An approach with Lagrange identity of the mixed problem in theory of strain gradient thermoelasticity

Abstract

In our paper we first define the mixed initial-boundary values problem in the theory of strain gradient thermoelasticity. With the help of an identity of Lagrange’s type, we then prove some theorems regarding the uniqueness of the solution of this mixed problem and also two results regarding the continuous dependence of solutions on initial data and on the charges. We must ouline that we obtain these qualitative results without recourse to any laws of conservation of energy and without recourse to any boundedness assumptions on the coefficients. It is equally important to note that we do not impose restrictions on the elastic coefficients regarding their positive definition.

Keywords

• lagrange’s identity
• strain gradient thermoelasticity
• uniqueness
• continuous dependence results