The linear hyperbolic initial and boundary value problems in a domain with corners

TL;DR

This paper investigates linear hyperbolic initial and boundary value problems in polygonal domains, employing semigroup methods to establish well-posedness and identifying key elementary modes through diagonalization.

Contribution

It introduces a semigroup approach for hyperbolic IBVPs in polygonal domains and characterizes the elementary modes of the system.

Findings

Well-posedness established using semigroup methods

Identification of hyperbolic and elliptic modes

Applicable to constant and variable coefficient cases

Abstract

In this article, we consider linear hyperbolic Initial and Boundary Value Problems (IBVP) in a rectangle (or possibly curvilinear polygonal domains) in both the constant and variable coefficients cases. We use semigroup method instead of Fourier analysis to achieve the well-posedness of the linear hyperbolic system, and we find by diagonalization that there are only two elementary modes in the system which we call hyperbolic and elliptic modes. The hyperbolic system in consideration is either symmetric or Friedrichs-symmetrizable.