Sharp bounds for ground state eigenfunctions on domains with Horns and cusps
The authors consider the first eigenfunction of the Laplacian under Dirichlet boundary conditions for special ``horn shaped'' domains \(\Omega\): a) \(\Omega_p =\{(x,y) \in \mathbb{R} \times \mathbb{R}^{N-1};\;x>0,\;|y|<\rho(x)\}\), where \(\rho(x) \downarrow 0\) for \(x\to \infty\), \(\rho\in C'[0, \infty)\), \(\lim_{x\to\infty} |\rho' |=0\). b) The finite case: \(\Omega_\nu =\{(x,y) \in\mathbb{R} \times \mathbb{R}^{N-1}\), \(0<x<1\), \(|y|<\nu(x)\}\), where \(\nu(x) \downarrow 0\) for \(x\to 0\) and \(\lim_{x\to 0} |\nu' |=0\). They then prove lower bounds for the first eigenfunction \(\varphi(x)\) of \(\Omega\) in terms of the first eigenfunction of the unit ball of \(\mathbb{R}^{N-1}\).
scientific article; zbMATH DE number 1053991