I've to admit that my original argument had a (stupid) mistake. However the argument below should work.

First let me relabel: call the two functions $u_1,u_2$ by $u_0,u_1$ so as to be compatible with $u_t:=tu_1+(1-t)u_0$.

Note that given any $u\in H^1(\Omega)$, the infimum in $$F(u)=\inf_{v\in \mathcal V}\{\int_\Omega|\nabla u|^2v^2 dx+\int_\Omega(|\nabla v|^2+(1-v)^2)dx\}$$ is obtained: take a minimizing sequence for $F(u)$ and, using the Rellich-Kondrachov theorem, find $v\in \mathcal V$ and a subsequence $v_k$ so that $v_k\to v$ in $L^2$, $\nabla v_k\rightharpoonup \nabla v$ weakly in $L^2$, and $v_k\to v$ pointwise almost everywhere. Then
\begin{align*}
F(u)\le &\int_\Omega|\nabla u|^2v^2 dx+\int_\Omega(|\nabla v|^2+(1-v)^2)dx\\
\le &\liminf_{k\to\infty}\left[\int_\Omega|\nabla u|^2v_k^2 dx+\int_\Omega(|\nabla v_k|^2+(1-v_k)^2)dx\right]=F(u).
\end{align*}

Now let $w_j$ be a function in $\mathcal V$ for which the infimum $F(u_j)$ is attained, $j=0,1$, and let $w_t=tw_1+(1-t)w_0$. By the edit below $$t\mapsto w_t^2|\nabla u_t|^2$$ is convex so we may estimate
\begin{align*}
F(u_t)\le & \int_\Omega |\nabla u_t|^2w_t^2 dx+\int_\Omega(|\nabla w_t|^2+(1-w_t)^2)dx\\
\le & t\int_\Omega|\nabla u_1|^2w_1^2dx+(1-t)\int_\Omega |\nabla u_0|^2w_0^2dx\\
&\ \ + t\int_\Omega(|\nabla w_1|^2+(1-w_1)^2)dx+(1-t)\int_\Omega(|\nabla w_0|^2+(1-w_0)^2)dx\\
=&tF(u_1)+(1-t)F(u_0).
\end{align*}
Thus $F$ is convex and this gives you the properties you want.

**EDIT:** Denote $$f(t)=f_x(t)=|\nabla u_t(x)|^2, g(t)=g_x(t)=w_t(x)^2.$$ Both are convex functions, but this alone doesn't guarantee the convexity of their product. Instead we use the fact that they are essentially quadratic.

Let us show that $(fg)''(t)\ge 0$. Compute
\begin{align*}
f'(t)=&2\nabla(u_1-u_0)\cdot \nabla u_t\\
g'(t)=&2(w_1-w_0)w_t\\
f''(t)=& 2|\nabla(u_1-u_0)|^2\\
g''(t)=& 2(w_1-w_0)^2.
\end{align*}
Next check that
\begin{align*}
(fg)''(t)&=f''(t)g(t)+2f'(t)g'(t)+g''(t)f(t)\\
=&2[|\nabla(u_1-u_0)|^2w_ t^2+4w_t\nabla(u_1-u_0)\cdot (w_1-w_0)\nabla u_t+(w_1-w_0)^2|\nabla u_t|^2]\\
\ge& 2(|\nabla(u_1-u_0)|w_t-|w_1-w_0||\nabla u_t|)^2-4|w_1-w_0|w_t|\nabla(u_1-u_0)||\nabla u_t|\\
\ge & 0.
\end{align*}

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