A boundary formula for reproducing kernel Hilbert spaces of real harmonic functions in Lipschitz domains. Issue 1 (2nd January 2021)

• Record Type: Journal Article
• Authors: Chaira, Abdellatif; Touhami, Soumia
• Is Part Of: Complex variables and elliptic equations. Volume 66:Issue 1(2021)
• Page Start: 94
• Page End: 117
• Publication Date: 2021-01-02
• Subjects: M. Lanza de Cristoforis; Reproducing kernel Hilbert spaces -- Lipschitz domains -- harmonic spaces -- trace spaces and Moore–Penrose pseudo-inverse; 46E22 -- 35Cxx; Functions of complex variables -- Periodicals; Differential equations, Elliptic -- Periodicals; 515.905
• DOI: 10.1080/17476933.2019.1709967
• Languages: English
• ISSNs: 1747-6933

Abstract

Abstract : This paper develops a new Hilbert space method to characterize a family of reproducing kernel Hilbert spaces of real harmonic functions in a bounded Lipschitz domain Ω ⊂ R d, d ≥ 2 . Such method involves some families of positive self-adjoint operators and makes use of characterizations of their trace data and of a special inner product on H 1 ( Ω ) . We also establish boundary representation results for this family in terms of the L 2 -Bergman kernel. In particular, a boundary integral representation for the very weak solution of the Dirichlet problem for Laplace's equation with L 2 -boundary data is provided. Reproducing kernels and orthonormal bases for the harmonic spaces are also found.