Calculus of Variations and Geometric Measure Theory

G. Buttazzo - B. Velichkov

Shape optimization problems on metric measure spaces

created by velichkov on 02 Oct 2012
modified on 21 Apr 2018


Published Paper

Inserted: 2 oct 2012
Last Updated: 21 apr 2018

Journal: J. Funct. Anal.
Volume: 264
Number: 1
Pages: 1--33
Year: 2013
Doi: 10.1016/j.jfa.2012.09.017


We consider shape optimization problems of the form

$\min\left\{J(\Omega):\ \Omega\subset X,\ m(\Omega)\le c\right\},$

where $X$ is a metric measure space and $J$ is a suitable shape functional. We adapt the notions of $\gamma$-convergence and weak $\gamma$-convergence to this new general abstract setting to prove the existence of an optimal domain. Several examples are pointed out and discussed.

Keywords: shape optimization, capacity, eigenvalues, Sobolev spaces, metric spaces