The Dirichlet Problem for the Slab with Entire Data and a Difference Equation for Harmonic Functions

It is shown that the Dirichlet problem for the slab (a, b) × ℝd with entire boundary data has an entire solution. The proof is based on a generalized Schwarz reflection principle. Moreover, it is shown that for a given entire harmonic function g, the inhomogeneous difference equation h(t + 1, y) − h(t, y) = g(t, y) has an entire harmonic solution h.

Further properties of the Bergman spaces of slice regular functions

We continue the study of Bergman theory for the class of slice regular functions. In the slice regular setting there are two possibilities to introduce the Bergman spaces, that are called of the first and of the second kind. In this paperwe mainly consider the Bergman theory of the second kind, by providing an explicit description of the Bergman kernel in the case of the unit ball and of the half space. In the case of the unit ball, we study the Bergman-Sce transform. We also show that the two Bergman theories can be compared only if suitableweights are taken into account. Finally,we use the Schwarz reflection principle to relate the Bergman kernel with its values on a complex half plane.

Minimal Surfaces with Only Horizontal Symmetries

The Schwarz reflection principle states that a minimal surface S in ℝ3 is invariant under reflections in the plane of its principal geodesics and also invariant under 180°-rotations about its straight lines. We find new examples of embedded triply periodic minimal surfaces for which such symmetries are all of horizontal type.