The Dirichlet-to-Neumann operator for divergence form problems

Abstract

We present a way of defining the Dirichlet-to-Neumann operator on general Hilbert spaces using a pair of operators for which each one's adjoint is formally the negative of the other. In particular, we define an abstract analogue of trace spaces and are able to give meaning to the Dirichlet-to-Neumann operator of divergence form operators perturbed by a bounded potential in cases where the boundary of the underlying domain does not allow for a well-defined trace. Moreover, a representation of the Dirichlet-to-Neumann operator as a first-order system of partial differential operators is provided. Using this representation, we address convergence of the Dirichlet-to-Neumann operators in the case that the appropriate reciprocals of the leading coefficients converge in the weak operator topology. We also provide some extensions to the case where the bounded potential is not coercive and consider resolvent convergence.

Fibre homogenisation

“…That is, (N θγk ) k∈{1,...,n} = n r=1 d s=1 N (rs) θk γ rs k∈{1,...,n} , where N (rs) θ uniquely solves (6). Furthermore, since e ı θ,· C d C n×d ⊆ P (θ), (19) implies that β, η C n×d = e ı θ,· C d β, e ı θ,· C d η = ι * θ aι θ (e ı θ,· C d γ + grad θ N θγ ), e ı θ,· C d η = a(grad θ N θγ + e ı θ,· C d γ), e ı θ,…”

“…Let (θ k ) k∈N be convergent in Θ to some limit θ. We need to prove that for f ∈ [L 2 (Y )] 6 , then the sequence (T (θ k )f ) k∈N is weakly convergent in [L 2 (Y )] 6 to the limit T (θ)f . By Lemma 2.5, sup…”

Nonlocal H-convergence

“…We obtain that (u n ) n is bounded in dom(B); see also [40,Lemma 2.12] for the precise argument. Possibly choosing a subsequence (not relabelled) of (u n ) n , we may assume that u n ⇀ u in dom(B) for some u.…”

Continuous Dependence on the Coefficients II

“…The corresponding elliptic equation is then intensively studied. Even though it might be hidden in the derivations above, the 'study of the elliptic problem' essentially boils down to addressing the limit behaviour of a n as n → ∞; see [37,132]. Consequently, generalisations of the periodic case have been introduced.…”