Calculus of Variations and Geometric Measure Theory

D. Bucur - G. Buttazzo - B. Velichkov

Shape Optimization Problems with Internal Constraint

created by buttazzo on 12 Sep 2011
modified by velichkov on 23 Apr 2014


Published Paper

Inserted: 12 sep 2011
Last Updated: 23 apr 2014

Journal: Ann. I. H. Poincare' Anal. Non Lineaire
Volume: 30
Number: 3
Pages: 467--495
Year: 2013
Links: The final publication is available at


We consider shape optimization problems with internal inclusion constraints, of the form

$\min\big\{J(\Omega)\ :\ D\subset\Omega\subset\mathbb{R}^d,\

where the set $D$ is fixed, possibly unbounded, and $J$ depends on $\Omega$ via the spectrum of the Dirichlet Laplacian. We analyze the existence of a solution and its qualitative properties, and rise some open questions.

Keywords: Concentration-compactness, shape optimization, capacity, eigenvalues