Is it true that $\xi \in \partial G (v)$ implies $\frac{\xi}{F'(\phi (v))} \in \partial \phi (v)$?

Question

I am reading the introduction of Chapter 10 in the book Gradient Flows by Ambrosio and his coauthors.

As we have seen in Section 1.4, in the classical theory of subdifferential calculus for proper, lower semicontinuous functionals $\phi: X \rightarrow(-\infty,+\infty]$ defined in a Hilbert space $X$, the Fréchet Subdifferential $\partial \phi: X \rightarrow 2^X$ of $\phi$ is a multivalued operator defined as $$ \xi \in \partial \phi(v) \Longleftrightarrow v \in D(\phi), \quad \liminf _{w \rightarrow v} \frac{\phi(w)-\phi(v)-\langle\xi, w-v\rangle}{|w-v|} \geq 0, \qquad (*) $$ which we will also write in the equivalent form for $v \in D(\phi)$ $$ \xi \in \partial \phi(v) \Longleftrightarrow \phi(w) \geq \phi(v)+\langle\xi, w-v\rangle+o(|w-v|) \quad \text { as } w \rightarrow v $$

Above $D(\phi) := \{v \in X : \phi(v) \neq + \infty\}$ is the proper domain of $\phi$. Now we assume that $\phi$ is non-negative. We consider another function $F: [0, +\infty) \to [0, +\infty)$ with the convention that $F(+\infty) = +\infty$. We assume that $F$ is convex differentiable such that $F'(x) > 0$ for all $x \in [0, +\infty)$.

Let $G := F \circ \phi$. I would like to ask if the following "chain rule" holds, i.e., $$ \xi \in \partial G (v) \implies \frac{\xi}{F'(\phi (v))} \in \partial \phi (v). $$

Thank you so much for your elaboration!

Answer 1

$\renewcommand{\p}{\partial}\newcommand{\ep}{\varepsilon}\newcommand{\de}{\delta}\newcommand{\ip}[2]{\langle #1,#2\rangle}$The answer is yes.

Indeed, suppose the contrary: for some $v\in X$, some $\zeta\in\p G(v)$, some sequence $(w_n)$ in $X$ converging to $v$, some real $\ep>0$, and all $n$ we have
\begin{equation} \phi(w_n)\le\phi(v)+\ip\xi{w_n-v}-\ep|w_n-v|, \tag{1}\label{1} \end{equation} where \begin{equation} \xi:=\frac{\zeta}{F'(\phi(v))}. \end{equation} The condition $F'>0$ implies that $F$ is increasing. So, \eqref{1} and the mean value theorem imply \begin{equation} F(\phi(w_n))\le F(\phi(v)+\ip\xi{w_n-v}-\ep|w_n-v|) \\ =F(\phi(v))+F'(c_n)(\ip\xi{w_n-v}-\ep|w_n-v|) \end{equation} for some sequence $(c_n)$ converging to $\phi(v)$. Also, since $F$ is convex and differentiable, we see that $F'$ is continuous and hence $F'(c_n)\to F'(\phi(v))$. So, without loss of generality, for some real $\de>0$ -- say, for $\de=F'(\phi(v))\ep/2$ -- and all $n$ we have $F(\phi(w_n))\le F(\phi(v))+\ip\zeta{w_n-v}-\de|w_n-v|$, that is,
\begin{equation} G(w_n)\le G(v)+\ip\zeta{w_n-v}-\de|w_n-v|, \end{equation} which contradicts the condition $\zeta\in\p G(v)$. $\quad\Box$