Laplace equation, superposition of harmonic functions, and method of images

We introduce an Almgren frequency function of the sub-$p$-Laplace equation on the Heisenberg group to establish a doubling estimate under the assumption that the frequency function is locally bounded. From this, we obtain some partial results on unique continuation for the sub-$p$-Laplace equation.

Download Full-text

Analytic Solution for the Dirichlet Problem in 2-D

Journal of Computational and Theoretical Nanoscience ◽

10.1166/jctn.2018.7133 ◽

2018 ◽

Vol 15 (2) ◽

pp. 611-615

Author(s):

Nurcan Baykuş Savaşanerİl ◽

Havva Delİbaş

Keyword(s):

Dirichlet Problem ◽

Laplace Equation ◽

Harmonic Functions ◽

Analytic Solution ◽

Broad Class ◽

Applied Mathematics ◽

Fundamental Equation ◽

Elliptic Functions ◽

Physical Problems ◽

The Green Function

A broad class of steady-state physical problems can be reduced to finding the harmonic functions that satisfy certain boundary conditions. A fundamental equation of applied mathematics is Laplace equation. This equation models important phenomena in engineering and physics, Laplace equation with satisfied boundary values is known as the Dirichlet problem. In this study, an alternative method is presented for the solution of the Dirichlet problem in a cut-ring region and the solution function of the problem is based on the Green function, and therefore on elliptic functions.

Download Full-text

On Invariant Perforation in an Infinite Strip

Journal of Applied Mechanics ◽

10.1115/1.4012055 ◽

1959 ◽

Vol 26 (3) ◽

pp. 422-431

Author(s):

Chih-Bing Ling

Keyword(s):

Finite Group ◽

Stress Function ◽

Harmonic Functions ◽

Infinite Group ◽

The Other ◽

Infinite Strip ◽

Method Of Images ◽

Numerical Examples ◽

Sigma Function ◽

Working Out

Abstract The invariant perforation in an infinite strip can be classified into two groups. One is the finite group and the other is the infinite group. There are five cases in the finite group and nine cases in the infinite group. All the cases can be solved by the method of images. This method has, in fact, been used by the author to solve the stresses in an infinite strip containing either an unsymmetrically located single hole or a series of uniformly distributed equal holes. The solution is illustrated by working out in detail one of the cases in the infinite group, in which the strip contains two series of equal holes symmetrically staggered along the strip. The stress function is constructed by using a class of periodic harmonic functions derived from Weierstrass’ sigma function. Numerical examples also are given to show the effect of such a perforation on the stresses in the strip.

Download Full-text

NON-SELF-ADJOINT EXTENSIONS OF THE LÉVY–LAPLACIAN AND THE LÉVY–LAPLACE EQUATION

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025706002238 ◽

2006 ◽

Vol 09 (01) ◽

pp. 67-76

Author(s):

OLEG O. OBREZKOV

Keyword(s):

Hilbert Space ◽

Eigenvalue Problem ◽

Gauge Field ◽

Laplace Equation ◽

Complex Plane ◽

Harmonic Functions ◽

Yang Mills ◽

Rigged Hilbert Space ◽

Lévy Laplacian

The aim of this paper is twofold. First, we continue the research of Accardi and Smolyanov5,8 and investigate the eigenvalue problem for the Lévy–Laplacian. We show that the spectrum of the Lévy–Laplacian coincides with the whole complex plane and that the Lévy–Laplacian admits various non-self-adjoint extensions — the latter result is due to the structure of the rigged Hilbert space, on which the Lévy–Laplacian is defined. Second, we obtain solutions of the Lévy–Laplace equation in a wide set of functions defined on the rigged Hilbert space. This result is especially interesting due to the recent finding of the relation between Yang–Mills gauge field and Lévy harmonic functions.4

Download Full-text

Classes of harmonic functions in 2D generalized Poincaré geometry

Filomat ◽

10.2298/fil2101287b ◽

2021 ◽

Vol 35 (1) ◽

pp. 287-297

Author(s):

Gabriel Bercu ◽

Mircea Crasmareanu

Keyword(s):

Half Plane ◽

Laplace Equation ◽

Harmonic Functions ◽

Separation Of Variables ◽

Ricci Soliton ◽

Totally Geodesic ◽

Upper Half Plane ◽

Bochner Formula ◽

Flat Metric

By using the additive and multiplicative separation of variables we find some classes of solutions of the Laplace equation for a generalization of the Poincar? upper half plane metric. Non-constant totally geodesic functions implies the flat metric and several examples are studied including the Hamilton?s cigar Ricci soliton. The Bochner formula is discussed for our generalized Poincar? metric and for its important particular cases.

Download Full-text