L~(1,p)-COERCITIVITY AND ESTIMATES OF THE GREEN FUNCTION OF THE NEUMANN PROBLEM IN A CONVEX DOMAIN

摘要

We consider the Neumann problem for the Poisson equation in an arbitrary convex bounded n-dimensional domain. We obtain a coercive estimate for the solution to the Neumann problem and establish the unique solvability of the Neumann problem in the Sobolev space L~(1,p). We also obtain sharp pointwise estimates for the Green function and the gradient of the solution. We describe the eigensubspace corresponding to the least positive eigenvalue λ = n - 1 of the Neumann-Laplace operator in a convex subdomain of the unit (n - 1)-dimensional sphere. Bibliography: 17 titles.