If the minimum problem (${𝒫}_{\mathrm{\infty }}$) is the limit, in a variational sense, of a sequence of minimum problems with obstacles of the type $$\underset{u\ge {\varphi }_h}{\min }{\int }_{\mathrm{\Omega }}\left[f_h(x,Du)+a(x,u)\right]dx,$$ then (${𝒫}_{\mathrm{\infty }}$) can be written in the form $${𝒫}_{\infty }\underset{u}{min}\left\{{\int }_{\Omega }\left[f_{\infty }(x,Du)+a(x,u)\right]dx+{\int }_{\overline{\Omega }}{g}_{\infty }(x,\overline{u}\left(x\right))d{\mu }_{\infty }\left(x\right)\right\}$$ without any additional constraint.

Se il problema di minimo $({𝒫}_{\mathrm{\infty }})$ è il limite, in senso variazionale, di una successione di problemi di minimo con ostacoli del tipo $$\underset{u\ge {\psi }_h}{\min }{\int }_A\left[f_h(x,u,Du)+b(x,u)\right]dx,$$ allora $({𝒫}_{\mathrm{\infty }})$ può essere scritto nella forma $$\underset{u}{\min }\left\{{\int }_A\left[f_{\mathrm{\infty }}(x,u,Du)+b(x,u)\right]dx+{\int }_A{g}_{\mathrm{\infty }}(x,\tilde{u}(x))d{\lambda }_{\mathrm{\infty }}(x)\right\}$$ dove $\tilde{u}$ è un conveniente rappresentante di $u$ e ${\lambda }_{\mathrm{\infty }}$ è una misura non negativa.

The aim of the paper is to provide a linearization approach to the $\mathrm{See\; PDF}$-control problems. We begin by proving a semigroup-type behaviour of the set of constraints appearing in the linearized formulation of (standard) control problems. As a byproduct we obtain a linear formulation of the dynamic programming principle. Then, we use the $\mathrm{See\; PDF}$ approach and the associated linear formulations. This seems to be the most appropriate tool for treating $\mathrm{See\; PDF}$ problems in continuous and lower semicontinuous setting.

The aim of the paper is to provide a linearization approach to the ${𝕃}^{\infty }$See PDF-control problems. We begin by proving a semigroup-type behaviour of the set of constraints appearing in the linearized formulation of (standard) control problems. As a byproduct we obtain a linear formulation of the dynamic programming principle. Then, we use the ${𝕃}^p$See PDF approach and the associated linear formulations. This seems to be the most appropriate tool for treating ${𝕃}^{\infty }$See PDF problems in continuous and lower semicontinuous...

The subject of this paper is the rigorous derivation of reduced models for a thin plate by means of Γ-convergence, in the framework of finite plasticity. Denoting by ε the thickness of the plate, we analyse the case where the scaling factor of the elasto-plastic energy per unit volume is of order ε2α−2, with α ≥ 3. According to the value of α, partially or fully linearized models are deduced, which correspond, in the absence of plastic deformation, to the Von Kármán plate theory and the linearized...

We provide a rigorous justification of the classical linearization approach in plasticity. By taking the small-deformations limit, we prove via $\Gamma$-convergence for rate-independent processes that energetic solutions of the quasi-static finite-strain elastoplasticity system converge to the unique strong solution of linearized elastoplasticity.

We give necessary and sufficient conditions which characterize the Young measures associated to two oscillating sequences of functions, un on ${\omega }_1\times {\omega }_2$ and vn on ${\omega }_2$ satisfying the constraint $v_n\left(y\right)=\frac{1}{|{\omega }_1|}{\int }_{{\omega }_1}{u}_n(x,y)dx$. Our study is motivated by nonlinear effects induced by homogenization. Techniques based on equimeasurability and rearrangements are employed.

The notion of local completeness is extended to locally pseudoconvex spaces. Then a general version of the Borwein-Preiss variational principle in locally complete locally pseudoconvex spaces is given, where the perturbation is an infinite sum involving differentiable real-valued functions and subadditive functionals. From this, some particular versions of the Borwein-Preiss variational principle are derived. In particular, a version with respect to the Minkowski gauge of a bounded closed convex...

Given the probability measure $\nu$ over the given region $\Omega \subset {ℝ}^n$, we consider the optimal location of a set $\Sigma$ composed by $n$ points in $\Omega$ in order to minimize the average distance $\Sigma \mapsto {\int }_{\Omega }\mathrm{dist}(x,\Sigma )\mathrm{d}\nu$ (the classical optimal facility location problem). The paper compares two strategies to find optimal configurations: the long-term one which consists in placing all $n$ points at once in an optimal position, and the short-term one which consists in placing the points one by one adding at each step at most one point and preserving...

Given the probability measure ν over the given region $\Omega \subset {ℝ}^n$, we consider the optimal location of a set Σ composed by n points in Ω in order to minimize the average distance $\Sigma \mapsto {\int }_{\Omega }\mathrm{dist}(x,\Sigma )\mathrm{d}\nu$ (the classical optimal facility location problem). The paper compares two strategies to find optimal configurations: the long-term one which consists in placing all n points at once in an optimal position, and the short-term one which consists in placing the points one by one adding at each step at most one point and preserving...

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...