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Two‐Phase Free Boundary Problems for Parabolic Operators: Smoothness of the Front
Author(s) -
Ferrari Fausto,
Salsa Sandro
Publication year - 2014
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.21490
Subject(s) - lipschitz continuity , mathematics , smoothness , boundary (topology) , mathematical analysis , class (philosophy) , stability (learning theory) , computer science , machine learning , artificial intelligence
We continue to develop the regularity theory of general two‐phase free boundary problems for parabolic operators. In a 2010 paper, we establish the optimal (Lipschitz) regularity of a viscosity solution under the assumptions that the free boundary is locally a flat Lipschitz graph and a nondegeneracy condition holds. Here, on one side we improve this result by removing the nondegeneracy assumption; on the other side we prove the smoothness of the front. The proof relies in a crucial way on a local stability result stating that, for a certain class of operators, under small perturbations of the coefficients flat free boundaries remain close and flat. © 2013 Wiley Periodicals, Inc.