Dirichlet-to-Neumann maps on bounded Lipschitz domains

Abstract

The Dirichlet-to-Neumann map associated to an elliptic partial differential equation becomes multivalued when the underlying Dirichlet problem is not uniquely solvable. The main objective of this paper is to present a systematic study of the Dirichlet-to-Neumann map and its inverse, the Neumann-to-Dirichlet map, in the framework of linear relations in Hilbert spaces. Our treatment is inspired by abstract methods from extension theory of symmetric operators, utilizes the general theory of linear relations and makes use of some deep results on the regularity of the solutions of boundary value problems on bounded Lipschitz domains.

The Dirichlet-to-Neumann operator for divergence form problems

“…In the present work we also consider removing the coercivity condition on m. That is to say, we define the abstract analogue of the Dirichlet-to-Neumann graph with m being possibly not coercive. We note here that these results are the abstract counterpart of results developed in [BE1] and [AEKS]. In the case that the potentials m are not coercive we consider resolvent convergence for Dirichlet-to-Neumann operators.…”

Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by <i>d</i>-sets

“…We will discuss it in details in the next section. From [9], we also have that Ker A = {0} , since 0 is the eigenvalue of the Neumann eigenvalue problem for the Laplacian. For the Weil asymptotic formulas for the distribution of the eigenvalues of the Dirichlet-to-Neumann operator there are results for bounded smooth compact Riemannian manifolds with C ∞ boundaries [18], for polygons [19] and more general class of plane domains [17] and also for a bounded domain with a fractal boundary [39].…”

Hilbert transform for the three-dimensional Vekua equation

“…Intimately related to the Hilbert transform is the Dirichlet-to-Neumann (D-N) operator [7], which plays a fundamental role in the study of elliptic partial differential equations. In the rest of this article we restrict to the case p = 2 and work in domains Ω with Lipschitz boundary.…”

“…By(20),tr + F ∂Ω [h 0 ] = α 0 + H[α 0 ]. Therefore T 0,Ω [ g] + − → F 1,∂Ω [h 0 ] is monogenic.Following the same argument used in the proof of Theorem 4.4 of[20], we see thatD w = DT Ω [−g 0 + g] − D(T 0,Ω [ g] + − → F 1,∂Ω [h 0 ]) = −g 0 + g.The function w is purely vectorial becauseF ∂Ω [ α − H[α 0 ]] = −F ∂Ω [h 0 ], Sc F ∂Ω [ α − H[α 0 ]] = −T 0,Ω [ g] = −F 0,∂Ω [h 0 ].Note that(7) applied to the function T Ω [ g], and the fact thatDT Ω [ g] = g yield F ∂Ω [α 0 + α] = 0, so tr F ∂Ω [ α − H[α 0 ]] = − tr F ∂Ω [α 0 + H[α 0 ]] = −α 0 − H[α 0 ],…”