Rectifiability and Elliptic Partial Differential Equations

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This project lies at the interface of geometric measure theory and partial differential equations, and it will utilize techniques from harmonic analysis. In geometric measure theory, one studies geometric properties of sets via the behavior of some measure on them (the concept of "measure" generalizes the notions of length, area, and volume). In this project, the sort of set under consideration is typically the boundary (i.e., the enclosing perimeter) of some region in space, and for this a specific measure -- namely, "harmonic measure"-- plays a central role. Harmonic measure and its generalizations may be used to construct solutions in some region, with prescribed boundary values, to partial differential equations of so-called "elliptic" type. Such equations govern various steady-state phenomena in the real world, including electrostatics and steady-state temperature distributions. For example, harmonic measure may be used to determine the steady-state temperature distribution within some region, given a known temperature distribution on the perimeter of the region. Unsurprisingly, the geometry of the region and its boundary greatly affect one's ability to carry out such a program in practice. A principal goal of this project is to quantify, in a natural sense, the connection between geometry and the behavior of harmonic measure and its generalizations.The project has two main areas of focus. First, the principal investigator plans to find an intrinsically geometric characterization of quantitative absolute continuity of harmonic measure with respect to surface measure, on the boundary of an open set in d-dimensional Euclidean space. He expects that such a characterization should comprise two parts: a quantitative rectifiablity property of the boundary, plus scale invariant nontangential accessibility to ample portions of the boundary. Related to this work, the principal investigator also plans to investigate the analogous question concerning quantitative absolute continuity of elliptic-harmonic measure associated to a more general second-order elliptic operator. Second, the principal investigator will study the solvability of other boundary value problems for elliptic equations, in several different settings. More precisely, he intends to work toward an improved understanding of the Neumann problem in domains more general than Lipschitz domains and for nonsymmetric divergence form operators. As a first step, he plans to consider a certain family of transmission problems, for which the Neumann problem is an endpoint case. He also plans to treat boundary-value problems for certain higher-order divergence-form elliptic operators.