Weighted Gagliardo-Nirenberg inequalities via Optimal Transport Theory and Applications

2022/05/18 by Balogh, Zoltán M., Don, Sebastiano, Kristály, Alexandru · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2205.09051

Abstract

We prove Gagliardo-Nirenberg inequalities with three weights -- verifying a joint concavity condition -- on open convex cones of \mathbb Rⁿ. If the weights are equal to each other the inequalities become sharp and we compute explicitly the sharp constants. For a certain range of parameters we can characterize the class of extremal functions; in this case, we also show that the sharpness in the main three-weighted Gagliardo-Nirenberg inequality implies that the weights must be equal up to some constant multiplicative factors. Our approach uses optimal mass transport theory and a careful analysis of the joint concavity condition of the weights. As applications we establish sharp weighted p-log-Sobolev, Faber-Krahn and isoperimetric inequalities with explicit sharp constants.

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• The weighted isoperimetric inequality and Sobolev inequality outside convex sets

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