as a linear subspace of $L^2(\mu)$ which obeys certain properties. A particular nonlinear operator $D: H \to L^2(\mu)$ has the role of the modulus of the weak gradient in H and the rst eigenvalue of the Dirichlet laplacian is defined as $\lambda_1(O) = \inf \{\int {\vert Du\vert}^2d\mu\ \; u \in H, \vert\vert u\vert\vert_{L^2}=1, \;u=0 \ \;{\textrm{ a.e. on }} \Omega^c\}.$
We adapt the classical $\gamma$-convergence techniques to this general abstract setting to prove an existence result for the problem $(1)$.|
We apply the existence result to the case of a metric measure space $(X,d,\mu)$. We apply the existence result to the case of a metric measure space $H^{1,2}(X)$, as de ned by Cheeger, satis es the properties of $H$, under some classical hypothesis on $X$ (doubling, supporting weak Poincare inequality) and the assumption that the inclusion $H^{1,2}\hookrightarrow L^2$ is compact. The particular case of a
Carnot-Carathéodory space is also discussed.