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Fractals, trees and the Neumann Laplacian. (English) Zbl 0799.46041

Two sided estimates are given for the quantity \[ \beta (E) = \inf \bigl\{ \| E-P \| : P \in K \bigr\}, \] where \(E\) is the natural embedding of \(W^{1,p} (\Omega)\) into \(L^ p (\Omega)\) and \(K\) denotes the set of compact linear operators between these spaces, for a class of domains in \(R^ n\) whose boundaries are fractal in the sense that their Hausdorff, or Minkowski, dimensions lie in \((n-1,n]\). In particular, necessary and sufficient conditions for the compactness of \(E\) are included and this for domains which may not have the extension property. The domains considered have a central spine which is a tree and the method of approach involves the reduction of the problem to an equivalent one on this tree. Implications for the eigenvalues of the Neumann Laplacian are considered.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
35P20 Asymptotic distributions of eigenvalues in context of PDEs
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References:

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