Improved interpolation inequalities, relative entropy and fast diffusion equations
Jean Dolbeault
Ceremade (UMR CNRS no. 7534), Université Paris-Dauphine, Place de Lattre de Tassigny, F-75775 Paris Cédex 16, FranceGiuseppe Toscani
University of Pavia Department of Mathematics, Via Ferrata 1, 27100 Pavia, Italy
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Abstract
We consider a family of Gagliardo–Nirenberg–Sobolev interpolation inequalities which interpolate between Sobolevʼs inequality and the logarithmic Sobolev inequality, with optimal constants. The difference of the two terms in the interpolation inequalities (written with optimal constant) measures a distance to the manifold of the optimal functions. We give an explicit estimate of the remainder term and establish an improved inequality, with explicit norms and fully detailed constants. Our approach is based on nonlinear evolution equations and improved entropy–entropy production estimates along the associated flow. Optimizing a relative entropy functional with respect to a scaling parameter, or handling properly second moment estimates, turns out to be the central technical issue. This is a new method in the theory of nonlinear evolution equations, which can be interpreted as the best fit of the solution in the asymptotic regime among all asymptotic profiles.
Cite this article
Jean Dolbeault, Giuseppe Toscani, Improved interpolation inequalities, relative entropy and fast diffusion equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 5, pp. 917–934
DOI 10.1016/J.ANIHPC.2012.12.004