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

Abstract

We prove the concavity of p-Rényi entropy power for positive solutions to the doubly nonlinear diffusion equations on 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 R n . Moreover, two improvement of L p -Gagliardo-Nirenberg inequalities are derived. (2010). Primary 58J35, 35K92; Secondary 35B40,35K55 Mathematics Subject Classification

On the p-Renyi entropy power for nonlinear diffusion equations on Riemannian manifolds

“…Motivated by the works of [8,6,3,4], in [11] and [10], the first author and coauthors studied the concavities of the p-Shannon entropy power and the p-Rényi entropy power for positive solutions to the p-heat equations ∂ t u p−1 = (p − 1) p−1 ∆ p u and the doubly nonlinear diffusion equations on Riemannian manifolds with nonnegative Ricci curvature, respectively. More precisely, let u be a positive solution to the doubly nonlinear diffusion equation…”

“…on compact Riemanian manifolds without boundary, where According to [2] and [10], we define the entropy functional…”

“…Remark 1.4. The mainly difference between [10] and this paper is that we introduce two A-trace-free tensors B f and T f in the variational formula of p-Renyi entropy power.…”

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

“…To end this paper, we would like to mention that Wang et al [48,49] proved the concavity of the Renyi entropy power for the p-Laplacian equation and the double nonlinear diffusion equations on compact Riemannian manifolds. Our work on the entropy power concavity inequalities in Theorem 2.1, Theorem 2.2 and Theorem 2.3 has been done in our 2017 preprint [24] and is independent of [48,49].…”