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In this paper we complete the characterization of those $f$, $\mu$ and $\nu$ such that $${\parallel w\parallel }_{H^1(\mathrm{\Omega })}^2+{\int }_{B}f(x,w(x))d\mu +\nu (B)$$ is $\mathrm{\Gamma }(L^2{(\mathrm{\Omega })}^{-})$ limit of a sequence of obstacles ${\parallel w\parallel }_{H^1(\mathrm{\Omega })}^2+{\mathrm{\Phi }}_h(w,B)$ where

In this paper we complete the characterization of those $f$, $\mu$ and $\nu$ such that $${\parallel w\parallel }_{H^1(\mathrm{\Omega })}^2+{\int }_{B}f(x,w(x))𝑑\mu +\nu (B)$$ is $\mathrm{\Gamma }(L^2{(\mathrm{\Omega })}^{-})$ limit of a sequence of obstacles ${\parallel w\parallel }_{H^1(\mathrm{\Omega })}^2+{\mathrm{\Phi }}_h(w,B)$ where

In this paper we show that all limits of sequences of minimum problems of the type $$\min \left\{\int {|Du|}^2+\int {|u|}^2:u\ge {\psi }_h\mathit{\hspace{1em}}\text{ on }B\right\}$$ are minimum problems of the type $$\min \left\{\int {|Du|}^2+\int {|u|}^2+{\int }_{B}f(x,u)𝑑\mu +\nu (B)\right\}$$ where $\mu$, $\nu$ are positive Radon measures, $\mu \in H^{-1}$ and $f(x,u)$ is convex and non-increasing in the variable $u$.