Refined Rellich boundary inequalities for the derivatives of a harmonic function

Agrawal, Siddhant; Alazard, Thomas

doi:10.1090/proc/16277

Refined Rellich boundary inequalities for the derivatives of a harmonic function
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by Siddhant Agrawal and Thomas Alazard;

Proc. Amer. Math. Soc. 151 (2023), 2103-2113

DOI: https://doi.org/10.1090/proc/16277

Published electronically: February 10, 2023

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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 $1<p\leq 2$ by using a Riemann mapping and interpolation argument.

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