Sharp L^p-entropy inequalities on manifolds

\small{In 2004, Del Pino and Dolbeault \cite{DPDo} and Gentil \cite{G} investigated, independently, best constants and extremals associated to sharp Euclidean $L^p$-entropy inequalities. In this work, we present some important advances in the Riemannian context. Namely, let $(M,g)$ be a compact Riemannian manifold of dimension $n\ge 3$. For $1<p\le 2$, we prove that the sharp Riemannian $L^p$-entropy inequality $${\int }_M\mathrm{\mid }u{\mathrm{\mid }}^p\log (\mathrm{\mid }u{\mathrm{\mid }}^p)d{v}_g\le \frac{n}{p}\log ({\mathcal{A}}_{opt}{\int }_M\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}_g^{p}d{v}_g+\mathcal{B})$$ \n holds on all functions $u\in H^{1,p}(M)$ such that $\mathrm{\mid }\mathrm{\mid }u\mathrm{\mid }{\mathrm{\mid }}_{L^p(M)}=1$. Moreover, we show that the first best Riemannian constant ${\mathcal{A}}_{opt}$ is equal to the corresponding Euclidean one. Our approach is inspired on the Bakry, Coulhon, Ledoux and Sallof-Coste’s idea \cite{Ba} of getting Euclidean entropy inequalities as a limit case of suitable Gagliardo-Nirenberg inequalities. It is conjectured that the above inequality sometimes fails for $p>2$.}