### 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 |
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Pages (from-to) | 1-11 |

Number of pages | 11 |

Journal | Complex Variables and Elliptic Equations |

DOIs | |

Publication status | Accepted/In press - Mar 19 2017 |

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### 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

*Complex Variables and Elliptic Equations*, 1-11. https://doi.org/10.1080/17476933.2017.1303052

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

Research output: Contribution to journal › Article

*Complex Variables and Elliptic Equations*, pp. 1-11. https://doi.org/10.1080/17476933.2017.1303052

}

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

SP - 1

EP - 11

JO - Complex Variables and Elliptic Equations

JF - Complex Variables and Elliptic Equations

SN - 1747-6933

ER -