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We obtain the minimal energy solution of the Plateau problem with elastic boundary as a variational limit of the minima of the Kirchhoff-Plateau problems with a rod boundary when the cross-section of the rod vanishes. The limit boundary is a framed curve that can sustain bending and twisting.

We study the equilibrium problem of a system consisting of several Kirchhoff rods linked in an arbitrary way and tied by a soap film, using techniques of the Calculus of Variations. We prove the existence of a solution with minimum energy, which may be quite irregular, and perform experiments confirming the kind of surface predicted by the model.

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 our results, we introduce a particular class of subbodies, namely the sets with curvature measure.

Some nonlinear evolution problems arising in the theory of elastic beams with linear memory are considered. Under proper assumptions on the memory kernel the existence of uniform absorbing sets and of a global attractor is achieved.

An approach to weak balance laws in Continuum Mechanics is presented, involving densities with only divergence measure, which relies on the balance of power. An equivalence theorem between Cauchy powers and Cauchy fluxes is proved. As an application of this method, the construction of the stress tensor when the body is an orientable differential manifold is achieved under very general assumptions.

In this paper we investigate the asymptotic behavior, as time tends to infinity, of the solutions of an non-autonomous integro-partial differential equation describing the heat flow in a rigid heat conductor with memory. Existence and uniqueness of solutions is provided. Moreover, under proper assumptions on the heat flux memory kernel and on the magnitude of nonlinearity, the existence of uniform absorbing sets and of a global uniform attractor is achieved. In case of quasiperiodic dependence of time of the external heat supply, the above attractor is shown to have finite Hausdorff dimension.

Asymptotic behavior in time for a model of nonlinear plane air waves when a viscosity term is present is investigated. The existence of a uniform bound of several norms is established, and in particular the continuity of the solution, which implies the nonexistence of shock waves.

A “saddle point” (or maximum-minimum) principle is set up for the quasi-static boundary-value problem in linear viscoelasticity. The appropriate class of convolution-type functionals for it is taken in terms of bilinear forms with a weight function involving the Fourier transform. The “minimax” property is shown to hold as a direct consequence of thermodynamic restrictions on the relaxation function. This approach can be extended to further linear evolution problems where initial data are not prescribed.

Consider the nonlinear parabolic equation in a cylinder with homogeneous Dirichlet boundary conditions on ▽ω. We show that the L 2-norm of u is equivalent to C t −1/(p−2) where C is a constant which can be related to an “eigenvalue” of the p-laplacian. This results is also generalized in various directions.

This paper contains some results on asymptotic behaviour of solutions to strongly damped abstract nonlinear wave equations. After reviewing sufficient hypotheses for existence and uniqueness, uniform time estimates are given and a global attractor for the trajectories of the associated dynamical system is constructed. Finally, applications are made to nonlinear wave equations such as Sine–Gordon equation, proving the finite dimensionality of the corresponding attractors.

In this paper we show the existence, uniqueness and regularity of the solutions to the thermoelastic transmission problem. Moreover, when the solutions are symmetrical we show that the energy decays exponentially as time goes to infinity, no matter how small is the size of the thermoelastic part.

This paper is devoted to existence, uniqueness and asymptotic behavior, as time tends to infinity, of the solutions of an integro-partial differential equation arising from the theory of heat conduction with memory, in presence of a temperature-dependent heat supply. A linearized heat flux law involving positive instantaneous conductivity is matched with the energy balance, to generate an autonomous semilinear system subject to initial history and Dirichlet boundary conditions. Existence and uniqueness of solution is provided. Moreover, under proper assumptions on the heat flux memory kernel, the existence of absorbing sets in suitable function spaces is achieved.

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.

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.