Nonlinear Partial Differential Equations

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

Abstract Evans I will continue my ongoing research in nonlinear PDE, with emphasis on these topics: (i) PDE methods for Hamiltonian dynamics: The goal here is to use the (nonsmooth) solution of the appropriate cell PDE to study Hamiltonian dynamics on the Aubry-Mather set, for problems with many degrees of freedom. (ii) Regularity for optimal Lipschitz extensions: The boundary value problem for the ``infinity-Laplacian'' is a highly degenerate nonlinear PDE, which is extremely interesting since it is a sort of Euler-Lagrange equation for a calculus of variations problem ``in the sup-norm''. I intend to study carefully the possible regularity of weak solutions. (iii) Relaxation approximations for Hamilton-Jacobi equations: Formal asymptotics strongly suggest that perturbing a Hamilton-Jacobi PDE by a wave operator, with large enough wave speed, should in the limit produce a viscosity solution. A proof of this would dramatically extend the applicability of viscosity solution methods, to cover approximations with no maximum principle. (iv) Mass transfer problems: I continue to be interested in PDE methods for Monge-Kantorovich mass transfer problems, especially those occurring on fast time scales, coupled with slower dynamics. Partial differential equations (PDE) occur as mathematical descriptions of an extremely wide variety of physical and other phenomena, and nonlinear PDE are especially difficult, since they do not permit us to decompose complicated solutions into a superposition of simpler solutions. Many important nonlinear PDE do however have a variational structure, meaning that they correspond to a sort of optimization principle. The overall goal of the calculus of varitations is finding ways to exploit these variational principles, to help us understand the nature of solutions to the corresponding nonlinear PDE. In this project I will (mostly) continue to study several important classes of variational problems, to understand more about (i) certain nondissipative dynamical systems, (ii) optimal extension problems, (iii) wave-like approximations to dissipative phenomena, and (iv) optimal mass reallocation problems.