Boundary Value Problems
Volume 2010 (2010), Article ID 526917, 11 pages
Research Article

On a Mixed Problem for a Constant Coefficient Second-Order System

Via Millaures 12, 10146 Turin, Italy

Received 2 July 2010; Accepted 1 December 2010

Academic Editor: Peter W. Bates

Copyright © 2010 Rita Cavazzoni. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The paper is devoted to the study of an initial boundary value problem for a linear second-order differential system with constant coefficients. The first part of the paper is concerned with the existence of the solution to a boundary value problem for the second-order differential system, in the strip Ω 𝐴 = R 𝑑 1 × ( 0 , 𝐴 ) , where 𝐴 is a suitable positive number. The result is proved by means of the same procedure followed in a previous paper to study the related initial value problem. Subsequently, we consider a mixed problem for the second-order constant coefficient system, where the space variable varies in Ω 𝐴 and the time-variable belongs to the bounded interval ] 0 , 𝑇 [ , with 𝑇 sufficiently small in order that the operator satisfies suitable energy estimates. We obtain by superposition the existence of a solution 𝑢 𝐿 2 ( [ 0 , 𝑇 ] × [ 0 , 𝐴 ] , 𝐻 3 ( R 𝑑 1 ) ) , by studying two related mixed problems, whose solutions exist due to the results proved for the Cauchy problem in a previous paper and for the boundary value problem in the first part of this paper.