Inequalities for the Minimal Eigenvalue of the Laplacian in an Annulus
نویسندگان
چکیده
We discuss the behavior of the minimal eigenvalue λ of the Dirichlet Laplacian in the domainD1\D2 := D (an annulus) whereD1 is a circular disc andD2 ⊂ D1 is a smaller circular disc. It is conjectured that the minimal eigenvalue λ has a maximum value when D2 is a concentric disc. If h is a displacement of the center of the disc D2 and λ(h) is the corresponding minimal eigenvalue, then dλ(h) dh < 0 so that λ(h) is minimal when ∂D2 touches ∂D1, where ∂D is the boundary of D. Numerical results are given to back the conjecture. Upper and lower bounds are given for λ(h). The above conjecture is proved.
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