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 $L^2(0,T,V^*)$?

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 $L^2(0,T,V^*)$?

I want to know, if there is any result, that generalizes the boundedness of the derivative of the Moreau-Enveloppe of a convex, lower semicontinuous functional. One can prove the boundedness from $V$ to $V^*$, but I need the boundedness from $L^2(0,T,V)$ to it's dual $L^2(0,T,V^*)$.

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 $L^2(0,T,V^*)$?