i have got a problem with some assumptions to solve a parabolic variational inequality. My Problem is: Find a function $u$ with


\begin{align}
u\in L^2(0,T;V),~ u' \in L^2(0,T;V') \\
(u'(t),v-u(t)) + a(u(t),v-u(t)) + \psi(v) -\psi(u(t)) \geq (f(t),v-u(t)) \\
u(0) = u\_0.
\end{align}


We want to approximate $\psi$ with its Yosida-Approximation $\psi\_j$ (it is convex and lower-semicontinuous.) Now the Duvaut & Lions postulate, that there is a bounded sequence $\lbrace \varphi\_j\rbrace \subset V$, so that
\begin{align}
\psi\_j'(\varphi\_j) = 0 ~\forall j \in \mathbb{N}.
\end{align}


Is there any result, that proves this postulation? Thanks in advance, FFoDWindow



Derivative of Yosida-Approximation
Since no one else has written an answer, I'm copying my comment:


If $\psi$ is convex and bounded from below (which I'm assuming since you didn't state anything), the Moreau envelope $\psi\_j$ admits a minimizer (in fact, the same as $\psi$) for any $j$. Since the Moreau envelope is differentiable, this minimizer -- call it $\phi\_j$ -- must satisfy the first-order necessary optimality condition $\psi\_j'(\phi\_j)=0$.


For the mentioned properties (and many, many more) of the Moreau envelope (and the Yosida approximation of subdifferentials), see Bauschke and Combette, *Convex Analysis and Monotone Operators in Hilbert Spaces*, Springer 2011.


