Journal
JOURNAL OF SPECTRAL THEORY
Volume 13, Issue 1, Pages 383-394Publisher
EUROPEAN MATHEMATICAL SOC-EMS
DOI: 10.4171/JST/446
Keywords
Laplace eigenfunctions; spherical harmonics; symmetry conjecture; Legendre polynomials; Bessel functions
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We prove that the 2-sphere does not have symmetry of Lp norms of eigenfunctions of the Laplacian for p > 6, which addresses a question posed by Jakobson and Nadirashvili. In other words, there exists a sequence of spherical eigenfunctions *n, with eigenvalues An approaching infinity as n approaches infinity, such that the ratio of the Lp norms of the positive and negative parts of the eigenfunctions does not converge to 1 when p > 6. Our proof relies on fundamental properties of the Legendre polynomials and Bessel functions of the first kind.
We show that the 2-sphere does not exhibit symmetry of Lp norms of eigenfunctions of the Laplacian for p > 6, which answers a question of Jakobson and Nadirashvili. In other words, there exists a sequence of spherical eigenfunctions *n, with eigenvalues An oo as n oo, such that the ratio of the Lp norms of the positive and negative parts of the eigenfunctions does not tend to 1 as n oo when p > 6. Our proof relies on fundamental properties of the Legendre polynomials and Bessel functions of the first kind.
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