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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.

L 1,p –Coercitivity and Estimates of the Green Function of the Neumann Problem in a Convex Domain