A General Approach to Stability in Free Boundary Problems

Abstract

Every solution of a linear elliptic equation on a bounded domain may be considered as an equilibrium of a free boundary problem. The free boundary problem consists of the corresponding parabolic equation on a variable unknown domain with free boundary conditions prescribing both Dirichlet and Neumann data. We establish a rigorous stability analysis of such equilibria, including the construction of stable and unstable manifolds. For this purpose we transform the free boundary problem to a fully nonlinear and nonlocal parabolic problem on a fixed domain with fully nonlinear lateral boundary conditions and we develop the general theory for such problems. As an illustration we give two examples, the second being the focussing flame problem in combustion theory. Academic Press

Smooth solutions to a class of free boundary parabolic problems

“…The decomposition (2.7) may look a bit strange; it is similar to the decompositions used in the papers [3,4,5] and in others to study stability problems. It is crucial in our analysis because it lets us decouple system (2.8), expressing s in terms of w. Indeed, the boundary condition u − g 0 = 0 on ∂Ω t is rewritten as…”

“…In particular, C 2+α initial data near any regular stationary solution (Ω, U) with bounded Ω were considered in the paper [3], where we studied stability of regular stationary solutions, establishing a linearized stability principle for (1.1) in the time-independent case. Since self-similar solutions to (1.4) become stationary solutions to a problem of the type (1.1) after a suitable change of coordinates, we could also consider initial data for License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use…”

“…Indeed, the uniqueness results available up to now concern only particular situations, such as radially symmetric solutions of (1.4), studied in [10], and solutions in cylinders, for initial data which are monotonic in the direction of the axis of the cylinder (see [15] in dimension N = 2 and [12] in any dimension), or in the direction orthogonal to the axis of the cylinder (see [1], for the two-phase case). Moreover, the above mentioned papers [3,5] also give uniqueness results in parabolic Hölder spaces, but only for solutions close to the special solutions considered.…”

“…Below we briefly describe our approach, which is similar to the approach of [3]. We look for ∂Ω t as the range of the function ξ → ξ + s(t, ξ)ν(ξ), defined for ξ ∈ ∂Ω 0 , where ν(ξ) is the exterior unit normal vector to ∂Ω 0 at ξ, and the unknown s(t, ξ) is the signed distance of the point x = ξ + s(t, ξ)ν(ξ) from ∂Ω t .…”

“…This is crucial for our analysis: since the principal part of A coincides with the principal part of L at t = 0, the linearized problem near t = 0, w 0 = 0 is a good linear parabolic problem, for which optimal Hölder regularity results and estimates are available; we need such estimates to solve the nonlinear problem in a standard way, using the contraction theorem in a ball of the parabolic Hölder space C It is clear now why we lose regularity: not only because of the change of coordinates, but also because w has the same space regularity of Du 0 , due to the splitting (1.5). This problem may be avoided for initial data near smooth (at least C 3+α ) stationary solutions, where it is possible to linearize around the stationary solution itself; see [3].…”