Connections Between the Dirichlet and the Neumann Problem for Continuous and Integrable Boundary Data
We present results concerning the 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. We show that the representation holds in the case of integrable boundary data, thus providing an explicit solution of the generalized solution of the Neumann problem.
KeywordsDirichlet problem Dirichlet-to-Neumann operator Infinite-dimensional Laplace operator Laplace operator Neumann problem
1991 Mathematics Subject Classification31B05 31B10 42B37 35J05 35J25
The first author acknowledges support from the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-III-P4-ID-PCE-2016-0372. 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.
- 1.D.H. Armitage, S.J. Gardiner, Classical Potential Theory. Springer Monographs in Mathematics (Springer, London, 2001)Google Scholar
- 9.L.L. Helms, Introduction to Potential Theory, Pure and Applied Mathematics, vol. XXII. (Robert E. Krieger Publishing Co., Huntington, 1975)Google Scholar
- 10.V. Isakov, Inverse Problems for Partial Differential Equations. Applied Mathematical Sciences, vol. 127, 2nd edn. (Springer, New York, 2006)Google Scholar