A Class of Free Boundary Problems with Onset of a new Phase

Details A Class of Free Boundary Problems with Onset of a new Phase A Class of Free Boundary Problems with Onset of a new Phase

2006-10-26

arxiv.org/abs/math/0610771v1

Mathematics

Analysis of PDEs

Scientific paper

A class of diffusion driven Free Boundary Problems is considered which is characterized by the initial onset of a phase and by an explicit kinematic condition for the evolution of the free boundary. By a domain fixing change of variables it naturally leads to coupled systems comprised of a singular parabolic initial boundary value problem and a Hamilton-Jacobi equation. Even though the one dimensional case has been thoroughly investigated, results as basic as well-posedness and regularity have so far not been obtained for its higher dimensional counterpart. In this paper a recently developed regularity theory for abstract singular parabolic Cauchy problems is utilized to obtain the first well-posedness results for the Free Boundary Problems under consideration. The derivation of elliptic regularity results for the underlying static singular problems will play an important role.

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