Fractals, trees and the Neumann Laplacian (Q1318071)

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.