We consider the integral functional $\int f\left(x,Du\right)dx$ under non standard growth assumptions of $\left(p,q\right)$-type: namely, we assume that ${\left|z\right|}^{p\left(x\right)}\le f\left(x,z\right)\le L\left(1+{\left|z\right|}^{p\left(x\right)}\right)$, a relevant model case being the functional $\int {\left|Du\right|}^{p\left(x\right)}dx$. Under sharp assumptions on the continuous function $p\left(x\right)>1$ we prove regularity of minimizers both in the scalar and in the vectorial case, in which we allow for quasiconvex energy densities. Energies exhibiting this growth appear in several models from mathematical physics.

We describe a general axiomatic way to define functions of class Ck, k ∈ N∪{∞} on topological abelian groups. In the category of Banach spaces, this definition coincides with the usual one. The advantage of this axiomatic approach is that one can dispense with the notion of norms and limit procedures. The disadvantage is that one looses the derivative, which is replaced by a local linearizing factor. As an application we use this approach to define C∞ functions in the setting of graded/super manifolds....

The distributional $k$-dimensional Jacobian of a map $u$ in the Sobolev space $W^{1,k-1}$ which takes values in the sphere $S^{k-1}$ can be viewed as the boundary of a rectifiable current of codimension $k$ carried by (part of) the singularity of $u$ which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary $M$ of codimension $k$ can be realized as Jacobian of a Sobolev map valued in $S^{k-1}$. In case $M$ is polyhedral, the map we construct...

We study a class of functions which differ essentially from those which are the sum of a convex function and a regular one and which have interesting properties related to $\mathrm{\Gamma }$-convergence and to problems with non-convex constraints. In particular some results are given for the associated evolution equations.

La semiconcavità è una nozione che generalizza quella di concavità conservandone la maggior parte delle proprietà ma permettendo di ampliarne le applicazioni. Questa è una rassegna dei punti più salienti della teoria delle funzioni semiconcave, con particolare riguardo allo studio dei loro insiemi singolari. Come applicazione, si discuterà una formula di rappresentazione per la soluzione di un modello dinamico per la materia granulare.