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Some aspects of mean curvature flow

Course Type: 
PhD Course
Academic Year: 
April - July
20 h

The aim of this course is to show some connections between the asymptotic behaviour of the parabolic scalar Ginzburg-Landauequations (also called Allen-Cahn equations) and mean curvature flow of a hypersurface. We shall discuss also theasymptotic behaviour of the stationary points of these equations and some connections with minimalsurfaces and prescribed mean curvature surfaces.Contents of the lectures.

  • A possible motivation: two conjectures by De Giorgi. The Modica-Mortola theorem.
  • The signed distance function; second fundamental form and mean curvature vector.
  • Scalar parabolic Ginzburg-Landau equations. Well-posedness.
  • Inner and outer asymptotic expansions: matching conditions.
  • Construction of sub/supersolutions to the scalar parabolic Ginzburg-Landau equations.
  • Convergence to mean curvature flow for short times and for well-prepared initial data.
  • The elliptic case: asymptotic expansions. Construction of sub/supersolutions, and convergence to prescribed mean curvature hypersurfaces.
A-136; Room A-005 on June 8
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