A general approach to stability in free boundary problems

Every solution of a linear elliptic equation on a bounded domain may be con sidered as an equilibrium of a tree boundary problem. The free boundary pro blem consists of the corresponding parabolic equation on a variable unknown domain with free boundary conditions prescribing both Dirichlet and Neuman n data. We establish a rigorous stability analysis of such equilibria, incl uding the construction of stable and unstable manifolds. For this purpose w e transform the free boundary problem to a fully nonlinear and nonlocal par abolic problem on a fixed domain with fully nonlinear lateral boundary cond itions and we develop the general theory for such problems. As an illustrat ion we give two examples, the second being the focussing flame problem in c ombustion theory. (C) 2000 Academic Press.