Continuity estimates for Riesz potentials on polygonal boundaries

Xavier Claeys, Muhammad Hassan () and Benjamin Stamm
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Xavier Claeys: Sorbonne Université, Université Paris-Diderot SPC, CNRS
Muhammad Hassan: Sorbonne Université, CNRS, Université de Paris
Benjamin Stamm: University of Stuttgart

Partial Differential Equations and Applications, 2024, vol. 5, issue 3, 1-20

Abstract: Abstract Riesz potentials are well known objects of study in the theory of singular integrals that have been the subject of recent, increased interest from the numerical analysis community due to their connections with fractional Laplace problems and proposed use in certain domain decomposition methods. While the $$\textrm{L}^p$$ L p -mapping properties of Riesz potentials on flat geometries are well-established, comparable results on rougher geometries for Sobolev spaces are very scarce. In this article, we study the continuity properties of the surface Riesz potential generated by the $$1/\sqrt{x}$$ 1 / x singular kernel on a polygonal domain $$\Omega \subset {\mathbb {R}}^2.$$ Ω ⊂ R 2 . We prove that this surface Riesz potential maps $$\textrm{L}^{2}(\partial \Omega )$$ L 2 ( ∂ Ω ) into $$\textrm{H}^{1/2}(\partial \Omega ).$$ H 1 / 2 ( ∂ Ω ) . Our proof is based on a careful analysis of the Riesz potential in the neighbourhood of corners of the domain $$\Omega .$$ Ω . The main tool we use for this corner analysis is the Mellin transform which can be seen as a counterpart of the Fourier transform that is adapted to corner geometries.

Keywords: Riesz potentials; Polygonal domains; Continuity estimates; Mellin transform; Sobolev spaces; 45P05; 47G10; 47G30; 65R99 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s42985-024-00280-4

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