The concavity of $p$-Rényi entropy power for doubly nonlinear diffusion equations and $L^p$-Gagliardo-Nirenberg-Sobolev inequalities

Abstract

We prove the concavity of $p$-Rényi entropy power for positive solutions to the doubly nonlinear diffusion equations on ${\mathbb{R}}^n$ or compact Riemannian manifolds with nonnegative Ricci curvature. As applications, we give new proofs of the sharp $L^p$-Sobolev inequality and $L^p$-Gagliardo-Nirenberg inequalities on ${\mathbb{R}}^n$. Moreover, two improvement of $L^p$-Gagliardo-Nirenberg inequalities are derived.

References in corpus (1)

• The concavity of $p$-entropy power and applications in functional inequalities

Cited by in corpus (1)

• On the Renyi entropy power and the Gagliardo-Nirenberg-Sobolev inequality on Riemannian manifolds