Abstract
We give a representation of the solution of the Neumann problem for the Laplace operator on the n-dimensional unit ball in terms of the solution of an associated Dirichlet problem. The representation is extended to other operators besides the Laplacian, to smooth simply connected planar domains, and to the infinite-dimensional Laplacian on the unit ball of an abstract Wiener space, providing in particular an explicit solution for the Neumann problem in this case. As an application, we derive an explicit formula for the Dirichlet-to-Neumann operator, which may be of independent interest.
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The first author acknowledges the support from the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0045. The second author kindly acknowledges the support by a grant of the Romanian National Authority for Scientific Research, CNCS - UEFISCDI, project number PNII-ID-PCCE-2011-2-0015. The last author acknowledges the support by a grant of the College of Science and Mathematics at Kennesaw State University.
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Beznea, L., Pascu, M.N. & Pascu, N.R. An Equivalence Between the Dirichlet and the Neumann Problem for the Laplace Operator. Potential Anal 44, 655–672 (2016). https://doi.org/10.1007/s11118-015-9524-z
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DOI: https://doi.org/10.1007/s11118-015-9524-z
Keywords
- Dirichlet problem
- Neumann problem
- Laplace operator
- Infinite-dimensional Laplace operator
- Dirichlet-to-Neumann operator