Parabolic methods for ultraspherical interpolation inequalities

Jean Dolbeault; An Zhang

doi:10.3934/dcds.2022080

Article Contents

2023, Volume 43, Issue 3&4: 1347-1365. Doi: 10.3934/dcds.2022080

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Parabolic methods for ultraspherical interpolation inequalities

Jean Dolbeault^1, and
An Zhang^2, ,

1.
Ceremade, UMR CNRS n° 7534, Université Paris-Dauphine, PSL Research University, Place de Lattre de Tassigny, 75775 Paris Cedex 16, France
2.
School of Mathematical Sciences, Beihang University, No. 37 Xueyuan Road, Haidian, Beijing 100191, China

^* Corresponding author: An Zhang
^* Corresponding author: An Zhang

It is with great pleasure that we dedicate this paper to Juan Luis Vázquez on the occasion of his 75^th birthday.

Received: February 2022

Early access: June 22, 2022

Published online: June 22, 2022

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Abstract

The carré du champ method is a powerful technique for proving interpolation inequalities with explicit constants in presence of a non-trivial metric on a manifold. The method applies to some classical Gagliardo-Nirenberg-Sobolev inequalities on the sphere, with optimal constants. Very nonlinear regimes close to the critical Sobolev exponent can be covered using nonlinear parabolic flows of porous medium or fast diffusion type. Considering power law weights is a natural question in relation with symmetry breaking issues for Caffarelli-Kohn-Nirenberg inequalities, but regularity estimates for a complete justification of the computation are missing. We provide the first example of a complete parabolic proof based on a nonlinear flow by regularizing the singularity induced by the weight. Our result is established in the simplified framework of a diffusion built on the ultraspherical operator, which amounts to reduce the problem to functions on the sphere with simple symmetry properties.

Keywords:

Gagliardo-Nirenberg-Sobolev inequalities,
Caffarelli-Kohn-Nirenberg inequalities,
interpolation,
sphere,
flows,
optimal constants,
weights,
ultraspherical operator,
carré du champ method,
entropy methods,
nonlinear parabolic equations,
porous media,
fast diffusion,
regularity.

Mathematics Subject Classification: Primary: 58J35, 26D10; Secondary: 35K65, 47D07, 35B65.

Citation:

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Figure 1. Admissible ranges of $ m $ for $ n = 0.25 $, $ 1 $, $ 1.8 $ (first line, from the left to the right) and $ n = 2 $, $ 3 $, $ 4 $ (second line). The dotted and dashed curves correspond respectively to $ m = n/(n+2) $ and $ m = (n-2)/n $. In Lemma 4, Condition (21) amounts to $ (n-2)/n<m<1 $ and $ m\neq n/(n+2) $

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