I am having a cost function and I want to know whether convex or not. Could you explain help me my problem? My problem is that given a cost function such as $$F(u)=\int_{0 \le u(x) \le 1,\Omega}g^2(x)u(x)dx+\int \|\nabla u(x) \|dx$$
where $g: \Omega \to \mathbb R$,$u(x)>0$ and $\sum_i u_i(x) =1$ the second term looks like total variance, then it is convex function.
My problem is that how to prove $F(u)$ is a convex function?
In addition, if I have two function: first is convex and second is also convex function. Is sum of two functions convex or not?