Learn More
1 Overview This article is an extremely rapid survey of the modern theory of partial differential equations (PDEs). Sources of PDEs are legion: mathematical physics, geometry, probability theory, continuum mechanics, optimization theory, etc. Indeed, most of the fundamental laws of the physical sciences are partial differential equations and most papers(More)
We construct a unique weak solution of the nonlinear PDE which asserts each level set evolves in time according to its mean curvature. This weak solution allows us then to define for any compact set Γ o a unique generalized motion by mean curvature, existing for all time. We investigate the various geometric properties and pathologies of this evolution.
Our intention in this paper is to publicize and extend somewhat important work of Plotnikov [P] on the asymptotic limits of solutions of viscous regularizations of an nonlinear diffusion PDE with a cubic nonlinearity. Since the formal limit PDE is in general ill–posed, we expect that the limit solves instead a corresponding diffusion equation with(More)
We regard the limit as p ! 1 of the ow governed by the p-Laplacian as providing a simplistic model for the \collapse of an initially unstable sandpile." Upon rescaling to stretch out the initial layer we obtain some simple dynamics and provide fairly explicit solutions in certain cases. In particular we note that such models entail \instanta-neous" mass(More)
We discuss a quantum analogue of Mather's minimization principle for La-grangian dynamics, and provide some formal calculations suggesting the corresponding Euler– Lagrange equation. We then rigorously construct from the dual eigenfunctions of a certain non-selfadjoint operator a candidate ψ for a minimizer, and recover aspects of " weak KAM " theory in the(More)