This note serves as a basic introduction on the analysis of infinity harmonic functions, a subject that has received considerable interests very recently. The author discusses its connection with absolute minimal Lipschitz extension, present several equivalent characterizations of infinity harmonic functions. He presents the celebrated theorem by R. Jensen [17] on the uniqueness of infinity har...

In this work we focus on approximations of continuous harmonic functions by discrete harmonic functions based on the discrete Laplacian in a triangulation of a point set. We show how the choice of edge weights based on generalized barycentric coordinates influences the approximation quality of discrete harmonic functions. Furthermore, we consider a varying point set to demonstrate that generali...

1. The class Q of complex solutions of a linear partial differential equation. Any two real harmonic functions U(x, y), V(x, y)—that is, solutions of the Laplace equation—can be combined to form a complex harmonic function U+iV. The class of all complex harmonic functions is of no interest, because in effect it possesses no special properties not already possessed by real harmonic functions. Ho...

From a monotonicity property, we derive several results characterizing positive harmonic functions in the unit ball in R and positive measures on the unit sphere Sn−1. Let Bn = {x ∈ Rn : |x| < 1}, n ≥ 2 be the unit ball in Rn and Sn−1 = ∂Bn be the unit sphere. From a monotonicity property, we derive several results characterizing positive harmonic functions in Bn and positive measures on Sn−1. ...

Response of a transversely isotropic 3-D half-space subjected to a surface time-harmonic excitation is presented in analytical form. The derivation of the fundamental solutions expressed in terms of displacements is based on the prefect series of displacement potential functions that have been obtained in the companion paper by the authors. First the governing equations are uncoupled in the cyl...

Harmonic and analytic functions have natural discrete analogues. Harmonic functions can be defined on every graph, while analytic functions (or, more precisely, holomorphic forms) can be defined on graphs embedded in orientable surfaces. Many important properties of the “true” harmonic and analytic functions can be carried over to the discrete setting.

In this paper, a necessary and sufficient coefficient are given for functions in a class of complex valued meromorphic harmonic univalent functions of the form f = h + ḡ using Salagean operator. Furthermore, distortion theorems, extreme points, convolution condition and convex combinations for this family of meromorphic harmonic functions are obtained. Keywords—Harmonic mappings, Meromorphic fu...