Logarithmic Harnack inequalities
Let \(M\) be a \(d-\)dimensional smooth connected compact Riemannian manifold with non-negative Ricci curvature and let \(\nabla\) denote the gradient associated with the Laplace-Beltrami operator \(\Delta.\) The authors prove the following logarithmic Harnack inequality: \[ |\nabla f|^2+\alpha f^2 \log f^2 \leq \alpha \sup(f^2\log f^2) \] for \(f\) satisfying \[ \Delta f=-\alpha f \log f^2.\tag \(*\) \] Further, equation \((*)\) is satisfied by the function \(f,\) achieving the smallest constant \(\alpha\) (log-Sobolev constant) in the logarithmic Sobolev inequality: \[ \int_M f^2(x)\log f^2(x) \leq \alpha \int_M |\nabla f(x)|^2 , \qquad \int_M |f|^2 =\text{vol } M. \] In addition, \(\alpha \geq \min \{\lambda_1/8e, d^{-1} D(M)^{-2}\},\) where \(\lambda_1\) is the first eigenvalue of the Laplacian and \(D(M)\) denotes the diameter of \(M.\) Similar results are proved for some graphs.
scientific article; zbMATH DE number 991916