On explicit representation and approximations of Dirichlet-to-Neumann semigroup
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In his book (Functional Analysis, Wiley, New York, 2002), P. Lax constructs an explicit representation of the Dirichlet-to-Neumann semigroup, when the matrix of electrical conductivity is the identity matrix and the domain of the problem in question is the unit ball in ℝ n . We investigate some representations of Dirichlet-to-Neumann semigroup for a bounded domain. We show that such a nice explicit representation as in Lax book, is not possible for any domain except Euclidean balls. It is interesting that the treatment in dimension 2 is completely different than other dimensions. Finally, we present a natural and probably the simplest numerical scheme to calculate this semigroup in full generality by using Chernoff’s theorem.
KeywordsDirichlet-to-Neumann operator Lax’s Dirichlet-to-Neumann semigroup γ-harmonic lifting
We wish to thank Professor Ralph deLaubenfels who was the instigator of this method, for his collaboration with the first author which ends up with this paper.
- 1.Arendt, W., Ter Elst, A.F.M.: The Dirichlet-to-Neumann operator on rough domains. arXiv:1010.1703 (2010)