Calculus of Variations and Geometric Measure Theory

B. Velichkov

Existence and regularity results for some shape optimization problems

created by velichkov on 26 Dec 2013
modified on 11 May 2015


Phd Thesis

Inserted: 26 dec 2013
Last Updated: 11 may 2015

Year: 2013


This Ph.D. thesis was prepared under the joint supervision of Giuseppe Buttazzo and Dorin Bucur and was discussed on 8 November 2013 at Scuola Normale Superiore.

The main topics are the existence and the regularity of optimal domains for spectral functionals of the form


under a perimeter or volume constraint and a possible presence of a geometric obstacle $D$, containing or contained in the competing domains.

For optimal sets in the entire Euclidean space we prove the Lipschitz regularity of the eigenfunctions, in the presence of measure constraint, and the $C^{1,\alpha}$ regularity of the free boundary, in the case of perimeter constraint.

We also consider spectral optimization problems in non-Euclidean settings, optimization problems for potentials and measures, as well as multiphase and optimal partition problems.