A free boundary problem for the localization of eigenfunctions / Guy David, Marcel Filoche, David Jerison, Svitlana Mayboroda.

Format:

Book

Author/Creator:

David, Guy, 1957- author.

Filoche, Marcel, author.

Jerison, David, 1953- author.

Mayboroda, Svitlana, 1981- author.

Series:

Astérisque ; 392.

Astérisque, 0303-1179 ; 392

Language:

English

French

Subjects (All):

Boundary value problems--Numerical solutions.

Boundary value problems.

Eigenfunctions.

Schrödinger operator.

Differential equations, Partial.

Physical Description:

ii, 203 pages ; 24 cm.

Place of Publication:

Paris : Société Mathématique de France, 2017.

Summary:

"We study a variant of the Alt, Caffarelli, and Friedman free boundary problem, with many phases and a slightly different volume term, which we originally designed to guess the localization of eigenfunctions of a Schrödinger operator in a domain. We prove Lipschitz bounds for the functions and some nondegeneracy and regularity properties for the domains"--Abstract.

Contents:

Introduction

Motivation for our main functional

Existence of minimizers

Poincaré inequalities and restriction to spheres

Minimizers are bounded

Two favorite competitors

Hölder-continuity of u inside [Omega]

Hölder-continuity of u on the boundary

The monotonicity formula

Interior Lipschitz bounds for u

Global Lipschitz bounds for u when [Omega] is smooth

A sufficient condition for [u] to be positive

Sufficient conditions for minimizers to be nontrivial

A bound on the number of components

The main non degeneracy condition; good domains

The boundary of a good region is rectifiable

Limits of minimizers

Blow-up limits with two phases

Blow-up limits with one phase

Local regularity when all the indices are good

First variation and the normal derivative.

Notes:

Includes bibliographical references (pages 201-203).

ISBN:

9782856298633

285629863X

OCLC:

1005935620