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NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2015, 6 (1), P. 46–56

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

Konstantin Pankrashkin – Laboratoire de mathématiques, Université Paris-Sud Bâtiment 425, 91405 Orsay Cedex, France; konstantin.pankrashkin@math.u-psud.fr

Let Ω⊂ R² be the exterior of a convex polygon whose side lengths are l₁, . . . , l_M. For a real constant α, let H_α^Ω denote the Laplacian in Ω, u → -Δu with the Robin boundary conditions ∂u/∂v = αu at ∂Ω, where v is the outer unit normal. We show that, for any fixed m∈N, the mth eigenvalue E_m^Ω(α) of H_α^Ω behaves as E_m^Ω(α) = -α² + μ_m^D + O(α^-1/2) as α → +∞ where μ_m^D stands for the mth eigenvalue of the operator D₁ ⊕ . . . ⊕ D_M and D_n denotes the one-dimensional Laplacian f → -f” on (0,l_n) with the Dirichlet boundary conditions.

Keywords: eigenvalue asymptotics, Laplacian, Robin boundary condition, Dirichlet boundary condition.

PACS 41.20.Cv, 02.30.Jr, 02.30.Tb

DOI 10.17586/2220-8054-2015-6-1-46-56

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