Partial Differential Equations and Harmonic Analysis

University Of North Carolina At Chapel Hill, Chapel Hill NC

Investigators

Abstract

March 11, 2002 PI: Michael Taylor, University of North Carolina, Chapel Hill DMS-0139726 ******************************************************** ABSTRACT FOR MICHAEL TAYLOR'S PROJECT Professor Taylor proposes to investigate problems in partial differential equations, particularly in the areas of wave propagation, boundary problems for elliptic PDE, and functional calculus for elliptic self-adjoint operators. Taylor will study wave propagation in fairly rough media, in which rays of geometrical optics are just barely defined, and in even rougher media, in which such rays are not uniquely defined. He will examine to what extent wave propagation in such media behaves like that in more regular media and to what extent it differs. He plans to pursue the effects of curvature bounds on a manifold on the nature of such wave propagation, and also investigate propagation, reflection, and diffraction of waves at a boundary for a rough medium. Taylor will also continue a program of studying elliptic equations with rough coefficients on Lipschitz domains. This will include nonlinear equations, such as arise in the study of harmonic maps between manifolds with rough metric tensors. In the area of functional calculus, one attack is to synthesize general functions of the Laplace operator from the solution operator to the wave equation. A detailed knowledge of wave propagation can then yield highly nontrivial information on such functional calculus. Taylor is pursuing a variety of projects in this area, ranging from detailed behavior of eigenfunction expansions to singular perturbation problems arising in the study of pattern formation. Wave motion, from sound passing through air to pressure and shear waves passing through the earth to radio waves traveling through space, is a ubiquitous phenomenon, whose mathematical analysis has for a long time been a major part of mathematics. A variety of partial differential equations have arisen to describe different aspects of wave motion, and their study continues to stimulate work on analytical tools to elucidate the behavior of solutions to such PDE. Analyzing a wave as a superposition of various frequencies gave rise to the area of harmonic analysis. Localizing the study of a wave simultaneously in space and in frequency is at the heart of the more recent area of microlocal analysis. Taylor has been active in developing tools in these areas and using them to produce results in PDE. Some of his efforts have focused on treating PDE with rough coefficients and PDE in domains with rough boundary. These arise to describe waves traveling through complicated media, whose boundaries have many corners and other irregularities. While harmonic analysis techniques have provided much insight into partial differential equations, the reverse direction has also led to significant results in harmonic analysis. Taylor is continuing to investigate how results on Fourier synthesis of singular functions can be explained in terms of propagation and focusing of waves.