Rate-independent problems are considered, where the stored energy density is a function of the gradient. The stored energy density may not be quasiconvex and is assumed to grow linearly. Moreover, arbitrary behaviour at infinity is allowed. In particular, the stored energy density is not required to coincide at infinity with a positively 1-homogeneous function. The existence of a rate-independent process is shown in the so-called energetic formulation.

We set a coupled boundary value problem between two domains of different dimension. The first one is the unit cube of Rn, n C [2,3], with a crack and the second one is the crack. this problem comes from Ciarlet et al. (1989), that obtained an analogous coupled problem. We show that the solution has singularities due to the crack. As in Grisvard (1989), we adapt the Hilbert uniqueness method of J.-L. Lions (1968,1988) in order to obtain the exact controllability of the associated wave equation with...

This paper is concerned with the global exact controllability of the semilinear heat equation (with nonlinear terms involving the state and the gradient) completed with boundary conditions of the form $\frac{\partial y}{\partial n}+f\left(y\right)=0$. We consider distributed controls, with support in a small set. The null controllability of similar linear systems has been analyzed in a previous first part of this work. In this second part we show that, when the nonlinear terms are locally Lipschitz-continuous and slightly superlinear, one...

In this paper, we consider probability measures μ and ν on a d-dimensional sphere in ${𝐑}^{d+1},d\ge 1,$ and cost functions of the form $c(𝐱,𝐲)=l\left(\frac{{|𝐱-𝐲|}^2}{2}\right)$ that generalize those arising in geometric optics where $l\left(t\right)=-logt.$ We prove that if μ and ν vanish on $(d-1)$-rectifiable sets, if |l'(t)|>0,${lim}_{t\to 0^{+}}l\left(t\right)=+\infty ,$ and $g\left(t\right):=t(2-t){\left(l^{\text{'}}\left(t\right)\right)}^2$ is monotone then there exists a unique optimal map To that transports μ onto $\nu ,$ where optimality is measured against c. Furthermore, ${inf}_{𝐱}|T_o𝐱-𝐱|>0.$ Our approach is based on direct variational arguments. In the special case when $l\left(t\right)=-logt,$ existence of optimal maps...

Here we present an approximation method for a rather broad class of first order variational problems in spaces of piece-wise constant functions over triangulations of the base domain. The convergence of the method is based on an inequality involving $W^{-1,p}$ norms obtained by Nečas and on the general framework of Γ-convergence theory.

Si discute il comportamento asintotico di energie di tipo Ginzburg-Landau, per funzioni da ${\mathbb{R}}^{n+k}$ in ${\mathbb{R}}^k$, e sotto l'ipotesi che l'esponente di crescita $p$ sia strettamente maggiore di $k$. In particolare, si illustra un risultato di compattezza e di $\mathrm{\Gamma }$-convergenza, rispetto a una opportuna topologia sui Jacobiani, visti come correnti $n$-dimensionali. L'energia limite è definita sulla classe degli $n$-bordi interi $M$, e la sua densità dipende localmente dalla molteplicità di $M$ tramite una famiglia di costanti di...

In the paper, a fractional continuous Roesser model is considered. Existence and uniqueness of a solution and continuous dependence of solutions on controls of the nonlinear model are investigated. Next, a theorem on the existence of an optimal solution for linear model with variable coefficients is proved.

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.