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ON THE ROBIN EIGENVALUES OF THE LAPLACIAN IN THE EXTERIOR OF A CONVEX POLYGON

Аннотация:

Let Ω ⊂ R 2 be the exterior of a convex polygon whose side lengths are `1, . . . , `M. For a real constant α, let HΩ α denote the Laplacian in Ω, u 7→ −∆u, with the Robin boundary conditions ∂u/∂ν = αu at ∂Ω, where ν is the outer unit normal. We show that, for any fixed m ∈ N, the mth eigenvalue EΩ m(α) of HΩ α behaves as EΩ m(α) = −α 2 + µ D m + O(α −1/2 ) as α → +∞, where µ D m stands for the mth eigenvalue of the operator D1⊕· · ·⊕DM and Dn denotes the one-dimensional Laplacian f 7→ −f 00 on (0, `n) with the Dirichlet boundary conditions. 

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