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.
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Translated from Problemy Matematicheskogo Analiza 73, October 2013, pp. 3–16.
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Alkhutov, Y., Maz’ya, V.G. L 1,p–Coercitivity and Estimates of the Green Function of the Neumann Problem in a Convex Domain. J Math Sci 196, 245–261 (2014). https://doi.org/10.1007/s10958-014-1656-y
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DOI: https://doi.org/10.1007/s10958-014-1656-y