General and physically privileged solutions to certain symmetric systems of linear PDE's with tensor functionals as unknowns
This paper develops a rather thorough study of linear systems of symmetric partial differential equations that model, under continuum thermodynamic hypotheses, the macroscopic behavior of physical bodies. Such equations carry, as unknown, functionals with values in the space of second-order tensors in three-dimensional Euclidean space. The set of general solutions for the considered equations is characterized. Further, a subset of physically consistent solutions is described. It is worth emphasizing the use of differential tools for functionals defined on convex nowhere dense sets, but without requiring their extension to open sets. The need of such a methodology is pointed out: mainly, it is required when one needs to consider fading-memory bodies that are physically distinct but globally equivalent, i.e. that exhibit the same behavior when the same thermodynamic conditions are imposed.
symmetric partial differential equations
scientific article; zbMATH DE number 1884303