Potential Theory on Metric Measure Spaces

University Of Cincinnati Main Campus, Cincinnati OH

Investigators

Abstract

The Principal Investigator proposes to continue the program of developing analysis on metric spaces via three projects. The first project will study the trace spaces related to a Sobolev-type function space on general metric measure spaces that may not have a Riemannian structure; the idea here is to find out how well data have to be determined on the boundary of a given domain in order to obtain a reliable prediction of a corresponding extension of the data to the interior of the domain. The second project is to explore the conformal Martin boundary for bounded domains in metric measure spaces of bounded geometry. The conformal Martin boundary is in general different from the classical Martin boundary corresponding to the Laplacian operator, and provides a more reliable gauge of the potential-theoretic boundary relevant to certain non-linear partial differential equations. The third project is to construct a distributional derivative structure on general metric spaces and thereby measure the boundary of domains and determine when spheres in such a metric space are rectifiable. Potential applications of the research proposed in this project include connections between stochastic processes and analysis in abstract metric spaces. Abstract metric spaces arise in applications in physics and engineering, and hence the questions addressed in this project have possible impact in physics and engineering as well. In addition, these projects unify the theory of subelliptic equations of divergence form in Riemannian manifolds with the theory of degenerate elliptic equations in the sub-Riemannian geometry of Carnot-Carath\'eodory spaces. Also, the construction and study of conformal Martin boundary is a new concept even in the Euclidean setting, and will provide a new tool in the study of boundary behavior of quasiconformal mappings.