A boundary formula for reproducing kernel Hilbert spaces of real harmonic functions in Lipschitz domains. Issue 1 (2nd January 2021) - Chaira, Abdellatif; Touhami, Soumia
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.