# Some free boundary problems recast as nonlocal parabolic equations

@article{ChangLara2019SomeFB,
title={Some free boundary problems recast as nonlocal parabolic equations},
author={H'ector A. Chang-Lara and Nestor Guillen and Russell W. Schwab},
journal={Nonlinear Analysis},
year={2019}
}
• Published 7 July 2018
• Mathematics
• Nonlinear Analysis
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## References

SHOWING 1-10 OF 68 REFERENCES
Regularity of the free boundary in parabolic phase-transition problems
• Mathematics
• 1996
In this paper we start the study of the regularity properties of the free boundary, for parabolic two-phase free boundary problems. May be the best known example of a parabolic two-phase free
From the free boundary condition for Hele-Shaw to a fractional parabolic equation
• Mathematics
• 2016
We propose a method to determine the smoothness of sufficiently flat solutions of one phase Hele-Shaw problems. The novelty is the observation that under a flatness assumption the free boundary
On the regularity theory of fully nonlinear parabolic equations
Recently M. Crandall and P. L. Lions [3] developed a very successful method for proving the existence of solutions of nonlinear second-order partial differential equations. Their method, called the
On the regularity theory of fully nonlinear parabolic equations: II
Recently M. Crandall and P. L. Lions [3] developed a very successful method for proving the existence of solutions of nonlinear second-order partial differential equations. Their method, called the
Min–max formulas for nonlocal elliptic operators
• Mathematics
Calculus of Variations and Partial Differential Equations
• 2019
In this work, we give a characterization of Lipschitz operators on spaces of $C^2(M)$ functions (also $C^{1,1}$, $C^{1,\gamma}$, $C^1$, $C^\gamma$) that obey the global comparison property-- i.e.
Regularity of Free Boundaries in Obstacle-type Problems
• Mathematics
• 2012
The regularity theory of free boundaries flourished during the late 1970s and early 1980s and had a major impact in several areas of mathematics, mathematical physics, and industrial mathematics, as
Neumann Homogenization via Integro-Differential Operators
• Mathematics
• 2014
In this note we describe how the Neumann homogenization of fully nonlinear elliptic equations can be recast as the study of nonlocal (integro-differential) equations involving elliptic
On the global existence for the Muskat problem
• Mathematics, Computer Science
• 2013
This work proves an L2(R) maximum principle, in the form of a new “log” conservation law which is satisfied by the equation (1) for the interface, and takes advantage of the fact that the bound ‖∂xf0‖L∞ < 1 is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law.
A free-boundary problem for the heat equation arising in flame propagation
• Mathematics
• 1995
We introduce a new free-boundary problem for the heat equation, of interest in combustion theory. It is obtained in the description of laminar flames as an asymptotic limit for high activation
On the existence of convex classical solutions to a generalized Prandtl-Batchelor free-boundary problem-II
Abstract. We give an analytical proof of the existence of convex classical solutions for the (convex) Prandtl-Batchelor free boundary problem in fluid dynamics. In this problem, a convex vortex core