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Nonconvexity and compact containment of mean value sets for second order uniformly elliptic operators in divergence form

Niles Armstrong, Kansas State University
Tuesday, May 28, 2019 - 1:30pm
PDL C-401

In his Fermi Lectures on the obstacle problem, Caffarelli stated a mean value theorem for second order uniformly elliptic divergence form operators with the form \$L:=D_i a^{ij}(x) D_j.\$ This theorem is a clear analog to the standard mean value theorem for Euclidean balls for the Laplacian, with the only difference being the sets over which the averages are taken. I will discuss the initial regularity results that were known for these sets, show a new compact containment result, and finally give an example of an operator with smooth coefficients and nonconvex mean value sets.

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