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.References
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Bibliographic Information
- Siddhant Agrawal
- Affiliation: Instituto de Ciencias Matemáticas (ICMAT), C/ Nicolás Cabrera, 13-15 (Campus Cantoblanco), 28049 Madrid, Spain
- MR Author ID: 1352311
- ORCID: 0000-0001-5554-2278
- Thomas Alazard
- Affiliation: Université Paris-Saclay, ENS Paris-Saclay, CNRS, Centre Borelli UMR9010, avenue des Sciences, F-91190 Gif-sur-Yvette, France
- MR Author ID: 749193
- ORCID: 0000-0001-8386-0476
- Received by editor(s): May 10, 2022
- Received by editor(s) in revised form: September 15, 2022, and September 16, 2022
- Published electronically: February 10, 2023
- Additional Notes: The work was supported by the National Science Foundation under Grant No. DMS-1928930. The first author received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program through the grant agreement 862342. The second author also acknowledges the SingFlows project (grant ANR-18-CE40-0027) of the French National Research Agency (ANR)
- Communicated by: Benoit Pausader
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 2103-2113
- MSC (2020): Primary 26D10, 35A23
- DOI: https://doi.org/10.1090/proc/16277
- MathSciNet review: 4556204