Initial-boundary value problems for second order systems ofpartial differential equations

Abstract

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