Estimates on the modulus of expansion for vector fields solving nonlinear equations

In this article, by extending the method of [AC] we prove a sharp estimate on the expansion modulus of the gradient of the logarithm of the parabolic kernel to the Schördinger operator with convex potential on a bounded convex domain. The result improves an earlier work of Brascamp-Lieb which asserts the log-concavity of the parabolic kernel. We also give an alternate proof to a corresponding estimate on the first eigenfunction of the Schördinger operator, obtained first by Andrews and Clutterbuck via the study of the asymptotics to a parabolic problem. Our proof is more direct via an elliptic maximum principle. An alternate proof of the fundamental gap theorem of [AC], by considering the quotient of moduli of continuity, is also obtained. Moreover we derive a Neumann eigenvalue comparison result and some other lower estimates on the first Neumann eigenvalue for Laplace operator with a drifting term, including an explicit estimate on a conjecture of P. Li.