Boundary regularity in the Dirichlet problem for the invariant Laplacians $\Delta _\gamma $ on the unit real ball

Abstract

Abstract. We study the boundary regularity in the Dirichlet problem of the differential operatorsOur main result is: if γ > −1/2 is neither an integer nor a half-integer not less than n/2 − 1, one cannot expect global smooth solutions of ∆γ u = 0; if u ∈ C ∞ (Bn) satisfies ∆γu = 0, then u must be either a polynomial of degree at most 2γ + 2 − n or a polyharmonic function of degree γ + 1.

On the rigidity and boundary regularity for Bakry-Emery-Kohn harmonic functions in Bergman metric on the unit ball in 𝐶ⁿ

“…The problem with the regularity of P [f ] was studied by many authors, including R. Graham [2,3], Ahern and Bruna [1], C. Liu and L. Peng [8], Li and Simon [6], Li and Wei [7], etc. The following striking theorem was proved by R. Graham [2], especially, the rigidity part.…”

Schwarz Lemma for Solutions of the $$\alpha $$-harmonic Equation

“…In [17], Liu and Peng considered the existence and the representation of the solutions to the following Dirichlet problem:…”

“…(see [17,Theorem 2.4]). Here and hereafter, H k (S n−1 , R n ) denotes the space of spherical harmonic mappings of degree k from S n−1 into R n .…”

Graham type theorem on classical bounded symmetric domains

“…Further information along this direction can be found in Li and Wei [11]. For more results on invariant harmonic functions and backgrounds we refer the reader to [1], [5], [8], [9], [12], [15]. However, the problem about whether Graham Theorem holds when M is a classical bounded symmetric domain with Bergman metric g is widely open.…”