Symmetry breaking in the minimization of the first eigenvalue for the composite clamped punctured disk

Claudia Anedda ¹   ; Fabrizio Cuccu ¹

¹ Mathematics and Computer Science Department University of Cagliari 09124 Cagliari (CA), Italy

Applicationes Mathematicae, Tome 42 (2015) no. 2-3, pp. 183-191

Résumé

Let $D_0=\{x\in \mathbb {R}^2: 0|x|1\}$ be the unit punctured disk. We consider the first eigenvalue $\lambda _1(\rho )$ of the problem $\Delta ^2 u =\lambda \rho u$ in $D_0$ with Dirichlet boundary condition, where $\rho $ is an arbitrary function that takes only two given values $0\alpha \beta $ and is subject to the constraint $\int _{D_0}\rho \,dx=\alpha \gamma +\beta (|D_0|-\gamma )$ for a fixed $0\gamma |D_0|$. We will be concerned with the minimization problem $\rho \mapsto \lambda _1(\rho )$. We show that, under suitable conditions on $\alpha ,\ \beta $ and $\gamma $, the minimizer does not inherit the radial symmetry of the domain.

Détail

DOI : 10.4064/am42-2-5

Keywords: mathbb unit punctured disk consider first eigenvalue lambda rho problem delta lambda rho dirichlet boundary condition where rho arbitrary function takes only given values alpha beta subject constraint int rho alpha gamma beta gamma fixed gamma concerned minimization problem rho mapsto lambda rho under suitable conditions alpha beta gamma minimizer does inherit radial symmetry domain