Monotonicity results for the principal eigenvalue of the generalized Robin problem. (English) Zbl 1089.35038

Giorgi, Tiziana; Smits, Robert G.

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.

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