On multiplicity of eigenvalues and symmetry of eigenfunctions of the p-Laplacian
Résumé
We investigate multiplicity and symmetry properties of higher eigenvalues and eigenfunctions of the p-Laplacian under homogeneous Dirichlet boundary conditions on certain symmetric domains Ω ⊂ R^N. By means of topological arguments, we show how symmetries of Ω help to construct subsets of W_0^(1,p)(Ω) with suitably high Krasnosel'ski˘ ı genus. In particular, if Ω is a ball B ⊂ R^N , we obtain the following chain of inequalities: λ_2(p; B) ≤ · · · ≤ λ_(N+1)(p; B) ≤ λ_eq(p; B). Here λ_i(p; B) are variational eigenvalues of the p-Laplacian on B, and λ_eq(p; B) is the eigenvalue which has an associated eigenfunction whose nodal set is an equatorial section of B. If λ_2(p; B) = λ(p; B), as it holds true for p = 2, the result implies that the multiplicity of the second eigenvalue is at least N. In the case N = 2, we can deduce that any third eigenfunction of the p-Laplacian on a disc is nonradial. The case of other symmetric domains and the limit cases p = 1, p = ∞ are also considered.
Domaines
Topologie géométrique [math.GT]Origine | Fichiers produits par l'(les) auteur(s) |
---|
Loading...