Refined Rellich boundary inequalities for the derivatives of a harmonic function
Siddhant Agrawal, Thomas Alazard · Proceedings of the American Mathematical Society · 2022
The classical Rellich inequalities imply that the L 2 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 L^p -estimate for 1 > p ≤ 2 1>p\leq 2 by using a Riemann mapping and interpolation argument.