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Pré-Publication, Document De Travail Année : 2022

Interpolation inequalities on the sphere: rigidity, branches of solutions, and symmetry breaking

Résumé

This paper is devoted to three Gagliardo-Nirenberg-Sobolev interpolation inequalities on the sphere. We are interested in branches of optimal functions when a scale parameter varies and investigate whether optimal functions are constant, or not. In the latter case, a symmetry breaking phenomenon occurs and our goal is to decide whether the threshold between symmetry and symmetry breaking is determined by a spectral criterion or not, that is, whether it appears as a perturbation of the constants or not. The first inequality is classical while the two other inequalities are variants which reproduce patterns similar to those observed in Caffarelli-Kohn-Nirenberg inequalities, for weighted inequalities on the Euclidean space. In the simpler setting of the sphere, it is possible to implement a parabolic version of the entropy methods associated to nonlinear diffusion equations, which is so far an open question on weighted Euclidean spaces.
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Dates et versions

hal-03834676 , version 1 (30-10-2022)

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  • HAL Id : hal-03834676 , version 1

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Esther Bou Dagher, Jean Dolbeault. Interpolation inequalities on the sphere: rigidity, branches of solutions, and symmetry breaking. 2022. ⟨hal-03834676⟩
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