# Remark on robin problem for Poisson equation

Heinrich Begehr, Saule Burgumbayeva, Bibinur Shupeyeva

Research output: Contribution to journalArticle

4 Citations (Scopus)

### Abstract

The two basic boundary value problems for the Poisson equation are the Dirichlet and the Neumann problems. Their related fundamental solutions are the harmonic Green and Neumann functions. A linear combination of these two boundary value problems is the Robin problem. The related fundamental solution, the Robin function, can be chosen as an interpolation between the Green and the Neumann functions. For plane domains this was done in [Begehr H, Vaitekhovich T. Modified harmonic Robin functions. Complex Variables Elliptic Equ. 2013;58:483–496] with complex notation. While the Dirichlet problem is unconditionally solvable this is in general not the case for the other ones. In explicit form, however, this was shown only for the unit disc. Here the solvability condition is given for any admissible plane domain. Proper alteration of the fundamental solution can make the solvability condition disappear. This is shown explicitly for the case of the upper half of the unit disc (Formula presented.).

Original language English 1-11 11 Complex Variables and Elliptic Equations https://doi.org/10.1080/17476933.2017.1303052 Accepted/In press - Mar 19 2017

### Fingerprint

Robin Problem
Poisson equation
Fundamental Solution
Poisson's equation
Neumann function
Solvability Conditions
Unit Disk
Boundary value problems
Boundary Value Problem
Harmonic functions
Complex Variables
Neumann Problem
Harmonic Functions
Notation
Dirichlet Problem
Dirichlet
Linear Combination
Green's function
Interpolation
Harmonic

### Keywords

• half disc
• harmonic Robin function
• parqueting-reflection principle
• plane domains
• Poisson equation
• solvability of Neumann and Robin problems

### ASJC Scopus subject areas

• Analysis
• Numerical Analysis
• Computational Mathematics
• Applied Mathematics

### Cite this

Remark on robin problem for Poisson equation. / Begehr, Heinrich; Burgumbayeva, Saule; Shupeyeva, Bibinur.

In: Complex Variables and Elliptic Equations, 19.03.2017, p. 1-11.

Research output: Contribution to journalArticle

@article{ef2d8e5be3e44ff08f6dc702691ca602,
title = "Remark on robin problem for Poisson equation",
abstract = "The two basic boundary value problems for the Poisson equation are the Dirichlet and the Neumann problems. Their related fundamental solutions are the harmonic Green and Neumann functions. A linear combination of these two boundary value problems is the Robin problem. The related fundamental solution, the Robin function, can be chosen as an interpolation between the Green and the Neumann functions. For plane domains this was done in [Begehr H, Vaitekhovich T. Modified harmonic Robin functions. Complex Variables Elliptic Equ. 2013;58:483–496] with complex notation. While the Dirichlet problem is unconditionally solvable this is in general not the case for the other ones. In explicit form, however, this was shown only for the unit disc. Here the solvability condition is given for any admissible plane domain. Proper alteration of the fundamental solution can make the solvability condition disappear. This is shown explicitly for the case of the upper half of the unit disc (Formula presented.).",
keywords = "half disc, harmonic Robin function, parqueting-reflection principle, plane domains, Poisson equation, solvability of Neumann and Robin problems",
author = "Heinrich Begehr and Saule Burgumbayeva and Bibinur Shupeyeva",
year = "2017",
month = "3",
day = "19",
doi = "10.1080/17476933.2017.1303052",
language = "English",
pages = "1--11",
journal = "Complex Variables and Elliptic Equations",
issn = "1747-6933",
publisher = "Taylor and Francis",

}

TY - JOUR

T1 - Remark on robin problem for Poisson equation

AU - Begehr, Heinrich

AU - Burgumbayeva, Saule

AU - Shupeyeva, Bibinur

PY - 2017/3/19

Y1 - 2017/3/19

N2 - The two basic boundary value problems for the Poisson equation are the Dirichlet and the Neumann problems. Their related fundamental solutions are the harmonic Green and Neumann functions. A linear combination of these two boundary value problems is the Robin problem. The related fundamental solution, the Robin function, can be chosen as an interpolation between the Green and the Neumann functions. For plane domains this was done in [Begehr H, Vaitekhovich T. Modified harmonic Robin functions. Complex Variables Elliptic Equ. 2013;58:483–496] with complex notation. While the Dirichlet problem is unconditionally solvable this is in general not the case for the other ones. In explicit form, however, this was shown only for the unit disc. Here the solvability condition is given for any admissible plane domain. Proper alteration of the fundamental solution can make the solvability condition disappear. This is shown explicitly for the case of the upper half of the unit disc (Formula presented.).

AB - The two basic boundary value problems for the Poisson equation are the Dirichlet and the Neumann problems. Their related fundamental solutions are the harmonic Green and Neumann functions. A linear combination of these two boundary value problems is the Robin problem. The related fundamental solution, the Robin function, can be chosen as an interpolation between the Green and the Neumann functions. For plane domains this was done in [Begehr H, Vaitekhovich T. Modified harmonic Robin functions. Complex Variables Elliptic Equ. 2013;58:483–496] with complex notation. While the Dirichlet problem is unconditionally solvable this is in general not the case for the other ones. In explicit form, however, this was shown only for the unit disc. Here the solvability condition is given for any admissible plane domain. Proper alteration of the fundamental solution can make the solvability condition disappear. This is shown explicitly for the case of the upper half of the unit disc (Formula presented.).

KW - half disc

KW - harmonic Robin function

KW - parqueting-reflection principle

KW - plane domains

KW - Poisson equation

KW - solvability of Neumann and Robin problems

UR - http://www.scopus.com/inward/record.url?scp=85015661444&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85015661444&partnerID=8YFLogxK

U2 - 10.1080/17476933.2017.1303052

DO - 10.1080/17476933.2017.1303052

M3 - Article

AN - SCOPUS:85015661444

SP - 1

EP - 11

JO - Complex Variables and Elliptic Equations

JF - Complex Variables and Elliptic Equations

SN - 1747-6933

ER -