Abstract
We consider the eigenvalue problem with Robin boundary condition ∆u + λu = 0 in Ω, ∂u/∂ν + αu = 0 on ∂Ω, where Ω ⊂ Rn , n ≥ 2 is a bounded domain with a smooth boundary, ν is the outward unit normal, α is a real parameter. We obtain two terms of the asymptotic expansion of simple eigenvalues of this problem for α → +∞. We also prove an estimate to the difference between Robin and Dirichlet eigenfunctions.
Highlights
We consider the eigenvalue problem with Robin boundary condition ∆u + λu = 0 in Ω, ∂u/∂ν + αu = 0 on ∂Ω, where Ω ⊂ Rn, n ≥ 2 is a bounded domain with a smooth boundary, ν is the outward unit normal, α is a real parameter
Let Ω ⊂ Rn, n ≥ 2 be a bounded domain with boundary Γ of class C3
These results indicate that the asymptotic behavior of λ1(α) for α → −∞ is strongly affected by the smoothness of the boundary and C1 class is optimal for the equality (1.8)
Summary
Let Ω ⊂ Rn, n ≥ 2 be a bounded domain with boundary Γ of class C3. Consider the eigenvalue problem. 1, and in particular when Ω ⊂ R2 is a triangle with inner half-angles α1, α2, α3 the authors have proved that lim α→−∞. These results indicate that the asymptotic behavior of λ1(α) for α → −∞ is strongly affected by the smoothness of the boundary and C1 class is optimal for the equality (1.8). With Hmax = max H(x) where H(x) is the mean curvature of the surface Γ orix∈Γ ented by inner normal at the point x It was proved in [17], that if, Γ ∈ C4, the reminder estimate O(|α|−4/3) can be replaced by O(|α|−3/2). Let us note the result of [7], where the lower estimate for the derivative of the first eigenvalue λ1(α) were obtained: lim inf λ1(α) ≥ 1. α→−∞ −α
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