NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2013, 4 (4), P. 474–483
ON THE ASYMPTOTICS OF THE PRINCIPAL EIGENVALUE FOR A ROBIN PROBLEM WITH A LARGE PARAMETER IN PLANAR DOMAINS
Konstantin Pankrashkin – Laboratoire de mathématiques – UMR 8628, Université Paris-Sud, Bâtiment 425, 91405 Orsay Cedex, France; konstantin.pankrashkin@math.u-psud.fr
http://www.math.u-psud.fr/~pankrash/
Let Ω⊂R2 be a domain having a compact boundary Σ which is Lipschitz and piecewise C4 smooth, and let ν denote the inward unit normal vector on Σ. We study the principal eigenvalue E(β) of the Laplacian in Ω with the Robin boundary conditions ∂f /∂ν + βf = 0 on Σ, where β is a positive number. Assuming that Σ has no convex corners, we show the estimate E(β) = −β2− γmaxβ + O(β⅔) as β → +∞, where γmax is the maximal curvature of the boundary.
Keywords: eigenvalue, Laplacian, Robin boundary condition, curvature, asymptotics.
PACS 41.20.Cv, 02.30.Jr, 02.30.Tb