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Refined Rellich boundary inequalities for the derivatives of a harmonic function

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Abstract

The classical Rellich inequalities imply that the L 2-norms of the normal and tangential derivatives of a harmonic function are equivalent. In this note, we prove several refined inequalities, which make sense even if the domain is not Lipschitz. For two-dimensional domains, we obtain a sharp L p-estimate for 1 < p ≤ 2 by using a Riemann mapping and interpolation argument.
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hal-03883558 , version 1 (03-12-2022)

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  • HAL Id : hal-03883558 , version 1

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Siddhant Agrawal, Thomas Alazard. Refined Rellich boundary inequalities for the derivatives of a harmonic function. 2022. ⟨hal-03883558⟩
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