A theory of integral representation, relaxation and homogenization for some types of variational functionals taking extended real values and possibly being not finite also on large classes of regular functions is presented. Some applications to gradient constrained relaxation and homogenization problems are given.

Vengono studiate proprietà di semicontinuità per integrali multipli $$u\in W^{k,1}\left(\mathrm{\Omega };{\mathbb{R}}^d\right)\mapsto {\int }_{\mathrm{\Omega }}f\left(x,u\left(x\right),\mathrm{\dots }{\nabla }^{k}u\left(x\right)\right)dx$$ quando $f$ soddisfa a condizioni di semicontinuità nelle variabili $\left(x,u,\mathrm{\dots },{\nabla }^{k-1}u\left(x\right)\right)$ e può non essere soggetta a ipotesi di coercitività, e le successioni ammissibili in $W^{k,1}\left(\mathrm{\Omega };{\mathbb{R}}^d\right)$ convergono fortemente in $W^{k-1,1}\left(\mathrm{\Omega };{\mathbb{R}}^d\right)$.

We give a new short proof of the Morrey-Acerbi-Fusco-Marcellini Theorem on lower semicontinuity of the variational functional ${\int }_{\Omega }F(x,u,\nabla u)dx$. The proofs are based on arguments from the theory of Young measures.

Studiamo un problema ellittico quasilineare concernente un dominio circondato da un rinforzo sottile di spessore variabile, in cui il coefficiente dell'equazione è (localmente) non costante. Esso concerne due diversi esponenti, uno nel dominio e l'altro nel rinforzo, una condizione di Dirichlelet sulla frontiera esterna e una condizione di trasmissione. Prediciamo il comportamento asintotico della soluzione quando lo spessore, insieme con il coefficiente nel rinforzo, tende a zero perché essi siano...

We deal with an optimal matching problem, that is, we want to transport two measures to a given place (the target set) where they will match, minimizing the total transport cost that in our case is given by the sum of two different multiples of the Euclidean distance that each measure is transported. We show that such a problem has a solution with an optimal matching measure supported in the target set. This result can be proved by an approximation procedure using a $p$-Laplacian system. We prove...

The limit behavior of a periodic assembly of a finite number of elasto-plastic phases is investigated as the period becomes vanishingly small. A limit quasi-static evolution is derived through two-scale convergence techniques. It can be thermodynamically viewed as an elasto-plastic model, albeit with an infinite number of internal variables.

Let $f$ be a function defined on the set ${𝐌}^{2\times 2}$ of all $2$ by $2$ matrices that is invariant with respect to left and right multiplications of its argument by proper orthogonal matrices. The function $f$ can be represented as a function $\tilde{f}$ of the signed singular values of its matrix argument. The paper expresses the ordinary convexity, polyconvexity, and rank 1 convexity of $f$ in terms of its representation $\tilde{f}.$

We study the functional $I_f\left(u\right)={\int }_{\Omega }f\left(u\left(x\right)\right)dx$, where u=(u₁, ..., uₘ) and each $u_j$ is constant along some subspace $W_j$ of ℝⁿ. We show that if intersections of the $W_j$’s satisfy a certain condition then $I_f$ is weakly lower semicontinuous if and only if f is Λ-convex (see Definition 1.1 and Theorem 1.1). We also give a necessary and sufficient condition on ${W_j}_{j=1,...,m}$ to have the equivalence: $I_f$ is weakly continuous if and only if f is Λ-affine.

We study the integral representation of relaxed functionals in the multi-dimensional calculus of variations, for integrands which are finite in a convex bounded set with nonempty interior and infinite elsewhere.

We consider periodic minimizers of the Lawrence–Doniach functional, which models highly anisotropic superconductors with layered structure, in the simultaneous limit as the layer thickness tends to zero and the Ginzburg–Landau parameter tends to infinity. In particular, we consider the properties of minimizers when the system is subjected to an external magnetic field applied either tangentially or normally to the superconducting planes. For normally applied fields, our results show that the resulting...

Si dimostra che il funzionale ${\int }_{\mathrm{\Omega }}f(u,Du)dx$ è semicontinuo inferiormente su $W_{loc}^{1,1}(\mathrm{\Omega })$, rispetto alla topologia indotta da $L_{loc}^1(\mathrm{\Omega })$, qualora l’integrando $f(s,p)$ sia una funzione non-negativa, misurabile in $s$, convessa in $p$, limitata nell’intorno dei punti del tipo $(s,0)$, e tale che la funzione $s\mapsto f(s,0)$ sia semicontinua inferiormente su $𝐑$.