We extend a celebrated identity by P. Pucci and J. Serrin, concerning C 2 solutions of Euler equations of functionals of the calculus of variations, to the case of C 1 solutions under the only additional assumption of strict convexity in the gradient. Some particular cases in which the mere convexity is sufficient are also considered.
We investigate the general constitutive relation of an isotropic linear fluid when the stress tensor can depend on higher-order spatial gradients of the velocity. We apply the results to the case of second-grade and third-grade fluids, be compressible or not. However, the expression of the general isotropic tensor can be a matter of interest also for other… (More)
Cauchy fluxes induced by locally summable tensor fields with divergence measure are characterized. The equivalence between integral formulations involving subsets of finite perimeter and much more restricted classes of subsets is proved.
Cauchy interactions between subbodies of a continuous body are introduced in the framework of Measure Theory, extending the class of previously admissible ones. A decomposition theorem into a volume and a surface interaction is proved, as well as characterizations of the single components. Finally, an extension result and a generalized balance law are given.
By means of balanced virtual powers, an axiomatic approach is developed, in the spirit of Noll, to second-gradient continua. The measure-theoretical formulation allows a considerable simplification since the existence of an edge stress density is regarded as a special case of a surface stress which is a singular measure with respect to the area. To prove… (More)
Balance laws of the type of entropy are treated in the framework of geometric measure theory, and a weak version, although conceptually simple, of the Second Law of Ther-modynamics is introduced, allowing extensions to measure-valued entropy productions and to sets of finite perimeter as subbodies.
We give a general formulation of the Principle of virtual powers in Continuum Mechanics from a distributional point of view, and study some of its relevant consequences in the field of balance equations.
A weak formulation of the stress boundary conditions in Continuum Mechanics is proposed. This condition has the form of a balance law, allows also singular measure data and is consistent with the regular case. An application to the Flamant solution in linear elasticity is shown.
By means of a perturbation argument devised by P. Bolle, we prove the existence of infinitely many solutions for perturbed symmetric polyharmonic problems with non–homogeneous Dirichlet boundary conditions. An extension to the higher order case of the estimate from below for the critical values due to K. Tanaka is obtained.