Helmholtz and dispersive equations with variable coefficients on exterior domains

TL;DR

This paper establishes smoothing estimates for Helmholtz, Schrödinger, and wave equations with variable coefficients on exterior domains, addressing trapping issues and extending classical results to less regular, long-range perturbations.

Contribution

It provides explicit conditions on variable coefficients to prevent trapping and extends smoothing estimates to exterior domains with variable, limited regularity coefficients.

Findings

Proves smoothing estimates for Helmholtz equations with variable coefficients.

Extends smoothing estimates to Schrödinger and wave flows on exterior domains.

Identifies conditions preventing trapping in variable coefficient settings.

Abstract

We prove smoothing estimates in Morrey-Campanato spaces for a Helmholtz equation $$ -Lu+zu=f, \qquad -Lu:=\nabla^{b}(a(x)\nabla^{b}u)-c(x)u, \qquad \nabla^{b}:=\nabla+ib(x) $$ with fully variable coefficients, of limited regularity, defined on the exterior of a starshaped compact obstacle in $\mathbb{R}^{n}$, $n\ge3$, with Dirichlet boundary conditions. The principal part of the operator is a long range perturbation of a constant coefficient operator, while the lower order terms have an almost critical decay. We give explicit conditions on the size of the perturbation which prevent trapping. As an application, we prove smoothing estimates for the Schr\"{o}dinger flow $e^{itL}$ and the wave flow $e^{it \sqrt{L}}$ with variable coefficients on exterior domains and Dirichlet boundary conditions.