\mbox{ and }\,\Sup_{x>0,y>0} \frac{|f(t,x)-f(t,y)|}{|x-y|(1+x^{-(\gamma+1)}+y^{-(\gamma+1)})}<\infty\Big\}.\end{aligned}$$ The following lemma gives some properties of the dual value function $\tilde v$.

\[classe\] We assume that there exists a solution the the dual problem . The following properties hold:\
1) The dual value function $\tilde v$ is convex in $y$,\
2) The dual value function $\tilde v$ satisfies the following growth condition $$\begin{aligned}
\label{growthc}
\Sup_{y>0} \frac{|\tilde v(t,y)|}{y+y^{-\gamma}}<\infty\end{aligned}$$ 3)The dual value function $\tilde v$ satisfies $$\begin{aligned}
\label{acc}
\Sup_{y_1>0,y_2>0} \frac{|\tilde v(t,y_1)-\tilde
  v(t,y_2)|}{|y_1-y_2|(1+y_1^{-(\gamma+1)}+y_2^{-(\gamma+1)})}<\infty.
 \end{aligned}$$

[**Proof.**]{} See Appendix.\
\

When we assume that there exists a solution the the dual problem , one can use the conjugate duality relation proved in theorem 5.1 of Mnif and Pham [@mnipham01].

Since the dual value function $\tilde v$ is locally bounded, the upper and the lower semi-continuous envelope of the function $\tilde v$ are well-defined. The definition of the upper and the lower semi-continuous envelope of a function $\phi$ are given as follows.

\(i) The upper semi-continuous envelope of a function $\phi$ is $$\begin{aligned}
\phi^*(t,y)= \Limsup_{\stackrel{(t^\prime,y^\prime)\rightarrow (t,y)}{t^\prime\in [0,T),\,y^\prime>0}}
\phi(t^\prime,y^\prime),\,\, \mbox{ for all }(t,y)\in [0,T)\times (0,\infty).\end{aligned}$$ (ii) The lower semi-continuous envelope of a function $\phi$ is $$\begin{aligned}
\phi_*(t,y)= \Liminf_{\stackrel{(t^\prime,y^\prime)\rightarrow (t,y)}{t^\prime\in [0,T),\,y^\prime>0}}
\phi(t^\prime,y^\prime),\,\, \mbox{ for all }(t,y)\in [0,T)\times (0,\infty).\end{aligned}$$

Since the continuity of the 