Remark on robin problem for Poisson equation

Heinrich Begehr, Saule Burgumbayeva, Bibinur Shupeyeva

Research output: Contribution to journalArticlepeer-review

6 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 languageEnglish
Pages (from-to)1-11
Number of pages11
JournalComplex Variables and Elliptic Equations
DOIs
Publication statusAccepted/In press - Mar 19 2017

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

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