Sharp \(L^p\)-entropy inequalities on manifolds. (English) Zbl 1323.58012
Summary: In 2003, M. Del Pino and J. Dolbeault [J. Funct. Anal. 197, No. 1, 151–161 (2003; Zbl 1091.35029)] and I. Gentil [J. Funct. Anal. 202, No. 2, 591–599 (2003; Zbl 1173.35424)] investigated, independently, best constants and extremals associated to Euclidean \(L^p\)-entropy inequalities for \(p>1\).
In this work, we present some contributions in the Riemannian context.
Namely, let \((M,g)\) be a compact Riemannian manifold of dimension \(n\geq 3\). For \(1<p\leq 2\), we establish the validity of the sharp Riemannian \(L^p\)-entropy inequality
\[
\int\limits_M|u|^p\log(|u|^p)dv_g\leq\frac{n}{p}\log\biggl(\mathcal A_{\mathrm{opt}} \int\limits_M |\nabla_g u|^pdv_g+\mathcal B\biggr)
\]
on all functions \(u\in H^{1,p}(M)\) such that \(\| u\|_{L^p(M)}=1\) for some constant \(\mathcal B\).
Moreover, we prove that the first best constant \(\mathcal A_{\mathrm{opt}}\) is equal to the corresponding Euclidean one.
Our approach is inspired on the D. Bakry et al.’s idea [Indiana Univ. Math. J. 44, No. 4, 1032–1074 (1995; Zbl 0857.26006)] of getting Euclidean entropy inequalities as a limit case of suitable subcritical interpolation inequalities.
It is conjectured that the inequality sometimes fails for \(p>2\).
MSC:
| 58J05 | Elliptic equations on manifolds, general theory |
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 41A44 | Best constants in approximation theory |
| 35J87 | Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators |
| 26D10 | Inequalities involving derivatives and differential and integral operators |
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