Uniform Rectifiability and Elliptic Equations

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

This project lies within the field of harmonic analysis and its application to, and interaction with, geometric measure theory and the theory of elliptic partial differential equations. Roughly speaking, in harmonic analysis one investigates properties of functions and "operators" (i.e., mappings that transform one function into another) by decomposing them into smaller, constituent pieces, which are easier to understand, and then reassembling the pieces. The name itself arose by analogy with the decomposition of a musical sound into its various frequency components ("harmonics"). Geometric measure theory involves the study of the relationship between geometric properties of sets, and their "measures" (the latter are generalizations of the notions of length, area, and volume). Partial differential equations and systems of elliptic type describe a wide variety of phenoma in the real world, including electrostatics, steady-state temperature distributions, and elastic deformations. A particular focus of the present project is to explore further the relationship between geometry and the behavior of solutions to these differential equations that arise in the physical world. The project has two main areas of focus. First, the principal investigator plans to characterize quantitative rectifiability properties of a closed set E, of codimension one in d-dimensional Euclidean space, in terms of the behavior of harmonic functions (and of solutions of linear and quasi-linear elliptic equations more generally) in the complement of E. In particular, a primary goal of the proposal is to prove theorems of F. and M. Riesz type, and converses, without imposing any connectivity assumptions on either the set E or its complement. In the classical F. and M. Riesz Theorem, and its modern descendants, one obtains existence of, and in some cases quantitative estimates for, the Poisson kernel (i.e., the Radon-Nykodym derivative of harmonic measure for a domain D, with respect to arclength or surface measure on the boundary of D), as a consequence of rectifiability properties of the boundary. In the presence of suitable connectivity hypotheses, say when the set E is the boundary of a domain D enjoying an appropriate quantitative version of path connectedness, the geometry of E can be characterized in terms of the behavior of the Poisson kernel associated to D. On the other hand, a counterexample of Bishop and Jones precludes such results in the absence of connectivity. Thus, to prove theorems of F. and M. Riesz type without connectivity hypotheses will entail finding estimates for solutions of elliptic equations, estimates that serve as appropriate substitutes for Poisson kernel regularity. In the second area of focus of this project, the principal investigator plans to continue to investigate solvability of elliptic boundary value problems, in particular, the Neumann problem, for divergence form, second-order elliptic equations with "radially"-independent coefficients in the half-space, without assuming self-adjointness of the coefficient matrix. Previous work of the principal investigaor and his coauthors has treated the Dirichlet and regularity problems in this setting.