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

Kevin Arfi; Anna Rozanova-Pierrat

doi:10.3934/dcdss.2019001

Article Contents

2019, Volume 12, Issue 1: 1-26. Doi: 10.3934/dcdss.2019001

This issue Previous Article Preface Next Article Dual graphs and modified Barlow-Bass resistance estimates for repeated barycentric subdivisions

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

Kevin Arfi and
Anna Rozanova-Pierrat^,

Laboratoire de Mathématiques et Informatique pour la Complexité et les Systèmes, CentralSupélec, Université Paris Saclay, Bȃtiment Bouygues, 3 rue Joliot-Curie, 91190 Gif-sur-Yvette, France

* Corresponding author: Anna Rozanova-Pierrat
* Corresponding author: Anna Rozanova-Pierrat

Received: January 31, 2017

Revised: May 31, 2017

Published: July 2018

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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.

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Mathematics Subject Classification: Primary: 35J25, 46E35; Secondary: 47A10.

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Figure 1. Example of the considered domains: $\Omega_0$ (the von Koch snowflake) is the bounded domain, bounded by a compact boundary $\Gamma$, which is a $d$-set (see Definition 2.3) with $d = \log 4/ \log 3>n-1 = 1$. The truncated domain $\Omega_S$ is between the boundary $\Gamma$ and the boundary $S$ (presented by the same von Koch fractal as $\Gamma$). The boundaries $\Gamma$ and $S$ have no an intersection and here are separated by the boundary of a ball $B_r$ of a radius $r>0$. The domain, bounded by $S$, is called $\Omega_1 = \overline{\Omega}_0\cup \Omega_S$, and the exterior domain is $\Omega = \mathbb{R}^n\setminus \overline{\Omega}_0$

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