Mathematics

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.