Symmetry breaking in the minimization of the first eigenvalue for the composite clamped punctured disk
Volume 42 / 2015
Applicationes Mathematicae 42 (2015), 183-191
MSC: 35J40, 47A75.
DOI: 10.4064/am42-2-5
Abstract
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.
