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

Abstract

In the framework of the Laplacian transport, described by a Robin boundary value problem in an exterior domain in R n , we generalize the definition of the Poincaré-Steklov operator to d -set boundaries, n − 2 < d < n , and give its spectral properties to compare to the spectra of the interior domain and also of a truncated domain, considered as an approximation of the exterior case. The well-posedness of the Robin boundary value problems for the truncated and exterior domains is given in the general framework of n -sets. The results are obtained thanks to a generalization of the continuity and compactness properties of the trace and extension operators in Sobolev, Lebesgue and Besov spaces, in particular, by a generalization of the classical Rellich-Kondrachov Theorem of compact embeddings for n and d -sets.

Optimal Absorption of Acoustic Waves by a Boundary

“…For the frequency model (2) it is possible to generalize the weak well-posedness result in domains with Lipschitz boundaries [25] to domains with a more general class of boundaries, named Ahlfors d-regular sets or simply d-sets [30] (see Appendix A), using functional analysis tools on "admissible domains" developed in [7]. The interest of this generalization is that this class of domains is optimal in the sense that it is the largest possible class [7] which keeps the Sobolev extension operators, for instance H 1 (Ω) to H 1 (R N ), continuous. In what follows, for our well-posedness result we take…”

“…We use, as in Refs. [7,40], the existence of a d-dimensional (0 < d ≤ N , d ∈ R) measure µ equivalent or equal to the Hausdorff measure m d on ∂Ω (see Definition 1) and a generalization of the usual trace theorem [30] (see Appendix A) and the Green formula [33,7] in the sense of the Besov space B 2,2 β (∂Ω)…”

“…It is described by the Helmholtz problem with a Robin boundary condition with a complex coefficient α, for which we give a well-posedness result on an admissible domain in R N in the sense of Ref. [7]. The existence of an optimal shape in the introduced acoustical framework is proved in Section 3.…”

“…In Section 2, we introduce the frequency model and its time-dependent analogue with a dissipation on the boundary. We analyze its dissipative properties and give the well-posedness results, due to [9,25] for at least Lipschitz boundaries, but we generalize the results for the Helmholtz problem in a larger class of domains with d-set boundaries using [7,40] (see Appendix A). This class, named in [7] "admissible domains" and containing for instance the Von Koch fractals, is composed of all Sobolev extension domains, thanks to results of [27], with boundaries on which it is possible to define a surjective linear continuous trace operator with linear continuous right inverse.…”

“…The existence of an optimal shape in the introduced acoustical framework is proved in Section 3. We recall useful results from [7] in Appendix A. Appendix B details how to obtain the damping parameter α in the Robin boundary condition that best approximates a given model with dissipation in the volume.…”