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A free boundary problem for the localization of eigenfunctions. (English) Zbl 1380.49001

Astérisque 392. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-863-3/pbk). ii, 203 p. (2017).
This work is concerned with the study of a variant of the Alt, Caffarelli, and Friedman free boundary problem. The model contains many phases and a slightly different volume term. The main purpose of the present work is to study the localization of eigenfunctions of a Schrödinger operator in a domain \(\Omega\subset{\mathbb R}^n\). The authors establish Lipschitz bounds for the functions and some nondegeneracy and regularity properties for the domains.
The content is divided into the following chapters: 1. Introduction; 2. Motivation for our main functional; 3. Existence of minimizers; 4. Poincaré inequalities and restriction to spheres; 5. Minimizers are bounded; 6. Two favorite competitors; 7. Hölder-continuity of \(u\) inside \(\Omega\); 8. Hölder-continuity of \(u\) on the boundary; 9. The monotonicity formula. 10. Interior Lipschitz bounds for \(u\); 11. Global Lipschitz bounds for \(u\) when \(\Omega\) is smooth; 12. A sufficient condition for \(|u|\) to be positive; 13. Sufficient conditions for minimizers to be nontrivial; 14. A bound on the number of components; 15. The main non degeneracy condition; good domains; 16. The boundary of a good region is rectifiable; 17. Limits of minimizers; 18. Blow-up limits are minimizers; 19. Blow-up limits with 2 phases; 20. Blow-up limits with one phase; 21. Local regularity when all the indices are good; 22. First variation and the normal derivative.

MSC:

49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
49Q20 Variational problems in a geometric measure-theoretic setting
49J40 Variational inequalities
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35P05 General topics in linear spectral theory for PDEs
49R05 Variational methods for eigenvalues of operators
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