What is it about?
The Robin boundary condition for the Laplace operator is formulated as a natural interpolation between Dirichlet and Neumann conditions. While the Dirichlet problem is unconditionally solvable the other ones are not. The solvability conditions, which were explicitly known just for particular domains, are formulated for arbitrary domains having a Green function. It is also explicitly shown how the Neumann function can be altered such that the Neumann problem becomes unconditionally solvable.
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Why is it important?
In order to get a harmonic Robin function interpolating the Green and Neumann functions a modification is required. But then the entire theory becomes homogeneous and esthetic, a general requirement for a persuading mathematical concept.
Perspectives
Although the theory of fundamental solutions to the Laplace operator is classical the knowledge about the third, the Robin boundary value problem is still not complete. The present paper is some contribution to get more mathematicians interested in this older subject.
Prof. Dr. Heinrich G.W. Begehr
Freie Universitat Berlin
Read the Original
This page is a summary of: Remark on Robin problem for Poisson equation, Complex Variables and Elliptic Equations, March 2017, Taylor & Francis,
DOI: 10.1080/17476933.2017.1303052.
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