Displaying similar documents to “Symmetrization of functions and principal eigenvalues of elliptic operators”

On the nodal set of the second eigenfunction of the laplacian in symmetric domains in R N

Lucio Damascelli (2000)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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We present a simple proof of the fact that if Ω is a bounded domain in R N , N ≥ 2 , which is convex and symmetric with respect to k orthogonal directions, 1 ≤ k ≤ N , then the nodal sets of the eigenfunctions of the laplacian corresponding to the eigenvalues λ 2 , ⋯ , λ k + 1 must intersect the boundary. This result was proved by Payne in the case N = 2 for the second eigenfunction, and by other authors in the case of convex domains in the plane, again for the second eigenfunction.

Eigenvalues of the p -Laplacian in 𝐑 N with indefinite weight

Yin Xi Huang (1995)

Commentationes Mathematicae Universitatis Carolinae

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We consider the nonlinear eigenvalue problem - div ( | ∇ u | p - 2 ∇ u ) = λ g ( x ) | u | p - 2 u in 𝐑 N with p > 1 . A condition on indefinite weight function g is given so that the problem has a sequence of eigenvalues tending to infinity with decaying eigenfunctions in W 1 , p ( 𝐑 N ) . A nonexistence result is also given for the case p ≥ N .