Pointwise and viscosity solutions for the limit of a two phase parabolic singular perturbation problem

CAFFARELLI, LUIS A.; LEDERMAN, CLAUDIA; WOLANSKI, NOEMÍ

INDIANA UNIVERSITY MATHEMATICS JOURNAL

Indiana Universtiy

Lugar: Bloomington; Año: 1997 vol. 46 p. 719 - 740

0022-2518

In this paper we are concerned with functions u which are obtained as the uniform limit, as  e -> 0, of solutions ue(x,t) of the equation (Pe): Δue- ute=be(u e) in D    where e> 0, be(s)=(1/e)b(s/e), support b=[0,1] and òb(s)ds = M .   We prove that the limit function u is a solution to                                        Δu - ut = 0      in   D /  ¶{u > 0} u = 0 , (u+v)2  - (u-v)2  = 2M   on  DÇ ¶{u > 0} in a viscosity sense and  in a pointwise sense at regular free boundary points. Here ν is the inward unit spatial normal to the free boundary ¶{u>0} , u+=max(u,0) and  u-=max(-u,0).