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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.

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.

If $\{{\phi }_h\}$ and $\{{\psi }_h\}$ are sequences of arbitrary functions from ${𝐑}^n$ into $\overline{𝐑}$, with ${\phi }_h\le {\psi }_h$, then there exist two subsequences $\{{\phi }_{h_k}\}$ and $\{{\psi }_{h_k}\}$, a function $f(x,u)$ convex in $u$, and two positive Radon measures $\mu$ and $\nu$, with $\mu \in H^{-1}({𝐑}^n)$, such that for every “admissible” open set $A$ and Borei set $B$, with $B\subseteq A$, and for every $g\in L^2(A)$, the sequences $\{m_k\}$ and $\{u_k\}$ of the minima and of the minimum points of the functional $${\int }_A\left[{|Du|}^2+{|u|}^2+gu\right]𝑑x,$$ with constraints of the type $\{{\phi }_{h_k}\}\le u\le \{{\psi }_{h_k}\}$ on $B$, converge respectively to the minimum $m_0$ and to the minimum point $u_0$ of the functional $${\int }_A\left[{|Du|}^2+{|u|}^2+gu\right]𝑑x+{\int }_{B}f(x,u)𝑑\mu +\nu (B),$$ without any additional...

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.

We introduce the space $GBD$ of generalized functions of bounded deformation and study the structure properties of these functions: the rectiability and the slicing properties of their jump sets, and the existence of their approximate symmetric gradients. We conclude by proving a compactness results for $GBD$, which leads to a compactness result for the space $GSBD$ of generalized special functions of bounded deformation. The latter is connected to the existence of solutions to a weak formulation of some variational...