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Stampa

SEMINARI DI ANALISI MATEMATICA

Evento 

Titolo:
Lipschitz continuity of the eigenfunctions on optimal sets
Quando:
29.10.2014 13.30 h
Dove:
Palazzo Campana - Torino
Aula:
Aula C
Relatore:
Dott. Bozhidar Velichkov
Afferenza:
Université de Grenoble
Proponente:
S. Terracini
Locandina:
Locandina

Descrizione

This talk is based on a recent joint work with Dorin Bucur, Dario Mazzoleni and Aldo Pratelli, in which we study the regularity up to the boundary of the eigenfunctions on the optimal sets \(\Omega^\ast\subset\mathbb{R}^d\), solutions of the shape optimization problem 

\[\min{\big\{F\big(\lambda_1(\Omega),\dots,\lambda_k(\Omega)\big)\;:\;\Omega\subset\mathbb{R}^d,\ |\Omega|=1\big\}},\] 

where \(F:\mathbb{R}^k\to\mathbb{R}\) is a Lipschitz continuous function strictly increasing and bi-Lipschitz in each variable and \(\lambda_1(\Omega),\dots,\lambda_k(\Omega)\) are the first \(k\) eigenvalues of the Dirichlet Laplacian on $\Omega$. The main result is the global (on $\R^d$) Lipschitz regularity of the eigenfunctions \(u_1,\dots,u_k\in H^1_0(\Omega^\ast)\) of the Dirichlet Laplacian on the optimal set $\Omega^*$.\\

We will concentrate our attention on the special case \[F\big(\lambda_1(\Omega),\dots,\lambda_k(\Omega)\big)=\lambda_k(\Omega).\]
Denoting with \(u_k\) the \(k\)th eigenfunction on the optimal domain \(\Omega_k\)
\[-\Delta u_k=\lambda_k(\Omega_k)u_k\quad\hbox{in}\quad\Omega_k,\qquad u_k=0\quad\hbox{on}\quad \mathbb{R}^d\setminus\Omega_k,\]
we will prove that \(u_k\) is Lipschitz continuous on \(\mathbb{R}^d\) by estimating the gradient \(|\nabla u_k|\) on the free boundary \(\partial\Omega_k\). The main difficulty comes from the fact that in the case of multiple eigenvalues \(\lambda_k(\Omega_k)=\lambda_{k-1}(\Omega_k)\) one cannot obtain an optimality condition involving only one prescribed eigenfunction \(u_k\). This forces us to use an approximation argument with optimal sets for functionals of the form \((1-\varepsilon)\lambda_k+\varepsilon\lambda_{k-1}\) and then search for uniform bounds on the gradients of the \(k\)th eigenfunctions.

Sede

Mappa
Sede:
Palazzo Campana   -   Sito web
Via:
Via Carlo Alberto, 10
Cap:
10123
Città:
Torino
Provincia:
To
Paese:
Paese: it

Descrizione

Dipartimento di Matematica, Università degli Studi di Torino

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