Si prova la maggior sommabilità del gradiente dei minimi locali di funzionali integrali della forma $${\int }_{\mathrm{\Omega }}f\left(Du\right)dx,$$ dove $f$ soddisfa l'ipotesi di crescita $${\left|\xi \right|}^p-c_1\le f\left(\xi \right)\le c\left(1+{\left|\xi \right|}^q\right),$$ con $1<p<q\le 2$. L'integrando $f$ è $C^2$ e $DDf$ ha crescita $p-2$ dal basso e $q-2$ dall'alto.

We deduce a macroscopic strain gradient theory for plasticity from a model of discrete dislocations. We restrict our analysis to the case of a cylindrical symmetry for the crystal under study, so that the mathematical formulation will involve a two-dimensional variational problem. The dislocations are introduced as point topological defects of the strain fields, for which we compute the elastic energy stored outside the so-called core region. We show that the $\Gamma$-limit of this energy (suitably rescaled),...

Let $\Lambda$ be a Lagrangian submanifold of $T^{*}X$ for some closed manifold X. Let $S(x,\xi )$ be a generating function for $\Lambda$ which is quadratic at infinity, and let W(x) be the corresponding graph selector for $\Lambda ,$ in the sense of Chaperon-Sikorav-Viterbo, so that there exists a subset $X_0\subset X$ of measure zero such that W is Lipschitz continuous on X, smooth on $X\setminus X_0$ and $(x,\partial W/\partial x(x\left)\right)\in \Lambda$ for $X\setminus X_0.$ Let H(x,p)=0 for $(x,p)\in \Lambda$. Then W is a classical solution to $H(x,\partial W/\partial x(x\left)\right)=0$ on $X\setminus X_0$ and extends to a Lipschitz function on the whole of X. Viterbo refers to W as a variational...

A unified framework for analyzing the existence of ground states in wide classes of elastic complex bodies is presented here. The approach makes use of classical semicontinuity results, Sobolev mappings and cartesian currents. Weak diffeomorphisms are used to represent macroscopic deformations. Sobolev maps and cartesian currents describe the inner substructure of the material elements. Balance equations for irregular minimizers are derived. A contribution to the debate about the role of the balance...

A unified framework for analyzing the existence of ground states in wide classes of elastic complex bodies is presented here. The approach makes use of classical semicontinuity results, Sobolev mappings and Cartesian currents. Weak diffeomorphisms are used to represent macroscopic deformations. Sobolev maps and Cartesian currents describe the inner substructure of the material elements. Balance equations for irregular minimizers are derived. A contribution to the debate about the role of the balance...

The paper is concerned with guaranteed and computable bounds of the limit (or safety) load, which is one of the most important quantitative characteristics of mathematical models associated with linear growth functionals. We suggest a new method for getting such bounds and illustrate its performance. First, the main ideas are demonstrated with the paradigm of a simple variational problem with a linear growth functional defined on a set of scalar valued functions. Then, the method is extended to...