2/16/2007

Yan Yan Li
Rutgers University

Some Liouville theorem and gradient estimate

The classical Liouville theorem says that a positive entire harmonic function must be a constant. We give a fully nonlinear version of it. This extension enables us to establish local gradient estimates of solutions to general conformally invariant fully nonlinear elliptic equations of second order. This talk will start from a proof of the classical Liouville theorem using only the comparison principle and the invariance of harmonicity under Mobius transformations and scalar multiplications. We will then outline the proof of the comparison principle used in establishing the new Liouville theorem. Finally we outline the proof of the gradient estimates via the Liouville theorem.