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, . . . , lM. For a real constant α , let HαΩ 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 EmΩ(α) of HαΩ behaves as EmΩ(α) = -α2 + μmD + O(α-1/2) as α → +∞ where μmD stands for the mth eigenvalue of the operator D1 ⊕ . . . ⊕ DM and Dn denotes the one-dimensional Laplacian f → -f” on (0,ln) 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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