Uniqueness of the blow-up at isolated singularities for the Alt-Caffarelli functional
| Autor: | Max Engelstein, Luca Spolaor, Bozhidar Velichkov |
|---|---|
| Přispěvatelé: | Massachusetts Institute of Technology (MIT), Calcul des Variations, Géométrie, Image (CVGI ), Laboratoire Jean Kuntzmann (LJK ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), VELICHKOV, Bozhidar |
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
General Mathematics Connection (vector bundle) Boundary (topology) [MATH] Mathematics [math] singular points 01 natural sciences free boundary Bernoulli's principle Mathematics - Analysis of PDEs 0103 physical sciences FOS: Mathematics [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] Uniqueness [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP] 0101 mathematics [MATH]Mathematics [math] Bernoulli problem Mathematics epiperimetric inequality 010102 general mathematics Isolated singularity 35J60 Cone (topology) monotonicity formula Gravitational singularity 010307 mathematical physics Analysis of PDEs (math.AP) 35R35 |
| Zdroj: | Duke Math. J. 169, no. 8 (2020), 1541-1601 |
| Popis: | In this paper we prove uniqueness of blow-ups and $C^{1,\log}$-regularity for the free-boundary of minimizers of the Alt-Caffarelli functional at points where one blow-up has an isolated singularity. We do this by establishing a (log-)epiperimetric inequality for the Weiss energy for traces close to that of a cone with isolated singularity, whose free-boundary is graphical and smooth over that of the cone in the sphere. With additional assumptions on the cone, we can prove a classical epiperimetric inequality which can be applied to deduce a $C^{1,\alpha}$ regularity result. We also show that these additional assumptions are satisfied by the De Silva-Jerison-type cones, which are the only known examples of minimizing cones with isolated singularity. Our approach draws a connection between epiperimetric inequalities and the \L ojasiewicz inequality, and, to our knowledge, provides the first regularity result at singular points in the one-phase Bernoulli problem. Comment: 37 pages. To appear in Duke Math Journal |
| Databáze: | OpenAIRE |
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