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Article Dans Une Revue International Mathematics Research Notices Année : 2019

Generalized logarithmic Hardy-Littlewood-Sobolev inequality

Résumé

This paper is devoted to logarithmic Hardy-Littlewood-Sobolev inequalities in the two-dimensional Euclidean space, in presence of an external potential with logarithmic growth. The coupling with the potential introduces a new parameter, with two regimes. The attractive regime reflects the standard logarithmic Hardy-Littlewood-Sobolev inequality. The second regime corresponds to a reverse inequality, with the opposite sign in the convolution term, that allows us to bound the free energy of a drift-diffusion-Poisson system from below. Our method is based on an extension of an entropy method proposed by E. Carlen, J. Carrillo and M. Loss, and on a nonlinear diffusion equation.
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Dates et versions

hal-02281279 , version 1 (09-09-2019)
hal-02281279 , version 2 (21-11-2019)
hal-02281279 , version 3 (24-12-2019)

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Jean Dolbeault, Xingyu Li. Generalized logarithmic Hardy-Littlewood-Sobolev inequality. International Mathematics Research Notices, inPress, ⟨10.1093/imrn/rnz324⟩. ⟨hal-02281279v3⟩
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