On the stability of the Caffarelli–Kohn–Nirenberg inequality

Wei, Juncheng; Wu, Yuanze

doi:10.1007/s00208-021-02325-0

Cited by 10 publications

(6 citation statements)

References 62 publications

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“…By using the finite dimensional reduction method, Ciraolo et al [7], Figalli and Glaudo [24], Deng et al [15] proved the sharp stability of profile decomposition for critical Sobolev inequality along Struwe's fundamental result in [45]. Then Wei and Wu [50] generalized the special case a = b = 0(Sobolev inequality) to (CKN) inequality in a proper parameter region and gave the remainder term. Piccione et al [43] Inspired by [20], we give an improved version of the nonlocal Sobolev inequality (1.3).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Clinical value of the emergency department in screening and diagnosis of COVID-19 in China

Zhang

Pan

Zhao

et al. 2020

J. Zhejiang Univ. Sci. B

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Studying the implications and characterizations of the excited state quantum phase transitions (ESQPTs) would enable us to understand various phenomena observed in quantum many body systems. In this work, we delve into the affects and characterizations of the ESQPTs in the anharmonic Lipkin-Meshkov-Glick (LMG) model by means of the entropy of the quantum work distribution. The entropy of the work distribution measures the complexity of the work distribution and behaves as a valuable tool for analyzing nonequilibrium work statistics. We show that the entropy of the work distribution captures salient signatures of the underlying ESQPTs in the model. In particular, a detailed analyses of the scaling behavior of the maximal entropy verifies that it acts as a witness of the ESQPTs. We further demonstrate that the entropy of the work distribution also reveals the features of the ESQPTs in the energy space and can be used to determine their critical energies. Our results provide further evidence of the usefulness of the entropy of the work distribution for investigating various phase transitions in quantum many body systems and open up a promising way for experimentally exploring the signatures of ESQPTs.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Clinical value of the emergency department in screening and diagnosis of COVID-19 in China

Zhang

Pan

Zhao

et al. 2020

J. Zhejiang Univ. Sci. B

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“…Combining inequalities ( 30), ( 33), ( 34), ( 35) and (36) proves inequality (29). Notice that the dependence in ε comes from the C µ norm of u(t, x)−B(t, x) /B(t, x) where the denominator is estimated by its supremum on the domain |x| ≤ 2 ρ(ε) R(t).…”

Section: 1mentioning

confidence: 55%

Constructive stability results in interpolation inequalities and explicit improvements of decay rates of fast diffusion equations

Bonforte¹,

Dolbeault²,

Nazaret³

et al. 2022

Preprint

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We provide a scheme of a recent stability result for a family of Gagliardo-Nirenberg-Sobolev (GNS) inequalities, which is equivalent to an improved entropy -entropy production inequality associated with an appropriate fast diffusion equation (FDE) written in self-similar variables. This result can be rephrased as an improved decay rate of the entropy of the solution of (FDE) for well prepared initial data. There is a family of Caffarelli-Kohn-Nirenberg (CKN) inequalities which has a very similar structure. When the exponents are in a range for which the optimal functions for (CKN) are radially symmetric, we investigate how the methods for (GNS) can be extended to (CKN). In particular, we prove that the solutions of the evolution equation associated to (CKN) also satisfy an improved decay rate of the entropy, after an explicit delay. However, the improved rate is obtained without assuming that initial data are well prepared, which is a major difference with the (GNS) case.

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“…Recently, Ciraolo et al [9], Figalli and Glaudo [22], and Deng et al [18] obtained sharp quantitative estimates of Struwe's decomposition (see [38]) for Sobolev inequality (1.1) by using the finite-dimensional reduction method. It is worth mentioning that Wang and Willem [41] obtained the remainder terms of Caffarelli-Kohn-Nirenberg inequalities (see [13]), Wei and Wu [42] established the stability of the profile decompositions to the Caffarelli-Kohn-Nirenberg inequalities and also gave the gradient-type remainder terms. Recently, there have been some important works worth mentioning regarding the study of Caffarelli-Kohn-Nirenberg inequalities.…”

Section: Motivationmentioning

confidence: 99%

Remainder terms of a nonlocal Sobolev inequality

Deng,

Tian,

Yang

et al. 2023

Mathematische Nachrichten

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In this note, we study a nonlocal version of the Sobolev inequality where is the best constant, * denotes the standard convolution and denotes the classical Sobolev space with respect to the norm . By using the nondegeneracy property of the extremal functions, we prove that the existence of the gradient type remainder term and a reminder term in the weak ‐norm of above inequality for all with .

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On the stability of the Caffarelli–Kohn–Nirenberg inequality

Cited by 10 publications

References 62 publications

Clinical value of the emergency department in screening and diagnosis of COVID-19 in China

Clinical value of the emergency department in screening and diagnosis of COVID-19 in China

Constructive stability results in interpolation inequalities and explicit improvements of decay rates of fast diffusion equations

Remainder terms of a nonlocal Sobolev inequality

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