arXiv subjects

Dario Mazzoleni

Publications and source records attributed to Dario Mazzoleni.

25 records · Page 2Linked to original sources

arXiv2018

Regularity of the free boundary for the vectorial Bernoulli problem

In this paper we study the regularity of the free boundary for a vector-valued Bernoulli problem, with no sign assumptions on the boundary data. More precisely, given an open, smooth set of finite measure $D\subset \mathbb{R}^d$, $Λ>0$ and $φ_i\in H^{1/2}(\partial D)$, we deal with \[ \min{\left\{\sum_{i=1}^k\int_D|\nabla v_i|^2+Λ\Big|\bigcup_{i=1}^k\{v_i\not=0\}\Big|\;:\;v_i=φ_i\;\mbox{on }\partial D\right\}}. \] We prove that, for any optimal vector $U=(u_1,\dots, u_k)$, the free boundary $\partial (\cup_{i=1}^k\{u_i\not=0\})\cap D$ is made of a regular part, which is relatively open and locally the graph of a $C^\infty$ function, a singular part, which is relatively closed and has Hausdorff dimension at most $d-d^*$, for a $d^*\in\{5,6,7\}$ and by a set of branching (two-phase) points, which is relatively closed and of finite $\mathcal{H}^{d-1}$ measure. Our arguments are based on the NTA structure of the regular part of the free boundary.

math.AP↗

arXiv2017

Some estimates for the higher eigenvalues of sets close to the ball

In this paper we investigate the behavior of the eigenvalues of the Dirichlet Laplacian on sets in $\mathbb{R}^N$ whose first eigenvalue is close to the one of the ball with the same volume. In particular in our main Theorem we prove that, for all $k\in\mathbb{N}$, there is a positive constant $C=C(k,N)$ such that for every open set $Ω\subseteq \mathbb{R}^N$ with unit measure and with $λ_1(Ω)$ not excessively large one has \begin{align*} |λ_k(Ω)-λ_k(B)|\leq C (λ_1(Ω)-λ_1(B))^β\,, && λ_k(B)-λ_k(Ω)\leq Cd(Ω)^{β'}\,, \end{align*} where $d(Ω)$ is the Fraenkel asymmetry of $Ω$, and where $β$ and $β'$ are explicit exponents, not depending on $k$ nor on $N$; for the special case $N=2$, a better estimate holds.

math.OC↗

arXiv2017

Regularity of the optimal sets for some spectral functionals

In this paper we study the regularity of the optimal sets for the shape optimization problem \[ \min\Big\{λ_1(Ω)+\dots+λ_k(Ω)\ :\ Ω\subset\mathbb{R}^d,\ \text{open}\ ,\ |Ω|=1\Big\}, \] where $λ_1(\cdot),\dots,λ_k(\cdot)$ denote the eigenvalues of the Dirichlet Laplacian and $|\cdot|$ the $d$-dimensional Lebesgue measure. We prove that the topological boundary of a minimizer $Ω_k^*$ is composed of a relatively open regular part which is locally a graph of a $C^{1,α}$ function and a closed singular part, which is empty if $d d^*$, where the natural number $d^*\in[5,7]$ is the smallest dimension at which minimizing one-phase free boundaries admit singularities. To achieve our goal, as an auxiliary result, we shall extend for the first time the known regularity theory for the one-phase free boundary problem to the vector-valued case.

math.AP↗

arXiv2016

Convex combinations of low eigenvalues, Fraenkel asymmetries and attainable sets

We consider the problem of minimizing convex combinations of the first two eigenvalues of the Dirichlet-Laplacian among open sets of $R^N$ of fixed measure. We show that, by purely elementary arguments, based on the minimality condition, it is possible to obtain informations on the geometry of the minimizers of convex combinations: we study, in particular, when these minimizers are no longer convex, and the optimality of balls. As an application of our results we study the boundary of the attainable set for the Dirichlet spectrum. Our techniques involve symmetry results à la Serrin, explicit constants in quantitative inequalities, as well as a purely geometrical problem: the minimization of the Fraenkel 2-asymmetry among convex sets of fixed measure.

math.AP↗

arXiv2014

A surgery result for the spectrum of the Dirichlet Laplacian

In this paper we give a method to geometrically modify an open set such that the first $k$ eigenvalues of the Dirichlet Laplacian and its perimeter are not increasing, its measure remains constant, and both perimeter and diameter decrease below a certain threshold. The key point of the analysis relies on the properties of the shape subsolutions for the torsion energy.

math.OC↗

arXiv2013

Lipschitz regularity of the eigenfunctions on optimal domains

We study the optimal sets $Ω^\ast\subset\mathbb{R}^d$ for spectral functionals $F\big(λ_1(Ω),\dots,λ_p(Ω)\big)$, which are bi-Lipschitz with respect to each of the eigenvalues $λ_1(Ω),\dots,λ_p(Ω)$ of the Dirichlet Laplacian on $Ω$, a prototype being the problem $$ \min{\big\{λ_1(Ω)+\dots+ λ_p(Ω)\;:\;Ω\subset\mathbb{R}^d,\ |Ω|=1\big\}}. $$ We prove the Lipschitz regularity of the eigenfunctions $u_1,\dots,u_p$ of the Dirichlet Laplacian on the optimal set $Ω^*$ and, as a corollary, we deduce that $Ω^*$ is open. For functionals depending only on a generic subset of the spectrum, as for example $λ_k(Ω)$ or $λ_{k_1}(Ω)+\dots+λ_{k_p}(Ω)$, our result proves only the existence of a Lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved.

math.AP↗

arXiv2011

Existence of minimizers for spectral problems

In this paper we show that any increasing functional of the first k eigenvalues of the Dirichlet Laplacian admits a (quasi-)open minimizer among the subsets of R^N of unit measure. In particular, there exists such a minimizer which is bounded, where the bound depends on k and N, but not on the functional. In the meantime, we show that the ratio λ_k(Ω)/λ_1(Ω) is uniformly bounded for sets Ω\in R^N.

math.FA↗