$\rho \in L^1(0,T;\mathbb{R})$ such that $$\limsup_{h\to 0^+} \frac{g(t+h) - g(t)}{h}\leq \rho(t) \;\;\; a.e.
\; t\in[0,T].$$ Then $$g(\beta)-g(\alpha) \leq \int_\alpha^\beta \limsup_{h\to 0^+}
\frac{g(t+h) - g(t)}{h} {\mathrm{d}}t\;\;\;\; \forall \;
0\leq\alpha\leq\beta\leq T.$$

We will denote by $B_R$ the open ball of radius $R$ centered at $0$ in $\mathcal{H}$.

The verification theorem
========================

We first introduce a set related to a subset of the superdifferential of a function in $C((0,T)\times\mathcal{H})$. Its definition is suggested by the definition of a sub/super solution. We recall that the superdifferential $D^{1,+}v(t,x)$ of $v \in
C((0,T)\times\mathcal{H})$ at $(t,x)$ is given by the pairs $(q,p)\in \mathbb{R}\times \mathcal{H}$ such that $v(s,y) - v(t,x)
 - \left\langle p, y-x \right\rangle - q(s-t) \leq o(\|x-y\| + |t-s|)$, and the subdifferential $D^{1,-}v(t,x)$ at $(t,x)$ is the set of all $(q,p)\in \mathbb{R}\times \mathcal{H}$ such that $v(s,y) - v(t,x) -
\left\langle p, y-x \right\rangle - q(s-t) \geq o(\|x-y\| + |t-s|)$.

\[defE\] Given $v\in C((0,T)\times\mathcal{H})$ and $(t,x)\in(0,T)\times \mathcal{H}$ we define $E^{1,+} v(t,x)$ as $$\begin{array}{ll}
E^{1,+}v(t,x)= \{ (q,p_1,p_2)\in \mathbb{R}\times D(A^*)
\times\mathcal{H} : &
\exists \varphi\in test1, \; g\in test2\; s.t.\\
& v-\varphi-g \text{ attains a local}\\
& \text{maximum at } (t,x),\\
& \partial_t(\varphi+g)(t,x)=q,\\
& D\varphi(t,x)=p_1, \;\; Dg(t,x)=p_2\\
& and \; v(t,x)= \varphi(t,x)+g(t,x) \}
\end{array}$$

If we define $$E^{1,+}_1v(t,x) =\{(q,p)\in 