Spectral problems on arbitrary open subsets of \(\mathbb R^n\) involving the distance to the boundary (Q2493963)

Let \(\Omega\) be an open set in \(\mathbb R^n\) and let \(\rho (x) =\text{dist} (x, \partial \Omega)\) be the distance of \(x \in \Omega\) to the boundary \(\partial \Omega\) (\(\not= \emptyset\)). Let \(W^1_p (\Omega; \mu, \theta)\) be the weighted Sobolev spaces normed by \[ \| f \, | W^1_p (\Omega; \mu, \theta) \| = \| \rho^\mu \nabla f \, | L_p (\Omega) \| + \| \rho^{- \theta} f \, | L_p (\Omega) \| , \quad 1 \leq p < \infty, \] where \(\nu, \theta \geq 0\). The authors study under which conditions (on \(\Omega\), \(\mu, \theta\)), the natural embedding \[ E_{\mu, \theta} : W^1_p (\Omega; \mu, \theta) \hookrightarrow L_p (\Omega) \] is compact and what can be said about upper and lower estimates for the corresponding approximation numbers. Let \(T_{\mu, \theta} (\Omega, N)\) be the self-adjoint operator (Neumann Laplacian) generated by the quadratic form \[ \| f \, | W^1_2 (\Omega; \mu, \theta) \| ^2 = \| \rho^\mu \nabla f \, | L_2 (\Omega) \| ^2 + \| \rho^{-\theta} f \, | L_2 (\Omega) \| . \] (Similarly for the Dirichlet Laplacian \(T_{\mu,\theta} (\Omega, D)\)). Applying their results about approximation numbers, the authors obtain estimates from above and from below for the distribution of the related eigenvalues (spectral counting function).