Two-Phase Free Boundary Problems for Parabolic Operators: Smoothness of the Front
Article first published online: 5 NOV 2013
Copyright © 2013 Wiley Periodicals, Inc.
Communications on Pure and Applied Mathematics
Volume 67, Issue 1, pages 1–39, January 2014
How to Cite
Ferrari, F. and Salsa, S. (2014), Two-Phase Free Boundary Problems for Parabolic Operators: Smoothness of the Front. Comm. Pure Appl. Math., 67: 1–39. doi: 10.1002/cpa.21490
- Issue published online: 8 NOV 2013
- Article first published online: 5 NOV 2013
- Manuscript Revised: 1 NOV 2012
- Manuscript Received: 1 DEC 2011
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.