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

TL;DR

This paper establishes the concavity of $p$-Rényi entropy power for certain nonlinear diffusion equations and uses this to provide new proofs and improvements of classical $L^p$-Sobolev and Gagliardo-Nirenberg inequalities.

Contribution

It proves the concavity of $p$-Rényi entropy power for nonlinear diffusion equations and derives new, sharper forms of $L^p$-Gagliardo-Nirenberg inequalities.

Findings

Proved concavity of $p$-Rényi entropy power for solutions to nonlinear diffusion equations.

Provided new proofs of sharp $L^p$-Sobolev and Gagliardo-Nirenberg inequalities.

Derived two improved versions of $L^p$-Gagliardo-Nirenberg inequalities.

Abstract

We prove the concavity of $p$-R\'enyi 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.