Hindawi Publishing Corporation Boundary Value Problems Volume 2010, Article ID 526917, 11 pages doi: Research Article On a Mixed Problem for a Constant Coefficient Second-Order System Rita Cavazzoni Via Millaures 12, 10146 Turin, Italy Correspondence should be addressed to Rita Cavazzoni, cavazzon@ Received 2 July 2010; Accepted 1 December 2010 Academic Editor: Peter W. Bates Copyright q 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. | Hindawi Publishing Corporation Boundary Value Problems Volume 2010 Article ID 526917 11 pages doi 2010 526917 Research Article On a Mixed Problem for a Constant Coefficient Second-Order System Rita Cavazzoni Via Millaures 12 10146 Turin Italy Correspondence should be addressed to Rita Cavazzoni cavazzon@ 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 A Rã-1 X 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 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 e L2 0 T X 0 A H3 Rã-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. 1. Introduction Consider the second-order linear differential operator ã d-1 d-1 Q ẰỐ2 - Sdt - Fada E dad Eã ãdã - G. a 1 a 1 y 1 The coefficients of the operator Q satisfy the following assumptions i A is a positive real number ii for all a y 1 . . ã S Fa G Ea y are ã X ã symmetric matrices with real entries iii .
