Dirichlet-to-Neumann or Poincar\'e-Steklov operator on fractals described by d -sets

TL;DR

This paper extends the Poincaré-Steklov operator to fractal boundaries modeled as d-sets, analyzing its spectral properties and well-posedness of Robin boundary problems in exterior domains with generalized trace and extension operators.

Contribution

It introduces a generalized Poincaré-Steklov operator for d-set boundaries and studies its spectral properties, expanding the analysis to fractal-like geometries.

Findings

Spectral properties of the generalized operator are characterized.

Well-posedness of Robin boundary problems is established for n-sets.

Trace and extension operators are generalized for fractal boundaries.

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 Poincar\'e-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.