Minimize integral subject to integral constraint

Question

I am trying to find the function $f:[0,1]^2\to\mathbb{R}$ which minimizes the integral $$\iint_{[0,1]^2}f_{x}(x,y)^2+f_{y}(x,y)^2\,dxdy,$$ subject to the constraint $\iint_{[0,1]^2} f(x,y)^2\,dxdy=1$. I think I am supposed to use the calculus of variations, but I'm not even sure how to start.

Answer 1

For every function we have that $$f_x^2(x,y)+f_y^2(x,y)\ge 0$$therefore$$\iint f_x^2(x,y)+f_y^2(x,y)dxdy\ge0$$so take $f(x,y)=1$ and this is the only funcions satisfying our problem.