\(L_p\)-spectral properties of the Neumann Laplacian on horns, comets and stars (Q1433406)

The author considers the Neumann Laplacian for a domain \(Q\subset \mathbb{R}^2\) and studies the spectral properties in \(L_p(\Omega)\). He considers e.g. domains \(\Omega\) given by \(\Omega = \{(x,y)\in\mathbb{R}^2| x > 0\), \((\varphi_1(x) < y < \varphi_2(x)\}\). Possible candidates would be \(\varphi_{1,2}(x) = \pm e^{-x}\). A general assumption is \(\lim_{x\to \infty}\varphi(x) = 0\), \(\varphi'\in c_0(L_{\infty})\). Other shapes are considered as well. He then proves results on the spectrum.