A note on strong-form stability for the Sobolev inequality

Neumayer, Robin

doi:10.1007/s00526-019-1686-x

Cited by 25 publications

(17 citation statements)

References 36 publications

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“…In the last two decades there has been an abundance of stability results for various functional inequalities. Examples include, for instance, isoperimetric inequalities [31,25,17,21], L p -Sobolev inequalities [11,27,37,28], fractional Sobolev inequalities [13], Gagliardo-Nirenberg inequalities [6], Brunn-Minkowski, concentration and rearrangement inequalities [24,23,26,15,30], eigenvalue inequalities [36,10,7,33,1], solutions to elliptic equations with critical exponents [12,22,18], Young's inequality [16], Hausdorff-Young inequality [14], etc. Many of these works use strategies inspired by the paper of Bianchi-Egnell and in essentially all works (exceptions being [28,23] and one version of a refined Hölder inequality in [10]) the remainder term is quadratic in the distance to the optimizers.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Degenerate stability of some Sobolev inequalities

Frank¹

2021

Preprint

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We show that on S 1 (1/ √ d − 2) × S d−1 (1) the conformally invariant Sobolev inequality holds with a remainder term that is the fourth power of the distance to the optimizers. The fourth power is best possible. This is in contrast to the more usual vanishing to second order and is motivated by work of Engelstein, Neumayer and Spolaor. A similar phenomenon arises for subcritical Sobolev inequalities on S d . Our proof proceeds by an iterated Bianchi-Egnell strategy.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Degenerate stability of some Sobolev inequalities

Frank¹

2021

Preprint

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“…The existence of C is obtained by concentration-compactness methods and by contradiction, hence comes with no estimate. Various refinements starting with [38] and the more recent results of [61,82,62] have been obtained, although without explicit and constructive estimates on C . By duality methods inspired by [77] and flows, quantitative and constructive results were obtained in weak norms in [45,56].…”

Section: Sobolev and Some Other Interpolation Inequalitiesmentioning

confidence: 99%

Functional inequalities: nonlinear flows and entropy methods as a tool for obtaining sharp and constructive results

Dolbeault¹

2021

Preprint

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Interpolation inequalities play an essential role in Analysis with fundamental consequences in Mathematical Physics, Nonlinear Partial Differential Equations (PDEs), Markov Processes, etc., and have a wide range of applications in various other areas of Science. Research interests have evolved over the years: while mathematicians were originally focussed on abstract properties (for instance appropriate notions of functional spaces for the existence of weak solutions in PDEs), more qualitative questions (for instance, bifurcation diagrams, multiplicity of the solutions in PDEs and their qualitative behaviour) progressively emerged. The use of entropy methods in nonlinear PDEs is a typical example: in some cases, the optimal constant in the inequality can be interpreted as an optimal rate of decay of an entropy for an associated evolution equation. Much more has been learned by adopting this point of view.This paper aims at illustrating some of these recent aspect of entropyentropy production inequalities, with applications to stability in Gagliardo-Nirenberg-Sobolev inequalities and symmetry results in Caffarelli-Kohn-Nirenberg inequalities. Entropy methods provide a framework which relates nonlinear regimes with their linearized counterparts. This framework allows to prove optimality results, symmetry results and stability estimates. Some emphasis will be put on the hidden structure which explain such properties. Related open problems will be listed.

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“…In the past years inequalities of geometric-functional type have been widely studied in the literature, a-far from complete-list is [21,24,26,28] (isoperimetric inequalities), [19,44] (anisotropic Wulff inequalities), [1,2,16] (Gaussian inequalities), [23,30,32] (Riesz inequalities), [13,14,15,27,45] (Sobolev inequalities), [6,22,29] (Faber-Krahn inequalities, see also [5] for an account on other quantitative spectral inequalities).…”

Section: Introductionmentioning

confidence: 99%