Harmonic Analysis and Linear Elliptic Equations

Brown University, Providence RI

Investigators

Abstract

ABSTRACT This proposal is concerned with research on singular integrals, the theory of weights, with applications to linear elliptic partial differential equations of second and of higher order. There are three main areas of current focus. First, we study the elliptic measure associated to a second order elliptic operator - both divergence and nondivergence form. Second, we are studying the boundary values of solutions to certain higher order elliptic operators in non-smooth domains. The third part of this proposal concerns the maximal functions and singular integrals associated to certain product domains - such operators involve nonisotropic dilations and results require a study of the geometry of families of rectangles arising from this dilation structure. In the middle of this century, revolutionary ideas in analysis were being forged on two fronts: the theory of singular integrals in harmonic analysis (the Calderon-Zygmund school of analysis) and the theory of elliptic equations (Di Giorgi, J. Nash). Each of these subjects has provided fascinating applications - the first in such areas as applied harmonic analysis/wavelets/Fourier analysis, and the second in the theory of nonlinear partial differential equations, of tremendous value in applications. But at the beginning of these studies, and for a period of twenty or so years, the development of each of these subjects proceeded independently, and with little interaction. However, since the early 1980's, the connections between these areas have been increasingly exploited and understood, to the great benefit of the fields of harmonic analysis and partial differential equations. This proposer's research is in this field of harmonic analysis, with an emphasis on its connection to and inspiration from partial differential equations.