Trying to solve this:
$$\min TC(A,a,q)= \int_M f(A,a,q)\,dx\, dy$$ $$s.t.$$ $$a\le\int_M g_i(A,q)\,dx\,dy$$ $$q\le \text{constant}$$ $$A,a,q\ge0$$
$(x,y)$ is omitted in $A(x,y), a(x,y), q(x,y)$ for simplicity.
I would like to find corner and interior solutions. q is separable in the objective function.
I have looked at calculus of variations but I could not find anything about nonnegativity and inequality integral constraints.