Initial-boundary value problems for second order systems of partial differential equations

TL;DR

This paper develops a well-posedness theory for second order PDE systems in bounded domains, effectively handling boundary phenomena like surface waves using pseudo-differential operators and mode analysis.

Contribution

It introduces a pseudo-differential operator approach that simplifies the analysis of second order systems, avoiding complications of larger first order systems and clarifying boundary condition relations.

Findings

Reduction to 2n equations via pseudo-differential operators

Local analysis of Cauchy and half-plane problems

Broader class of problems treatable compared to previous methods

Abstract

We develop a well-posedness theory for second order systems in bounded domains where boundary phenomena like glancing and surface waves play an important role. Attempts have previously been made to write a second order system consisting of n equations as a larger first order system. Unfortunately, the resulting first order system consists, in general, of more than 2n equations which leads to many complications, such as side conditions which must be satisfied by the solution of the larger first order system. Here we will use the theory of pseudo-differential operators combined with mode analysis. There are many desirable properties of this approach: 1) The reduction to first order systems of pseudo-differential equations poses no difficulty and always gives a system of 2n equations. 2) We can localize the problem, i.e., it is only necessary to study the Cauchy problem and halfplane problems with constant coefficients. 3) The class of problems we can treat is much larger than previous approaches based on "integration by parts". 4) The relation between boundary conditions and boundary phenomena becomes transparent.