This work is concerned with the reformulation of evolutionary problems in a weak form enabling consideration of solutions that may exhibit evolving microstructures. This reformulation is accomplished by expressing the evolutionary problem in variational form, i.e., by identifying a functional whose minimizers represent entire trajectories of the system. The particular class of functionals under consideration is derived by first defining a sequence of time-discretized minimum problems and subsequently...

This work is concerned with the reformulation of evolutionary problems in a weak form enabling consideration of solutions that may exhibit evolving microstructures. This reformulation is accomplished by expressing the evolutionary problem in variational form, i.e., by identifying a functional whose minimizers represent entire trajectories of the system. The particular class of functionals under consideration is derived by first defining a sequence of time-discretized minimum problems and...

This paper provides a convergent numerical approximation of the Pareto optimal set for finite-horizon multiobjective optimal control problems in which the objective space is not necessarily convex. Our approach is based on Viability Theory. We first introduce a set-valued return function V and show that the epigraph of V equals the viability kernel of a certain related augmented dynamical system. We then introduce an approximate set-valued return function with finite set-values as the solution of...

In the article an optimal control problem subject to a stationary variational inequality is investigated. The optimal control problem is complemented with pointwise control constraints. The convergence of a smoothing scheme is analyzed. There, the variational inequality is replaced by a semilinear elliptic equation. It is shown that solutions of the regularized optimal control problem converge to solutions of the original one. Passing to the limit in the optimality system of the regularized problem...

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.

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(x), we prove Γ-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(x) to be discontinuous on nice sets.

Diamo condizioni sulle funzioni $f$, $g$ e sulla misura $\mu$ affinché il funzionale $$F(u)={\int }_{\mathrm{\Omega }}f(x,u,Du)dx+{\int }_{\overline{\mathrm{\Omega }}}g(x,u)d\mu$$ sia $L^1(\mathrm{\Omega })$-semicontinuo inferiormente su $W^{1,1}(\mathrm{\Omega })\cap C^0(\overline{\mathrm{\Omega }})$. Affrontiamo successivamente il problema del rilassamento.

New $L^1$-lower semicontinuity and relaxation results for integral functionals defined in BV($\Omega$) are proved, under a very weak dependence of the integrand with respect to the spatial variable $x$. More precisely, only the lower semicontinuity in the sense of the $1$-capacity is assumed in order to obtain the lower semicontinuity of the functional. This condition is satisfied, for instance, by the lower approximate limit of the integrand, if it is BV with respect to $x$. Under this further BV dependence, a...

New L1-lower semicontinuity and relaxation results for integral functionals defined in BV(Ω) are proved, under a very weak dependence of the integrand with respect to the spatial variable x. More precisely, only the lower semicontinuity in the sense of the 1-capacity is assumed in order to obtain the lower semicontinuity of the functional. This condition is satisfied, for instance, by the lower approximate limit of the integrand, if it is BV with respect to x. Under this further BV dependence, a...

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.