×

Front propagation and phase field theory. (English) Zbl 0785.35049

Summary: The connection between the weak theories for a class of geometric equations and the asymptotics of appropriately rescaled reaction- diffusion equations is rigorously established. Two different scalings are studied. In the first, the limiting geometric equation is a first-order equation; in the second, it is a generalization of the mean curvature equation. Intrinsic definitions for the geometric equations are obtained, and uniqueness under a geometric condition on the initial surface is proved. In particular, in the case of the mean curvature equation, this condition is satisfied by surfaces that are strictly star-shaped, that have positive mean curvature, or that satisfy a condition that interpolates between the positive mean curvature and the starshape conditions.

MSC:

35K57 Reaction-diffusion equations
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI Link