Let $u$, $v\in C_c^\infty$ and $\Omega\subset \mathbb R^N$ is open bounded, smooth boundary. We also assume that $0\leq v\leq 1$.
Define $$ F(u,v):=\int_\Omega |\nabla u|^2v^2dx. $$ Do we have $F(u,v)$ is convex? i.e., for any $0<t<1$, do we have $$ F(tu_1+(1-t)u_2, tv_1+(1-t)v_2)\leq tF(u_1,v_1)+(1-t)F(u_2,v_2) $$
I know for instance that the product of two nonnegative, nondecreasing convex function is convex, but it does not apply to this case since $\nabla u$ is not always non-negative.
As noted in a comment by Rahul, this functional is not convex. Indeed, let $B_1,B_2$ be two disjoint closed balls in $\Omega$ and let $u,v$ be bump functions with support $B_1,B_2$ respectively. Then $F(u,v)=F(v,u)=0$, but $$F\left(\frac{u+v}{2},\frac{u+v}{2}\right) = F(u/2,u/2)+F(v/2,v/2)>0$$ showing that $F$ is not convex.
$F$ is separately convex in each variable, though.