Limiti di soluzioni di problemi variazionali con ostacoli bilaterali

Dal Maso, Gianni

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 69 (1980) no. 6, pp. 333-337

Résumé

If $\{\varphi_{h}\}$ and $\{\psi_{h}\}$ are sequences of arbitrary functions from $\mathbf{R}^{n}$ into $\bar{\mathbf{R}}$, with $\varphi_{h} \le \psi_{h}$, then there exist two subsequences $\{\varphi_{h_{k}}\}$ and $\{\psi_{h_{k}}\}$, a function $f(x,u)$ convex in $u$, and two positive Radon measures $\mu$ and $\nu$, with $\mu \in H^{-1}(\mathbf{R}^{n})$, 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 $\{\varphi_{h_{k}}\} \le u \le \{\psi_{h_{k}}\}$ 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 external constraint.