NEUMANN BOUNDARY CONDITIONS FOR THE INFINITY LAPLACIAN AND THE MONGE-KANTOROVICH MASS TRANSPORT PROBLEM
msra(2008)
摘要
In this note we review some recent results concerning the natural Neumann boundary condition for the 1-Laplacian and its relation with the Monge-Kantorovich mass transport problem. (1) We study the limit as p ! 1 of solutions ofpup = 0 in a domain with |Dup| p 2 @up/@� = g on @. We obtain a natural minimization problem that is verified by a limit point of {up} and a limit problem that is satisfied in the viscosity sense. It turns out that the limit variational problem is related to the Monge-Kantorovich mass transport problems when the measures are supported on @. (2) Next, we study the limit of Monge-Kantorovich mass transport prob- lems when the involved measures are supported in a small strip near the boundary of a bounded smooth domain, . Given an absolutely con- tinuous measure (with respect to the surface measure) supported on the boundary @ with zero mean value, by performing a suitable extension of the measures to a strip of width " near the boundary of the domain we consider the mass transfer problem for the extensions. Then we study the limit as " goes to zero of the Kantorovich potentials for the extensions and obtain that it coincides with a solution of the original mass transfer problem. (3) Also we present a Steklov like eigenvalue problem that appears as the limit of the usual Steklov eigenvalue problem for the p Laplacian as p ! 1.
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关键词
. quasilinear elliptic equations,neumann boundary conditions.
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