TY - BOOK ID - 8430136 TI - Lecture Notes on Mean Curvature Flow: Barriers and Singular Perturbations PY - 2013 SN - 8876424288 8876424296 PB - Pisa : Scuola Normale Superiore : Imprint: Edizioni della Normale, DB - UniCat KW - Mathematics KW - Physical Sciences & Mathematics KW - Geometry KW - Mathematics. KW - Geometry. KW - Euclid's Elements KW - Math KW - Science KW - Curvature. KW - Flows (Differentiable dynamical systems) KW - Differentiable dynamical systems KW - Calculus KW - Curves KW - Surfaces UR - https://www.unicat.be/uniCat?func=search&query=sysid:8430136 AB - The aim of the book is to study some aspects of geometric evolutions, such as mean curvature flow and anisotropic mean curvature flow of hypersurfaces. We analyze the origin of such flows and their geometric and variational nature. Some of the most important aspects of mean curvature flow are described, such as the comparison principle and its use in the definition of suitable weak solutions. The anisotropic evolutions, which can be considered as a generalization of mean curvature flow, are studied from the view point of Finsler geometry. Concerning singular perturbations, we discuss the convergence of the Allen–Cahn (or Ginsburg–Landau) type equations to (possibly anisotropic) mean curvature flow before the onset of singularities in the limit problem. We study such kinds of asymptotic problems also in the static case, showing convergence to prescribed curvature-type problems. ER -