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 |
|---|---|
| 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
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 journal › Article
}
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 -