A free boundary problem for the localization of eigenfunctions
Guy David (LM-Orsay), Marcel Filoche (PMC), David Jerison (MIT),, Svitlana Mayboroda (UMN-MATH)
TL;DR
This paper investigates a multi-phase free boundary problem inspired by eigenfunction localization in Schrödinger operators, establishing regularity and nondegeneracy properties of solutions and domains.
Contribution
It introduces a modified free boundary problem with volume considerations and proves key regularity results for the solutions and their domains.
Findings
Lipschitz bounds for the functions
Nondegeneracy properties of the solutions
Regularity properties of the free boundaries
Abstract
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 Schr\"odinger operator in a domain. We prove Lipschitz bounds for the functions and some nondegeneracy and regularity properties for the domains.
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Taxonomy
TopicsSpectral Theory in Mathematical Physics · Numerical methods in inverse problems · Holomorphic and Operator Theory