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An Epiperimetric Inequality for the Regularity of Some Free Boundary Problems: The 2‐Dimensional Case
Author(s) -
Spolaor Luca,
Velichkov Bozhidar
Publication year - 2019
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21785
Subject(s) - mathematics , scalar (mathematics) , boundary (topology) , dimension (graph theory) , inequality , mathematical analysis , component (thermodynamics) , boundary value problem , phase (matter) , sign (mathematics) , pure mathematics , geometry , physics , quantum mechanics
Using a direct approach, we prove a two‐dimensional epiperimetric inequality for the one‐phase problem in the scalar and vectorial cases and for the double‐phase problem. From this we deduce, in dimension 2, the C 1,α regularity of the free boundary in the scalar one‐phase and double‐phase problems, and of the reduced free boundary in the vectorial case, without any restriction on the sign of the component functions. Furthermore, we show that in the vectorial case each connected component of { | u | = 0 } might have cusps, but they must be a finite number. © 2018 Wiley Periodicals, Inc.