Includes bibliographical references (pages 201-203).
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.
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.
This resource is supported by the Institute of Museum and Library Services under the provisions of the Library Services and Technology Act as administered by State Library of Iowa.