Asymptotics of Eigenfunctions on Plane Domains
نویسندگان
چکیده
We consider a family of domains (ΩN )N>0 obtained by attaching an N×1 rectangle to a fixed set Ω0 = {(x, y) : 0 < y < 1, −φ(y) < x < 0}, for a Lipschitz function φ ≥ 0. We derive full asymptotic expansions, as N → ∞, for the mth Dirichlet eigenvalue (for any fixed m ∈ N) and for the associated eigenfunction on ΩN . The second term involves a scattering phase arising in the Dirichlet problem on the infinite domain Ω∞. We determine the first variation of this scattering phase, with respect to φ, at φ ≡ 0. This is then used to prove sharpness of results, obtained previously by the same authors, about the location of extrema and nodal line of eigenfunctions on convex domains.
منابع مشابه
MAT 280: Laplacian Eigenfunctions: Theory, Applications, and Computations Lectures 12+13: Laplacian Eigenvalue Problems for General Domains: IV. Asymptotics of the Eigenvalues
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