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Best constants in Poincar\'e inequalities for convex domains

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arxiv 1110.2960 v1 pith:3EI4BTMQ submitted 2011-10-13 math.AP

Best constants in Poincar\'e inequalities for convex domains

classification math.AP
keywords inequalitybestconstantsconvexdomainspoincaralonebound
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We prove a Payne-Weinberger type inequality for the $p$-Laplacian Neumann eigenvalues ($p\ge 2$). The inequality provides the sharp upper bound on convex domains, in terms of the diameter alone, of the best constants in Poincar\'e inequality. The key point is the implementation of a refinement of the classical P\'olya-Szeg\"o inequality for the symmetric decreasing rearrangement which yields an optimal weighted Wirtinger inequality.

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