The sharp Sobolev inequality in quantitative form

By Andrea Cianchi, N. Fusco, Francesco Maggi, Aldo Pratelli. 2009

171 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/2560ed062c24bd4d2a245eefcc3631852c203517

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Citations in the corpus

Remainder terms in the fractional Sobolev inequality (2012, unknown): “, [8,13]), optimal transportation (see, e.”
Some methods for studying stability in isoperimetric type problems (2008, background): “There are many relevant variational problems [PS, Ka] intimately related to the same symmetrization inequalities studied here from a quantitative point of view, and these ideas have already been shown to be flexible enough to be employed in the study of quantitative forms of other variational proble”
A Sharp Stability Result for the Relative Isoperimetric Inequality Inside Convex Cones (2012, background): “In general, sets for which τ(E) > 1 enjoy a nontrivial Sobolev-Poincaré type inequality (see [16, Lemma 3.1]).”
Quantitative logarithmic Sobolev inequalities and stability estimates (2014, result): “But ∫ R |(log f)′ − b|2dν is the relative Fisher information of ν with respect to the non-centered Gaussian dγb = e b.x−b2/2dγ which satisfies a logarithmic Sobolev inequality with constant 12 .”
Sharp gradient stability for the Sobolev inequality (2020, methodology): “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”
Stability Results for Logarithmic Sobolev and Gagliardo–Nirenberg Inequalities (2014, background): “Variants of the Pinsker-CsiszárKullback inequality can be found in Carrillo and Toscani [2000], Del Pino and Dolbeault [2002] and allow to control ‖u−BM,σ[u]‖L1(Rd) under additional assumptions.”
Stability for a GNS inequality and the Log-HLS inequality, with application to the critical mass Keller–Segel equation (2011, background): “Moreover, other recent proofs for stability based on optimal transportation [17] or symmetrization techniques [19, 20, 13, 18] did not produce (at least up to now) any results in this situation.”
New approach to the affine P\'olya-Szeg\"o principle and the stability version of the affine Sobolev inequality (2015, background): “We refer the reader to [2, 3, 4, 5, 10, 11, 12, 19, 21, 22, 23, 24, 25, 26] and references therein for more background and results in this direction.”
A fractional Yamabe flow and some applications (2011, background): “, [16, 15, 6, 25, 20, 31]), which give various lower bounds of the lef t-handed side of (12).”
Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence (2022, background): “In fact, stability has attracted a lot of attention in recent years and we refer to [44, 26, 39, 27, 23, 30, 21, 24, 38, 25, 61, 42, 43, 40, 10] and the references within for a list of works in this direction.”
Higher-order Sobolev embeddings and isoperimetric inequalities (2013, background): “The contributions to this field now constitute the corpus of a vast literature, which includes the papers [1, 4, 6, 10, 11, 15, 16, 20, 22, 24, 27, 31, 33, 39, 40, 45, 47, 49, 50, 52, 53, 54, 59, 60, 68, 89] and the monographs [17, 19, 21, 48, 67, 78].”