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May 7, 2018 | 2:45 p.m. - 4:00 p.m.
Category: Lecture
Location: Education, College of #179 | Map
5425 Gullen Mall
Detroit, MI 48202
Cost: Free

Speaker: Ivan Blank, Kansas State University

Title:  The Obstacle Problem and Mean Value Theorems for Divergence Form Elliptic PDEs

Abstract:  One of the basic properties of harmonic functions is the fact that at any point the value of the function is exactly equal to the average value of either a ball or a sphere centered at that point.  This theorem can be used to establish other basic properties of harmonic functions including the maximum principle, the Harnack inequality, and some important a priori estimates.  Standard proofs of this theorem rely on some of the smoothness and symmetry properties of the Laplacian and therefore do not generalize nicely to non-Euclidean spaces or more general elliptic operators.  On the other hand, in the Fermi lectures on the obstacle problem in 1998, Luis Caffarelli pointed out how one could use the solutions of specific obstacle problems in order to construct an elegant proof of the mean value theorem which does not depend on either the smoothness or the symmetry properties of the Laplacian.  With collaborators, I have given the details of this proof which shows that if L is a divergence form elliptic operator in Euclidean space or the Laplace-Beltrami operator on a Riemannian manifold, then at each point x_0, there is a nested family of mean value sets D_r(x_0) so that for any subsolution of Lu = 0 the average of u over the set is monotone increasing with r (just as the average of a subharmonic function over B_r(x_0) would be monotone increasing).  On the other hand, the study of the geometric, analytic, and even topological properties of these mean value sets is wide open.

For more information about this event, please contact Department of Mathematics at 3135772479 or math@wayne.edu.