Boundary value problrms for systems of first order pseudodifferential operators

A large group of problems for systems of partial differential equations of the first order is studied by common methods; for these problems the theorem on energy inequalities is proved, under the assumption that the system (or rather, its characteristic matrix) is symmetric, and also the theorem on the identity of the weak and the strong solutions; these two theorems are used to prove existence and uniqueness of the strong solution. The methods are applicable to a number of problems for symmetric hyperbolic systems of the first order and for symmetric stationary systems that need not be elliptic. Recently new possibilities of developing and applying these methods by using pseudodifferential operators have been discovered, and these are far from being exhausted at the present time. In § 1 the problem is stated and a brief survey of the literature is given. In §§ 2–6 the three theorems mentioned above are set out with proofs suitable for systems of pseudodifferential operators of the first order in a bounded domain. § 7 deals with boundary value problems for symmetrizable systems, more general than the symmetric systems.