Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor
The level set approach to the evolution of surfaces is based on characterizing the surfaces as level sets of the solution of certain fully nonlinear degenerate PDEs. The basic tool of this approach is the theory of viscosity solutions. In this paper, the authors extend an earlier work by \textit{L. C. Evans} and \textit{J. Spruck} [J. Differ. Geom. 33, No. 3, 635-681 (1991; Zbl 0726.53029)] as well as by \textit{Y.-G. Chen}, \textit{Y. Giga} and \textit{S. Goto} [ibid. 33, No. 3, 749-786 (1991; Zbl 0715.35037)] to the case where the normal velocity is a general continuous function of the normal and the curvature tensor. Applications in geometry include the case of Gaussian curvature. The main difficulty is that these problems lead to PDEs with singularities of higher order. The authors overcome this difficulty by extending the class of admissible test functions in the definition of viscosity solutions. They are able to establish a comparison principle and an existence result. Their method allows them to treat the generalized evolution of noncompact hypersurfaces.
scientific article; zbMATH DE number 794315