The Level Set Method for Systems of PDEs

We propose a level set method for systems of PDEs which is consistent with the previous research pursued by Evans (1996 Evans , L. C. ( 1996 ). A geometric interpretation of the heat equation with multivalued initial data . SIAM J. Math. Anal. 27 ( 4 ): 932 – 958 .Crossref], Web of Science ®] , Google Scholar]) for the heat equation and by Giga and Sato (2001 Giga , Y. , Sato , M.-H. ( 2001 ). A level set approach to semicontinuous viscosity solution for Cauchy problems . Comm. Partial Differential Equations 26 ( 5–6 ): 813 – 839 .Taylor & Francis Online], Web of Science ®] , Google Scholar]) 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 nonzero 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.