Stability for the Sobolev inequality with explicit constants

Abstract

A quantitative version of the Bianchi-Egnell inequality concerning the stability of the Sobolev inequality is proved for non-negative functions. For the proof we study a flow that interpolates continuously between a function and its symmetric decreasing rearrangement. introduction and main resultIn [9] Brezis and Lieb posed the question whether it is possible to bound the 'Sobolev deficit'from below in terms of some natural distance from the manifold of optimizers. Here d ≥ 3, 2 * = 2 d/(d − 2) is the 'Sobolev exponent', and, its distributional gradient is a square-summable function and it vanishes at infinity in the sense that |{x ∈ R d : |f (x)| > }| < ∞ for all > 0. Here |A| denotes the Lebesgue measure of a measurable set A. Throughout this paper we deal with real-valued functions. With minor additional effort our arguments can be extended to the case of complex-valued functions.Rodemich [39], Aubin [3] and Talenti [43] (see also [41]) proved that the Sobolev deficit is non-negative. Moreover, it was shown by Lieb [35], Gidas, Ni and Nirenberg [34] and Caffarelli, Gidas and Spruck [14] that the deficit vanishes if and only if the function f is of the form

On the sharp constant in the Bianchi-Egnell stability inequality

“…Theorem 1 shows that the study of the sharp constant c BE (s) cannot be reduced to a local analysis near the manifold M. This phenomenon is analogous to the situation for the planar isoperimetric inequality and its associated stability inequality; we refer to the introduction of [9] for more details and references about this case.…”

“…Since the proof of (1.4) in [4,6] proceeds by compactness, it yields no explicit lower bound on the constant c BE (s). For s = 1, a constructive lower bound on c BE (1) is proved in the recent preprint [9].…”

On the stability of the Caffarelli–Kohn–Nirenberg inequality

“…We also want to mention the paper [9], where Carlen develop a dual method to establish stability of functional inequalities. Finally we want to mention is that in the very recent papers [60][61][62], Konig proved that s BE is attainable which gives a positive answer to the open question proposed by Dolbeault, Esteban, Figalli, Frank and Loss in [28] and makes the key step in answering the long-standing open question of determining the best constant s BE . Konig's result on s BE has been generalized to c BE in the nondegenerate case in our very recent paper [79].…”

The optimal constant in the $L^{2}$ Folland-Stein inequality on the H-type group

“…We note that the eigenvalues of (1.8) play an important role in the study of stability for the Folland-Stein inequality (see [1,3,7] for the case of Euclidean space). In Lemma 3.2 we show that the embedding map…”