Nonlinear Partial Differential Equations

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AbstractEvansI 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 cellPDE to study Hamiltonian dynamics on the Aubry-Mather set, for problemswith many degrees of freedom. (ii) Regularity for optimal Lipschitz extensions:The boundary value problem for the ``infinity-Laplacian'' is a highlydegenerate nonlinear PDE, which is extremely interesting sinceit is a sort of Euler-Lagrange equation for a calculus of variationsproblem ``in the sup-norm''. I intend to study carefully the possibleregularity of weak solutions. (iii) Relaxation approximations for Hamilton-Jacobi equations: Formal asymptotics strongly suggest that perturbing a Hamilton-JacobiPDE by a wave operator, with large enough wave speed, should in the limitproduce a viscosity solution. A proof of this would dramatically extendthe applicability of viscosity solution methods, to cover approximationswith no maximum principle. (iv) Mass transfer problems:I continue to be interested in PDE methods for Monge-Kantorovich masstransfer problems, especially those occurring on fast time scales, coupled with slower dynamics.Partial differential equations (PDE) occur as mathematicaldescriptions of an extremely wide variety of physical and other phenomena,and nonlinear PDE are especially difficult, since they do not permit us to decomposecomplicated solutions into a superposition of simpler solutions. Many importantnonlinear PDE do however have a variational structure, meaning that they correspond to a sort of optimization principle. The overallgoal 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 variationalproblems, to understand more about (i) certain nondissipative dynamical systems, (ii) optimal extension problems, (iii) wave-like approximations to dissipativephenomena, and (iv) optimal mass reallocation problems.