Dean Baskin

Sharp high-frequency estimates for the Helmholtz equation and applications to boundary integral equations

We consider three problems for the Helmholtz equation in interior and exterior domains in

Radiation Fields on Schwarzschild Spacetime

\mathbb{R}^d

Radiation fields for semilinear wave equations

(

A parametrix for the fundamental solution of the Klein–Gordon equation on asymptotically de Sitter spaces

d=2,3

Asymptotics of radiation fields in asymptotically Minkowski space

): the exterior Dirichlet-to-Neumann and Neumann-to-Dirichlet problems for outgoing solutions, and the interior impedance problem. We derive sharp estimates for solutions to these problems that, in combination, give bounds on the inverses of the combined-field boundary integral operators for exterior Helmholtz problems.

Strichartz Estimates on Asymptotically de Sitter Spaces

In this paper we define the radiation field for the wave equation on the Schwarzschild black hole spacetime. In this context it has two components: the rescaled restriction of the time derivative of a solution to null infinity and to the event horizon. In the process, we establish some regularity properties of solutions of the wave equation on the spacetime. In particular, we prove that the regularity of the solution across the event horizon and across null infinity is determined by the regularity and decay rate of the initial data at the event horizon and at infinity. We also show that the radiation field is unitary with respect to the conserved energy and prove support theorems for each piece of the radiation field.

Resolvent estimates and local decay of waves on conic manifolds

We define the radiation fields of solutions to critical semilinear wave equations in R^3 and use them to define the scattering operator. We also prove a support theorem for the radiation fields with radial initial data. This extends the well known support theorem for the Radon transform to this setting and can also be interpreted as a Paley-Wiener theorem for the distorted nonlinear Fourier transform of radial functions.