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In this paper we study the lower semicontinuous envelope with respect to the $L^1$-topology of a class of isotropic functionals with linear growth defined on mappings from the $n$-dimensional ball into ${ℝ}^N$ that are constrained to take values into a smooth submanifold $𝒴$ of ${ℝ}^N$.

For vector valued maps, convergence in $W^{1,1}$ and of all minors of the Jacobian matrix in $L^1$ is equivalent to convergence weakly in the sense of currents and in area for graphs. We show that maps defined on domains of dimension $n\ge 3$ can be approximated strongly in this sense by smooth maps if and only if the same property holds for the restriction to a.e. 2-dimensional plane intersecting the domain.

In this paper we study the lower semicontinuous envelope with respect to the -topology of a class of isotropic functionals with linear growth defined on mappings from the -dimensional ball into  ${ℝ}^N$  that are constrained to take values into a smooth submanifold  $𝒴$  of  ${ℝ}^N$.

We study the integral representation properties of limits of sequences of integral functionals like $\int f(x,Du)\mathrm{d}x$ under nonstandard growth conditions of $(p,q)$-type: namely, we assume that $${\left|z\right|}^{p\left(x\right)}\le f(x,z)\le {L(1+|z|}^{p\left(x\right)}).$$ Under weak assumptions on the continuous function $p\left(x\right)$, we prove $\Gamma$-convergence to integral functionals of the same type. We also analyse the case of integrands $f(x,u,Du)$ depending explicitly on $u$; finally we weaken the assumption allowing $p\left(x\right)$ to be discontinuous on nice sets.

Let $𝒴$  be a smooth compact oriented riemannian manifoldwithout boundary, and assume that its $1$-homology group has notorsion. Weak limits of graphs of smooth maps $u_k:B^n\to 𝒴$  with equibounded total variation give riseto equivalence classes of cartesian currents in ${\mathrm{cart}}^{1,1}\left(B^n𝒴\right)$  for which we introduce a natural$BV$-energy.Assume moreover that the first homotopy group of  $𝒴$  iscommutative. In any dimension  $n$  we prove that every element $T$  in  ${\mathrm{cart}}^{1,1}\left(B^n𝒴\right)$  can be approximatedweakly in the sense of currents by a sequence of graphs...

Let $𝒴$ be a smooth oriented Riemannian manifold which is compact, connected, without boundary and with second homology group without torsion. In this paper we characterize the sequential weak closure of smooth graphs in $B^n\times 𝒴$ with equibounded Dirichlet energies, $B^n$ being the unit ball in ${ℝ}^n$. More precisely, weak limits of graphs of smooth maps $u_k:B^n\to 𝒴$ with equibounded Dirichlet integral give rise to elements of the space ${cart}^{2,1}(B^n\times 𝒴)$ (cf. [4], [5], [6]). In this paper we prove that every element $T$ in ${cart}^{2,1}(B^n\times 𝒴)$ is the weak limit...

We study the integral representation properties of limits of sequences of integral functionals like  $\int f(x,Du)\mathrm{d}x$  under nonstandard growth conditions of -type: namely, we assume that $${\left|z\right|}^{p\left(x\right)}\le f(x,z)\le {L(1+|z|}^{p\left(x\right)}).$$ Under weak assumptions on the continuous function , we prove -convergence to integral functionals of the same type. We also analyse the case of integrands depending explicitly on ; finally we weaken the assumption allowing to be discontinuous on nice sets.

We discuss a variational problem defined on couples of functions that are constrained to take values into the 2-dimensional unit sphere. The energy functional contains, besides standard Dirichlet energies, a non-local interaction term that depends on the distance between the gradients of the two functions. Different gradients are preferred or penalized in this model, in dependence of the sign of the interaction term. In this paper we study the lower semicontinuity and the coercivity of the energy...