Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by \(d\)-sets. (English) Zbl 1416.35095
Summary: 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.
MSC:
| 35J25 | Boundary value problems for second-order elliptic equations |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 47A10 | Spectrum, resolvent |
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