Barles, G.; Soner, H. M.; Souganidis, P. E. Front propagation and phase field theory. (English) Zbl 0785.35049 SIAM J. Control Optimization 31, No. 2, 439-469 (1993). 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. Cited in 2 ReviewsCited in 144 Documents MSC: 35K57 Reaction-diffusion equations 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 35A05 General existence and uniqueness theorems (PDE) (MSC2000) Keywords:viscosity solutions; mean curvature flow; weak propagation of fronts; phase field theory; signed distance function; reaction-diffusion equations PDFBibTeX XMLCite \textit{G. Barles} et al., SIAM J. Control Optim. 31, No. 2, 439--469 (1993; Zbl 0785.35049) Full Text: DOI Link