The sharp Gagliardo–Nirenberg–Sobolev inequality in quantitative form

Nguyen, Van Hoang

doi:10.1016/j.jfa.2019.02.016

Cited by 10 publications

(6 citation statements)

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“…Motivated by important applications to problems in the calculus of variations and evolution PDEs, in recent years there has been a growing interest around the understanding of quantitative stability for functional/geometric inequalities, see for instance [3,2,8,27,28,21,9,22,29,18,10,6,7,11,13,19,23,35,26,5,14,16,17,20,25,30,31,24,33,34], as well as the survey papers [15,26,17]. Following this line of research, in this paper we shall investigate the stability of minimizers to the classical Sobolev inequality.…”

Section: Introductionmentioning

confidence: 99%

Gradient stability for the Sobolev inequality: the case $p\geq 2$

Figalli¹,

Neumayer

2018

J. Eur. Math. Soc.

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Section: Introductionmentioning

confidence: 99%

Gradient stability for the Sobolev inequality: the case $p\geq 2$

Figalli¹,

Neumayer

2018

J. Eur. Math. Soc.

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“…This idea is used first by E. Carlen and A. Figalli in [18] and later expanded by F. Seuffert in [82] and V.H. Nguyen in [77]. Constants are anyway non-constructive as they rely on [7].…”

Section: Quantitative Stability Resultsmentioning

confidence: 99%

Hardy-Littlewood-Sobolev and related inequalities: stability

Dolbeault¹,

Esteban²

2022

Preprint

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The purpose of this text is twofold. We present a review of the existing stability results for Sobolev, Hardy-Littlewood-Sobolev (HLS) and related inequalities. We also contribute to the topic with some observations on constructive stability estimates for (HLS).It is with great pleasure that we dedicate this paper to Elliott Lieb on the occasion of his 90th birthday. A short review of some functional inequalitiesFunctional inequalities play a very important role in various fields of mathematics, ranging from geometry, analysis, and probability theory to mathematical physics. For many problems, the precise value of the best constants matters and was actively studied, often in relation with the explicit knowledge of the optimizers. A standard scheme goes as follows: by rearrangement and symmetrisation, optimality is reduced to a smaller class of functions, for instance, to radial functions. After proving that the equality case is achieved, the Euler-Lagrange equations are solved by ODE techniques, which allows to classify the optimal functions and compute the best constants. This is the strategy of E.H. Lieb in [72] for the Hardy-Littlewood-Sobolev inequality which can be written as

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“…We refer to the survey papers [56,60,71] for a general discussion and presentation of the results and we refer to [29,30,58,[61][62][63][64]69,72] for the study of the stability for isoperimetric inequalities and to [34, 37-39, 46, 81, 86-88] (see also the survey [31]) for the study of the stability of constant mean curvature hypersurfaces (i.e. the critical points of the classical isoperimetric inequality (1.7) and these results are motivated by the celebrated Alexandrov soap bubbles theorem in [1] and [2]); moreover we refer to [59,65,79,80] for the study of the stability of the Brunn-Minkowski inequality, to [23,53,96,103] for the stability of the Gagliardo-Nirenberg inequality and to [8,22,24,26,54,66,73,75] (besides the already cited papers) for further stability results related to the Sobolev inequality (in the fractional case or for p = 1).…”

Section: Quantitative Studiesmentioning

confidence: 99%