ON THE ASYMPTOTICS OF THE PRINCIPAL EIGENVALUE FOR A ROBIN PROBLEM WITH A LARGE PARAMETER IN PLANAR DOMAINS
Аннотация:
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 + Oβ 2 3as β → +∞, where γmax is the maximalcurvature of the boundary.
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