Let $V,H,V^\*$ be a Gelfand-Triple, $\phi\colon V \to \mathbb{R}$ convex, lower semicontinuous and proper. There exists a so called Moreau-Enveloppe $\phi\_j$, which is Gateâux-differentialable. It's derivative $\phi\_j'\colon V \to V^\*$ is demicontinuous and maps bounded sets to bounded sets in $V^\*$. Now my question: Is the mapping $\tilde{\phi\_j'} \colon L^2(0,T,V) \to L^2(0,T,V^\*)$ defined by $\tilde{\phi\_j'}(u)(t) \colon = \phi\_j'(u(t))$ well-defined? And are in $L^2(0,T,V)$-bounded sets mapped to bounded sets in $L^2(0,T,V^\*)$? 



Moreau-Enveloppe from $L^2(0,T;V) \to L^2(0,T;V^*)$
Your assumptions are not sufficient to consider this mapping property.


Consider $V = \mathbb{R}$ and the mapping $f(x) = x^3$. It maps bounded sets to bounded sets, but its Nemytskii operator $y \mapsto y^3$ is not well defined from $L^2(0,T)$ into itself, only from $L^{3p}(0,T)$ into $L^p(0,T)$, $p \ge 1$.


