University of Cambridge > Talks.cam > Geometric Analysis and Partial Differential Equations seminar > Green's function estimates and the Poisson equation.

Green's function estimates and the Poisson equation.

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The Green’s function of the Laplace operator has been widely studied in geometric analysis. Manifolds admitting a positive Green’s function are called nonparabolic. By Li and Yau, sharp pointwise decay estimates are known for the Green’s function on nonparabolic manifolds that have nonnegative Ricci curvature. The situation is more delicate when curvature is not nonnegative everywhere. While pointwise decay estimates are generally not possible in this case, we have obtained sharp integral estimates for the Green’s function on manifolds admitting a Poincare inequality and an appropriate (negative) lower bound on Ricci curvature. This has applications to solving the Poisson equation, and to the study of the structure at infinity of such manifolds.

This talk is part of the Geometric Analysis and Partial Differential Equations seminar series.

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