DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

LIPSCHITZ REGULARITY OF VISCOSITY SOLUTIONS
IN SOME NONLINEAR PARABOLIC-FREE
BOUNDARY PROBLEMS
EMMANOUIL MILAKIS
We study the regularity of solutions in Stefan-type free boundary problems. We prove that
viscosity solutions to a fully nonlinear free boundary problem are Lipschitz continuous
across the free boundary, provided that the free boundary is a Lipschitz graph in some
space direction.
Copyright © 2006 Emmanouil Milakis. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
We present the author’s result [4, 5] concerning the regularity of the solution to a class of
free boundary problems. More precisely, we study a two-phase Stefan-like free boundary
problem in which a fully nonlinear parabolic equation is verified by the solution
in the positive and the negative domains. These problems arise when a state variable,
v, (temperature, an enthalphy concentration) diffuses in any of two given states (solidliquid,
burnt-unburnt, etc.) but suffers a discontinuity in its behavior across some value
(e.g., v = 0) that indicates state transition. The case of the heat equation was studied by
Athanasopoulos et al. [1, 2].
We start our approach by giving some basic definitions and notations. Denote a point
in R n+1 by (x,t) = (x ′ ,x n ,t)andlet be the space of n × n symmetric matrices. Consider
an operator F : ⊆ R n×n → R to be smooth, concave, fully nonlinear, homogeneous of
degree 1, F(0) = 0, and uniformly elliptic. Denote by x n = f (x ′ ,t) a Lipschitz function
with Lipschitz constant L. We give the definition of a viscosity solution.
Definition 1.1. Let v be a continuous function in D 1 := B 1 (0) × (−1,1). Then v is called a
subsolution (supersolution) to a free boundary problem if
(i) F(D 2 v) − v t ≥ 0(≤ 0) in Ω + := D 1 ∩{v>0};
(ii) F(D 2 v − ) − (v − ) t ≤ 0(≥ 0) in Ω − := D 1 ∩{v ≤ 0} o ;
(iii) v ∈ C 1 (Ω + ) ∩ C 1 (Ω − );
(iv) for any (x,t) ∈ ∂Ω + ∩ D 1 , ∇ x v + (x,t) ≠ 0,
V ν ≥−G ( (x,t),ν,v ν + )
,vν
− (≤) (1.1)
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 787–793