My problem
I am trying to solve a chemistry problem stating it as a constrained variational problem. I am learning this subject by myself and I have problems imposing a non-integral constrain.
One constrain is of the form:
$$ \int \int c_1 f(x,y) dx dy = C_1$$
The funcitional to be minimized is
$$ \int \int \left[ c_2 f(x,y) g(x,y) + c_3 f^2(x,y) \right] dx dy$$
where $g(x,y)$ is known and both integrals share the same integration limits. $C_1, c_1, c_2, c_3$ are also known constants. So up to this point I wrote as my lagrangian as
$$ L = c_2 f(x,y) g(x,y) + c_3 f^2(x,y) + \lambda f(x,y)$$
There is not explicit dependence of $L$ with derivatives of $f(x,y)$. The problem appears when I want to impose:
$$ f(x,y) \ge 0$$
I am not 100% sure about how to do it, and, if I did it right, I do not know how use take advantage of that constrain while solving the Euler-Lagrange equations.
My efforts
I tried using a substitution $f(x,y)=u(x,y)^2$, but this leads me to the same solution than before.
I made some tries with the sign function, replacing $f(x,y)$ in my constrain with $c_1 f(x,y)sign(f(x,y))$. I am not sure that this is right and also leads me to an Euler-Lagrange equation that I cannot solve using my knowledge in math. I also tried to approximate the $sign$ function with similar results.
I tried adding $-\lambda(x,y) f(x,y)$ to the Lagrangian, but I am not sure if that is correct. I see no difference with imposing $ 0 \ge f(x,y) $ instead. In any case, I added a new E-L equation to be satisfied in which $\lambda(x,y)$ plays the roll of $f(x,y)$, but it trivially leads me to $f(x,y)=0$, so I am doing something wrong.
Question
How to impose $ f(x,y) \ge 0$, and, if I did it right, how to solve the Euler-Lagrange equation?
Thank you