Calculus of Variations and Geometric Measure Theory

G. Bellettini - M. Chermisi - M. Novaga

The level set method for systems of PDEs

created by belletti on 22 Nov 2005
modified by chermisi on 15 Apr 2010


Published Paper

Inserted: 22 nov 2005
Last Updated: 15 apr 2010

Journal: Comm. Partial Differential Equations
Year: 2007


We propose a level set method for systems of PDEs which is consistent with the previous research pursued by Evans for the heat equation and by Giga and Sato for Hamilton-Jacobi equations. Our approach follows a geometric construction related to the notion of barriers introduced by De Giorgi. The main idea is to force a comparison principle between manifolds of different codimension and require each sub-level of a solution of the level set equation to be a barrier for the graph of a solution of the corresponding system. We apply the method to a class of systems of first order quasi-linear equations. We compute the level set equation associated with suitable first order systems of conservation laws, with the mean curvature flow of a manifold of arbitrary codimension and with systems of reaction-diffusion equations. Finally, we provide a level set equation associated with the parametric curvature flow of planar curves.