Operator Cosine Functions and Boundary Value Problems

Abstract

For the first time, the theory of strongly continuous cosine operator functions (COF) has been applied to study the correct solvability of boundary value problems for second-order linear differential equations in a Banach space (elliptic case). The correct solvability of the Cauchy problem (hyperbolic case) is usually formulated in COF terms. The conditions on the order of COF growth are specified under which the Dirichlet boundary value problem is correct on a finite interval. An integral representation of the solution and its sharp estimate are given.