Giorgi, Tiziana; Smits, Robert G. Monotonicity results for the principal eigenvalue of the generalized Robin problem. (English) Zbl 1089.35038 Ill. J. Math. 49, No. 4, 1133-1143 (2005). Summary: We study domain monotonicity of the principal eigenvalue \(\lambda_1^\Omega(\alpha)\) corresponding to \(\Delta u=\lambda(\alpha) \, u \text{ in } \Omega, \frac{\partial u}{\partial \nu} =\alpha\, u \text{ on } \partial \Omega\), with \(\Omega \subset \mathbb{R}^n\) a \(C^{0,1}\) bounded domain, and \(\alpha\) a fixed real. We show that contrary to intuition domain monotonicity might hold if one of the two domains is a ball. Cited in 1 ReviewCited in 16 Documents MSC: 35P15 Estimates of eigenvalues in context of PDEs 35J25 Boundary value problems for second-order elliptic equations 49R50 Variational methods for eigenvalues of operators (MSC2000) Keywords:domain monotonicity; principal eigenvalue; bounded domain PDF BibTeX XML Cite \textit{T. Giorgi} and \textit{R. G. Smits}, Ill. J. Math. 49, No. 4, 1133--1143 (2005; Zbl 1089.35038) Full Text: Link OpenURL