Calculus of Variations and Geometric Measure Theory

D. Mazzoleni

Existence and regularity results for solutions of spectral problems

created by mazzoleni on 07 Nov 2014


Ph.D. Thesis

Inserted: 7 nov 2014
Last Updated: 7 nov 2014

Year: 2014


This Thesis is devoted to the study of some shape optimization problems for eigenvalues of the Dirichlet Laplacian. More precisely we consider the minimum problem \[ \min{\left\{F(\lambda_1(\Omega),\dots,\lambda_k(\Omega))\;:\;\Omega\subset\mathbb{R}^N,\mbox{ quasi-open, }
=1\right\}}, \] with $F\colon\mathbb{R}^k\rightarrow \mathbb{R}$ increasing in each variable and lower semicontinuous.

\medskip The first result of the Thesis is a proof of the existence of an optimal set for such a problem, thus extending a well-known result due to Buttazzo and Dal Maso to the ``unbounded'' setting. Moreover, under a slightly stronger assumption on $F$, it is possible to prove that all the minimizers have a diameter uniformly bounded by a constant depending only on $k,N$ (but \emph{not} on the functional). The main interest of this result is the very ``elementary'' techniques that are used. In fact the key point consists in showing that it is always possible to choose a minimizing sequence made of sets with uniformly bounded diameter, since getting rid of ``long tails'' decreases the first $k$ eigenvalues.

\medskip Then we focus on the study of the regularity of optimal sets, in particular a natural conjecture is that they should be open sets, at least. This kind of issue reveals to be quite hard to solve. With a ``direct'' approach we can prove, in the two dimensional setting, that minimizers for functionals like $\lambda_1(\cdot)+\dots+\lambda_k(\cdot)$ are open sets. Moreover we perform a finer analysis of the eigenfunctions of optimal sets (in generic dimension), employing techniques from the regularity of free boundary problems. In particular we prove that an optimal set $\Omega$ for the functional $\lambda_k(\cdot)$ has an eigenfunction, corresponding to the eigenvalue $\lambda_k(\Omega)$, which is Lipschitz continuous in $\mathbb{R}^N$.

\medskip At last we study the connectedness of optimal sets for convex combinations of the first three eigenvalues, and in particular we are able to prove that every minimizer for the problem \[ \min{\left\{\alpha\lambda_1(\Omega)+(1-\alpha)\lambda_2(\Omega)\;:\;\Omega\subset\mathbb{R}^N,\mbox{ (quasi-)open, }
=1\right\}}, \] is connected for all $\alpha\in (0,1]$.