Loss of derivatives for hyperbolic boundary problems with constant coefficients

TL;DR

This paper extends well-posedness results for symmetric and hyperbolic systems with constant coefficients and characteristic boundaries, using Fourier-Laplace methods, generalizing classical scalar hyperbolic boundary problem theorems.

Contribution

It generalizes the Kreiss-Sakamoto condition to symmetric hyperbolic systems with characteristic boundaries, providing explicit Fourier-Laplace solutions for these boundary problems.

Findings

Established well-posedness under weakened boundary conditions.

Extended classical results to symmetric hyperbolic systems with characteristic boundaries.

Provided explicit solution construction via Fourier-Laplace transform.

Abstract

Symmetric hyperbolic systems and constantly hyperbolic systems with constant coefficients and a boundary condition which satisfies a weakened form of the Kreiss-Sakamoto condition are considered. A well-posedness result is established which generalizes a theorem by Chazarain and Piriou for scalar strictly hyperbolic equations and non-characteristic boundaries from 1972. The proof is based on an explicit solution of the boundary problem by means of the Fourier-Laplace transform. As long as the operator is symmetric, the boundary is allowed to be characteristic.