Int}(D_i)$.

Let $\widetilde D_i\subset {\ensuremath{\mathbb{R}}}^N$ be such that $D_i\Subset \widetilde D_i$. Note that $\overline h_{i,\epsilon}(\cdot)$ can be continuously extended to $\widetilde D_i$. Without loss of generality, we may then assume that $\overline h_{i,\epsilon}(\cdot)$ is a continuous function on $\tilde D_i$ and there is $x_0\in {\rm Int}(D_i)$ such that $\overline h_{i,\epsilon}(x_0)=\sup_{x\in\widetilde D_i}\overline h_{i,\epsilon}(x)$. Observe that there is $\sigma>0$ and $\widetilde h_{i,\epsilon}(\cdot)\in C(\widetilde D_i)$ such that $B(x_0,\sigma)\Subset D_i$, $$\label{eq3}
0\leq\widetilde h_{i,\epsilon}(x)-\overline h_{i,\epsilon}(x)\le\frac{ \epsilon}{3}\quad \forall \,  x\in\widetilde D_i,$$ and $$\widetilde h_{i,\epsilon}(x)=\widehat h_{i,\epsilon}(x_0)\quad \forall \, x\in B(x_0,\sigma).$$

Let $$\eta(x)=\begin{cases} C\exp(\frac{1}{\|x\|^2-1})\quad &{\rm if}\,\ \|x\|<1,\cr\cr
0\quad &{\rm if}\,\ \|x\|\geq 1,
\end{cases}$$ where $C>0$ is such that $\int_{{\ensuremath{\mathbb{R}}}^N}\eta(x)dx=1$. For given $\delta>0$, set $$\eta_\delta(x)=\frac{1}{\delta^N}\eta\left(\frac{x}{\delta}\right).$$ Let $$h_{i,\epsilon,\delta}(x)=\int_{\widetilde D_i}\eta_\delta(y-x)\widetilde h_{i,\epsilon}(y)dy.$$ By [@Ev Theorem 6, Appendix C], $ h_{i,\epsilon,\delta}(\cdot)$ is in $C^\infty(\widetilde D_i)$ and when $0<\delta\ll 1$, $$\label{eq4}
|h_{i,\epsilon,\delta}(x)-\widetilde h_{i,\epsilon}(x)|<\frac{\epsilon}{3}\quad \forall \, x\in\bar D_i.$$ It is not difficulty to see that for $0<\delta\ll 1$, $$h_{i,\epsilon,\delta}(x)\leq
\widetilde h_{i,\epsilon}(x_0)\quad \forall x\in B(x_0,\sigma),$$ and $$h_{i,\epsilon,\delta}(x)=\widetilde h_{i,\epsilon}(x_0)\quad
\forall x\in B(x_0,\sigma/2).$$ Fix $0<\delta\ll 1$. Let $$\widehat h_{i, \epsilon}(x)=h_{i,\epsilon,\delta}(x).$$ Then $\widehat h_{i, \epsilon}(\cdot)$ attains its maximum at some $x_0\in{\rm Int}(D_i)$, and the partial derivatives of $\widehat h_{i,\epsilon}(\cdot)$ up to order $N-1$ at $x_0$ are zero. Let $$m_{i, \epsilon}(t, x)= \overline m_{i, \epsilon}(t, x)   +\widehat h_{i, \epsilon}(x)-\overline h_{i,\epsilon}(x).$$ 