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Nonlinear Analysis

On Green functions of second-order elliptic operators on Riemannian Manifolds: the critical case

Yehuda Pinchover: Technion-Israel Institute of Technology

Location:  Hill 425
Date & time: Wednesday, 08 February 2017 at 1:40PM - 1:41PM

Let P be a second-order, linear, elliptic operator with real coefficients which is defined on a noncompact and connected Riemannian manifold M. It is well known that the equation Pu=0 in M admits a positive supersolution which is not a solution if and only if P admits a unique positive minimal Green function on M, and in this case, P is said to be subcritical in M. If P does not admit a positive Green function but admits a global positive solution, then such a solution is called a ground state of P in M, and P is said to be critical in M.

We prove for a critical operator P in M, the existence of a Green function which is dominated above by the ground state of P away from the singularity. Moreover, in a certain class of Green functions, such a Green function is unique, up to an addition of a product of the ground states of P and P^*. This result extends and sharpens the celebrated result of Peter Li and Luen-Fai Tam concerning the existence of a symmetric Green function for the Laplace-Beltrami operator on a smooth and complete Riemannian manifold M.

This is a joint work with Debdip Ganguly.

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