On multiplicity of eigenvalues and symmetry of eigenfunctions of the $p$-Laplacian

Abstract

Abstract. 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 1,p 0 (Ω) with suitably high Krasnosel'skiȋ genus. In particular, if Ω is a ball B ⊂ R N , we obtain the following chain of inequalities:Here λi(p; B) are variational eigenvalues of the p-Laplacian on B, and λ (p; B) is the eigenvalue which has an associated eigenfunction whose nodal set is an equatorial section of B. If λ2(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.

Conformal upper bounds for the eigenvalues of the p‐Laplacian

“…Results concerning the first positive eigenvalue and Neumann boundary conditions are available in [16,22] for Euclidean domains. Concerning higher eigenvalues, results in the case of Dirichlet conditions for Euclidean domains can be found in [1,4,36] (continuity and limits with respect to p, multiplicity, etc.). As for Riemannian manifolds, we mention [12,30,32,35,41] for sharp estimates for the first positive eigenvalue in case of compact manifolds or domains and Neumann boundary conditions.…”

Conformal upper bounds for the eigenvalues of the $p$-Laplacian

“…Results concerning the first positive eigenvalue and Neumann boundary conditions are available in [16,22] for Euclidean domains. Concerning higher eigenvalues, results in the case of Dirichlet conditions for Euclidean domains can be found in [1,4,36] (continuity and limits with respect to p, multiplicity, etc.). As for Riemannian manifolds, we mention [12,30,32,35,41] for sharp estimates for the first positive eigenvalue in case of compact manifolds or domains and Neumann boundary conditions, under a given lower bound on the Ricci curvature.…”

Approximation of the second eigenvalue of the $p$-Laplace operator in symmetric domains

“…• The Courant theorem implies that in the linear case p = 2, the number of nodal domains of an eigenfunction associated to λ 2 is exactly 2. • In [2,3] is shown that the second eigenfunctions are not radial in ball. The limiting cases p → 1 and p → ∞ are more complicated and requires tools from non smooth critical point theory and the concept viscosity solutions.…”

“…The eigenvalues and the corresponding eigenfunctions of the p-Laplace operator have been much discussed in the literature, due to mathematical challenges, open questions cf. [3,13,21,22], and regarding related applications in image processing we refer to [14,16]. In particular, the second eigenvalue and eigenfunction have been studied extensively, see [1,11,24].…”