Uniqueness in a two phase free boundary problem

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 D ⊂ RN × (0, T) which satisfies the equation ∆u + Σ ai uxi − ut = 0 whenever u(x, t) = 0, i.e., in the subdomains D+ = {(x, t) ∈ D : u(x, t) > 0} and D− = {(x, t) ∈ D : u(x, t) 0 is a fixed constant, and the gradients are spatial sidederivatives 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 D. The problem admits 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 limit solution and 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.Fil: Lederman, Claudia Beatriz. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Vazquez, Juan Luis. Universidad Autonoma de Madrid; EspañaFil: Wolanski, Noemi Irene. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin