Uniqueness in a two phase free boundary problem

LEDERMAN, CLAUDIA; VÁZQUEZ, JUAN LUIS; WOLANSKI, NOEMÍ

ADVANCES IN DIFFERENTIAL EQUATIONS

Khayyam

Lugar: Athens; Año: 2001 vol. 6 p. 1409 - 1442

1079-9389

We investigate a two-phase free-boundary problem in heat propagation that in classical terms is formulated as follows: to find a continuous function $u(x,t)$ defined in a domain $caldsubset {Bbb R}^N imes(0,T)$ which satisfies the equation $$ Delta u+sum a_i,u_{x_i}-u_t=0quad $$ whenever $ u(x,t) e 0$, i.e., in the subdomains $cald_+={(x,t)in cald: u(x,t)>0}$ and ${cald}_-={(x,t)in cald: u(x,t)<0}$. Besides, we assume that both subdomains are separated by a smooth hypersurface, the free boundary, whose normal is never time-oriented and on which the following conditions are satisfied $$ u=0 ,quad | abla u^+|^2 - | abla u^-|^2 =2 M. $$ Here $M>0$ is a fixed constant, and the gradients are spatial side-derivatives in the usual two-phase sense. In addition, initial data are specified, as well as either Dirichlet or Neumann data on the parabolic boundary of $cald$. The problem admits {it classical} solutions only for good data and for small times. To overcome this problem several generalized concepts of solution have been proposed, among them the concepts of {it limit} solution and {it viscosity} solution. Continuing the work done for the one-phase problem we investigate conditions under which the three concepts agree and produce a unique solution for the two-phase problem.