General and physically privileged solutions to certain symmetric systems of linear P.D.E.s with tensor functionals as unknowns

Montanaro, Adriano ; Pigozzi, Diego

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 11 (2000) no. 4, pp. 245-276

Résumé

We characterize the general solutions to certain symmetric systems of linear partial differential equations with tensor functionals as unknowns. Then we determine the solutions that are physically meaningful in suitable senses related with the constitutive functionals of two simple thermodynamic bodies with fading memory that are globally equivalent, i.e. roughly speaking that behave in the same way along processes not involving cuts. The domains of the constitutive functionals are nowhere dense subsets of a suitable infinite-dimensional Hilbert space. By using the condition of material frame-indifference on the constitutive functionals and the theory [1] of differential calculus on convex sets (that may be nowhere dense), we give a rigorous meaning from a general point of view to the derivatives of these functionals, without assuming the possibility of extending them to an open set. Such results appear necessary for characterizing the couples of thermodynamic bodies with memory that are globally equivalent but are physically different; and such bodies exist.