David G., Filoche M., Jerison D., Mayboroda S. A free boundary problem for the localization of eigenfunctions

Paris: Société Mathématique de France, 2017. — 216 p.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.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 Ω
Hölder-continuity of u on the boundary
The monotonicity formula
Interior Lipschitz bounds for u
Global Lipschitz bounds for u when Ω 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 are 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
Bibliography