## Advances in Differential Equations

- Adv. Differential Equations
- Volume 6, Number 12 (2001), 1409-1442.

### Uniqueness in a two-phase free-boundary problem

Claudia Lederman, Juan Luis Vázquez, and Noemi Wolanski

#### Abstract

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
${{\mathcal D}}\subset {\mathbb R}^N\times(0,T)$ which satisfies the equation $$ \Delta
u+\sum a_i\,u_{x_i}-u_t=0\quad $$ whenever $ u(x,t)\ne 0$, i.e., in the subdomains
${{\mathcal D}}_+=\{(x,t)\in {{\mathcal D}}: u(x,t)>0\}$ and ${{{\mathcal
D}}}_-=\{(x,t)\in {{\mathcal D}}: u(x,t) <0\}$. 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: $$ u=0 ,\quad |\nabla
u^+|^2 - |\nabla u^-|^2 =2 M. $$ 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
${{\mathcal D}}$. 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.

#### Article information

**Source**

Adv. Differential Equations Volume 6, Number 12 (2001), 1409-1442.

**Dates**

First available in Project Euclid: 2 January 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1357139953

**Mathematical Reviews number (MathSciNet)**

MR1858427

**Zentralblatt MATH identifier**

1004.35123

**Subjects**

Primary: 35R35: Free boundary problems

#### Citation

Lederman, Claudia; Wolanski, Noemi; Vázquez, Juan Luis. Uniqueness in a two-phase free-boundary problem. Adv. Differential Equations 6 (2001), no. 12, 1409--1442.https://projecteuclid.org/euclid.ade/1357139953