Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method

By Matteo Bonforte, Jean Dolbeault, Bruno Nazaret, N. Simonov. 2020

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The purpose of this work is to establish a quantitative and constructive stability result for a class of subcritical Gagliardo-Nirenberg-Sobolev inequalities which interpolates between the logarithmic Sobolev inequality and the standard Sobolev inequality (in dimension larger than three), or Onofri's inequality in dimension two. We develop a new strategy, in which the flow of the fast diffusion equation is used as a tool: a stability result in the inequality is equivalent to an improved rate of convergence to equilibrium for the flow. The regularity properties of the parabolic flow allow us to connect an improved entropy - entropy production inequality during an initial time layer to spectral properties of a suitable linearized problem which is relevant for the asymptotic time layer. Altogether, the stability in the inequalities is measured by a deficit which controls in strong norms (a Fisher information which can be interpreted as a generalized Heisenberg uncertainty principle) the distance to the manifold of optimal functions. The method is constructive and, for the first time, quantitative estimates of the stability constant are obtained, including in the critical case of Sobolev's inequality. To build the estimates, we establish a quantitative global Harnack principle and perform a detailed analysis of large time asymptotics by entropy methods.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7963a9502cbbb70f950b3d8de956952bd9bda8ce

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

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.”