Shape optimization problems on metric measure spaces

Bozhidar Velichkov

(Scuola Normale Superiore, Pisa)

Event: ERC Workshop on Geometric Analysis on sub-Riemannian and Metric Spaces

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.