This mathematics research project by Matthew Blair deals with Fourier analysis and its applications to partial differential equations. Of particular interest here are space-time integrability bounds of Strichartz and local smoothing type for the wave and Schrödinger equations. The proposal also concerns closely related estimates on eigenfunctions of the Laplacian on a compact Riemannian manifold. These problems actually stem from those in classical Fourier analysis, such as the Stein-Tomas restriction theorem, and the relevant techniques involve closely related oscillatory integral estimates. While basic formulations of these inequalities exist for these equations when they are posed over Euclidean space and even Riemannian manifolds without boundary, less is known about their validity for initial boundary value problems. Here one considers a domain or more generally a Riemannian manifold with boundary, and the imposition of homogeneous boundary conditions affects the flow of energy in significant ways, sometimes even inhibiting dispersion. Consequently, the proofs of these estimates typically involve microlocal or phase space analysis, which provide avenues for understanding this phenomena. For smooth boundaries, Blair and his collaborators have achieved a great deal of success by realizing solutions as a superposition of wave packets, then showing that the resulting oscillatory integrals yield the desired inequalities. On domains with corners, even less is known about the validity of these estimates, but Blair and his collaborators have been able to obtain some results for polygonal domains.
This mathematics research project by Matthew Blair is in the area of Fourier analysis: this is a branch of mathematics that plays an important role in the development of mathematical and physical theories. The research pursued in this project provides the mathematical foundation for the study of light and sound waves. Fourier analysis continues to play a significant role in deepening our understanding of the equations which model this behavior. In particular, these investigations yield further insight as to how the presence of a hard boundary surface influences the development of waves. For example, if one listens to the symphony in an auditorium, the sounds heard are affected by the manner in which the acoustic waves reflect off the walls. In this sense, it can be important to understand how the shape of the hall influences its acoustics. While this is of course a classical problem, there is more to be understood in terms of how these interactions influence dispersive properties. Moreover, this line of work is important in the analysis of closely related nonlinear equations arising from fiber optics and water waves, where there is much to be done in understanding and limiting the various types of instabilities which can occur.
