Harmonic Analysis, Geometric Measure Theory and Partial Differential Equations

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

HARMONIC ANALYSIS, GEOMETRIC MEASURE THEORY AND PARTIAL DIFFERENTIAL EQUATIONS Abstract of Proposed Research Marius Mitrea This project will develop the systematic use of singular integral operators (SIO) under more general hypotheses than has been previously attained. The goal is to display the effectiveness of SIO-based methods in circumstances traditionally handled by other, more specialized, tools (such as variational methods, harmonic measure techniques, etc). It is well-recognized that there are subtle connections between the boundedness of singular integral operators and the geometric measure-theoretic properties of sets. A fundamental result in this direction is the boundedness of SIO with reasonable kernels on surfaces which are Ahlfors regular (i.e., behave like n-1 dimensional at all scales), and contain ``big pieces of Lipschitz surfaces'' in a uniform fashion (one calls such surfaces uniformly rectifiable). This earlier work involved geometric measure theory, but has not yet been applied to problems in Partial Differential Equations (PDE). The ultimate goal of this proposal is to explore the role that SIO may play in the treatment of boundary value problems under sharp geometric measure theoretic assumptions on the domain and its boundary. In particular, this work will develop the analysis of SIO on uniformly rectifiable surfaces for applications to problems in PDE, such as boundary problems for the Laplace operator, other second order elliptic operators and systems (such as the Lame, Stokes and Maxwell systems).