Bozhidar Velichkov

Faber–Krahn inequalities in sharp quantitative form

The classical Faber-Krahn inequality asserts that balls (uniquely) minimize the first eigenvalue of the Dirichlet-Laplacian among sets with given volume. In this paper we prove a sharp quantitative enhancement of this result, thus confirming a conjecture by Nadirashvili and Bhattacharya-Weitsman. More generally, the result applies to every optimal Poincare-Sobolev constant for the embeddings

Multiphase shape optimization problems

W^{1,2}_0(\Omega)\hookrightarrow L^q(\Omega)

Lipschitz Regularity of the Eigenfunctions on Optimal Domains

.

Spectral Optimization Problems for Potentials and Measures

This paper is devoted to the analysis of multiphase shape optimization problems, which can formally be written as

Existence and regularity results for some shape optimization problems

\min \{{g}((F_1(\Omega_1),\dots ,F_h(\Omega_h))+ m|\bigcup_{i=1}^h\Omega_i| :\ \Omega_i \subset D,\ \Omega_i\cap \Omega_j =\emptyset\},

Shape optimization problems for metric graphs

where

Regularity of the optimal sets for some spectral functionals

D\subseteq \mathbb{R}^d

A Shape Optimal Control Problem with Changing Sign Data

is a given bounded open set,

Optimal Potentials for Problems with Changing Sign Data

|\Omega_i|

Existence and Regularity of Minimizers for Some Spectral Functionals with Perimeter Constraint

is the Lebesgue measure of