Refined Rellich boundary inequalities for the derivatives of a harmonic function

Siddhant Agrawal , Thomas Alazard (1, 2)

1 CB - CB - Centre Borelli - UMR 9010
2 ENS Paris Saclay - Ecole Normale Supérieure Paris-Saclay

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.

Domaines: Equations aux dérivées partielles [math.AP]

Siddhant Agrawal, Thomas Alazard. Refined Rellich boundary inequalities for the derivatives of a harmonic function. Proceedings of the American Mathematical Society, 2023, 151 (5), pp.2103-2113. ⟨10.1090/proc/16277⟩. ⟨hal-03883558⟩