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

Claudia Anedda;  Fabrizio Cuccu — 2015

Applicationes Mathematicae

Let D₀=x∈ ℝ²: 0<|x|<1 be the unit punctured disk. We consider the first eigenvalue λ₁(ρ ) of the problem Δ² u =λ ρ u 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 ∫ D ₀ ρ d x = α γ + β ( | D ₀ | - γ ) for a fixed 0 < γ < |D₀|. We will be concerned with the minimization problem ρ ↦ λ₁(ρ). We show that, under suitable conditions on α, β and γ, the minimizer does not inherit the radial symmetry of the domain.

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