Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d -sets

Abstract

In the framework of the Laplacian transport, described by a Robin boundary value problem in an exterior domain in ${\mathbb{R}}^n$, 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.

References in corpus (2)

• Characterization of traces of smooth functions on Ahlfors regular sets
• Spectral geometry of the Steklov problem