In this paper, we prove that the second eigenfunctions of the p- Laplacian, p &gt; 1, are not radial on the unit ball in RN, for any N ≥ 2. Our proof relies on the variational characterization of the second eigenvalue and a variant of the deformation lemma. We also construct an infinite sequence of eigenpairs {τn,Ψn} such that Ψn is nonradial and has exactly 2n nodal domains. A few related open problems are also stated. © 2015 American Mathematical Society.

Anoop Thazhe Veetil

Department of Mathematics

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IIT Madras

Proceedings of the American Mathematical Society

In 1961, Payne and Weinberger showed that ‘among the class of membranes with given area, free along the interior boundaries and fixed along the outer boundary of given length, the concentric annulus has the highest fundamental frequency.’ In the present study, we extend this result for the first eigenvalue of the p-Laplacian, for p∈(1,∞), in higher dimensional domains where the outer boundary is a fixed sphere. As an application, we prove that the nodal set of the second eigenfunctions of the p-Laplacian (with mixed boundary conditions) on a ball cannot be a concentric sphere. © 2019 Elsevier Inc.

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Journal	Proceedings of the American Mathematical Society
Publisher	American Mathematical Society
ISSN	00029939
Open Access	Yes