中科院数学与系统科学研究院

数学研究所

中科院华罗庚数学重点实验室

华罗庚青年数学论坛

学术报告

报告人： 李林涵 博士（University of Minnesota）

题  目：Quantitative properties of the Green function for elliptic operators with non-smooth coefficients

时  间：2022.06.15（星期三），09:30-11:30

地  点：腾讯会议：897-960-843

摘  要：In the upper half-space, the distance function to the boundary is a positive solution to Laplace's equation that vanishes on the boundary, which can be interpreted as the Green function with pole at infinity for the Laplacian. We are interested in generalizing this result to a bigger class of operators and understanding the exact relations between the behavior of the Green function and the structure of the underlying operator. In joint work with G. David and S. Mayboroda, we obtain a precise and quantitative control of the proximity of the Green function and the distance function on the upper half-space by the oscillation of the coefficients of the operator. We shall see that the class of the operators that we consider is of the nature of the best possible for the Green function to behave like a distance function. We shall also discuss how this result can be generalized to other settings. If time permits, we shall discuss how this kind of results can be applied to derive optimal estimates for elliptic measures.