In this paper we consider on a bounded domain B⊂Rn the eigenvalue problem with Dirichlet boundary conditions: (1) (−Δ)Ku=λu in B, u=Du=⋯=DK−1u=0 on ∂B, where K is a positive integer. If B is a ball then we prove that for K=2,3 the eigenfunctions of (1) are radial, depending strictly monotonically on the radius, and are of one sign. In the second result the first eigenvalue λ of (1) can be estimated by λ≥(λ1μ2)m when K=2m, λ≥λ1(λ1μ2)m when K=2m+1, where λ1 is the first eigenvalue with Dirichlet data, and μ2 is the first nonzero eigenvalue with Neumann data for the Laplace operator −Δ in B, respectively.

Remarks on the first eigenspace for polyharmonic operators

PUCCI, Patrizia;
1989

Abstract

In this paper we consider on a bounded domain B⊂Rn the eigenvalue problem with Dirichlet boundary conditions: (1) (−Δ)Ku=λu in B, u=Du=⋯=DK−1u=0 on ∂B, where K is a positive integer. If B is a ball then we prove that for K=2,3 the eigenfunctions of (1) are radial, depending strictly monotonically on the radius, and are of one sign. In the second result the first eigenvalue λ of (1) can be estimated by λ≥(λ1μ2)m when K=2m, λ≥λ1(λ1μ2)m when K=2m+1, where λ1 is the first eigenvalue with Dirichlet data, and μ2 is the first nonzero eigenvalue with Neumann data for the Laplace operator −Δ in B, respectively.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11391/127916
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