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Lecture Notes on Mean Curvature Flow, Barriers and Singular Perturbations

  • Giovanni Bellettini

Part of the Publications of the Scuola Normale Superiore book series (PSNS, volume 12)

Table of contents

  1. Front Matter
    Pages i-xviii
  2. Giovanni Bellettini
    Pages 1-22
  3. Giovanni Bellettini
    Pages 23-35
  4. Giovanni Bellettini
    Pages 37-58
  5. Giovanni Bellettini
    Pages 59-68
  6. Giovanni Bellettini
    Pages 69-84
  7. Giovanni Bellettini
    Pages 103-126
  8. Giovanni Bellettini
    Pages 127-134
  9. Giovanni Bellettini
    Pages 135-163
  10. Giovanni Bellettini
    Pages 165-174
  11. Giovanni Bellettini
    Pages 175-185
  12. Giovanni Bellettini
    Pages 187-205
  13. Giovanni Bellettini
    Pages 207-215
  14. Back Matter
    Pages 297-329

About this book

Introduction

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.

Keywords

fattening mean curvature flow parabolic Allen-Cahn equation

Authors and affiliations

  • Giovanni Bellettini
    • 1
  1. 1.Dipartimento di MatematicaUniversità di Roma Tor VergataRomaItalia

Bibliographic information