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

CUCCU, FABRIZIO;ANEDDA, CLAUDIA

2015-01-01

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Abstract

Let D0={x∈R2:0<|x|<1} be the unit punctured disk. We consider the first eigenvalue λ1(ρ) of the problem Δ2u=λρu in D0 with Dirichlet boundary condition, where ρ is an arbitrary function that takes only two given values 0 <α<β and is subject to the constraint ∫d0ρdx="αγ+β(|D0|−γ)" for a fixed 0<γ<|d0|. we will be concerned with minimization problem ρ↦λ1(ρ). show that, under suitable conditions on α, β γ, minimizer does not inherit radial symmetry of domain. < div>