Professor Ashok Aryal (MSUM) will speak in the Tri-College Colloquium at NDSU on Tuesday, November 28.
Title: The Mean value theorem for Laplacian and its generalizations to arbitrary divergence form uniformly elliptic operators.
Abstract: The Mean Value Theorem (MVT) for sub and superharmonic functions is a cornerstone in the theory of elliptic PDE. Luis Caffarelli stated a generalization of this theorem for the solid case to arbitrary divergence form uniformly elliptic operators. More recently, Ivan Blank and Zheng Hao filled in the details of the proof. Currently, not much is known about the geometry and topology of the family of underlying sets given in this theorem. In this talk I will introduce the MVT for Laplacian. I will then sketch the proof of the MVT for weakly subharmonic functions and describe the generalized MVT. I will then examine some results extending our understanding of the underlying sets in the generalized MVT.