We review the level set methods for computing multi-valued solutions to a class of nonlinear first order partial differential equations,including Hamilton-Jacobi equations,quasi-linear hyperbolic equations,and conserv...We review the level set methods for computing multi-valued solutions to a class of nonlinear first order partial differential equations,including Hamilton-Jacobi equations,quasi-linear hyperbolic equations,and conservative transport equations with multi-valued transport speeds.The multivalued solutions are embedded as the zeros of a set of scalar functions that solve the initial value problems of a time dependent partial differential equation in an augmented space.We discuss the essential ideas behind the techniques,the coupling of these techniques to the projection of the interaction of zero level sets and a collection of applications including the computation of the semiclassical limit for Schr¨odinger equations and the high frequency geometrical optics limits of linear wave equations.展开更多
基金the National Science Foundation under Grant DMS05-05975Osher’s research was supported by AFOSR Grant FA9550-04-0143NSF DMS-0513394 and the Sloan Foundation。
文摘We review the level set methods for computing multi-valued solutions to a class of nonlinear first order partial differential equations,including Hamilton-Jacobi equations,quasi-linear hyperbolic equations,and conservative transport equations with multi-valued transport speeds.The multivalued solutions are embedded as the zeros of a set of scalar functions that solve the initial value problems of a time dependent partial differential equation in an augmented space.We discuss the essential ideas behind the techniques,the coupling of these techniques to the projection of the interaction of zero level sets and a collection of applications including the computation of the semiclassical limit for Schr¨odinger equations and the high frequency geometrical optics limits of linear wave equations.
