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

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
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