I had a question on the following problem.
Show that heat flow $$f_t=\Delta f$$ minimizes $$E(f)=\int_c^d\int_a^b|\nabla f|^2\ dx\ dy.$$ under von Neumann boundary conditions at the boundary of the rectangle $[a,b]\times[c,d]$.
I'm kind of stuck on solving this. My professor said to try starting by taking the derivative of $|\nabla f|^2$
What I have so far \begin{align} E(f)&=\int_c^d\int_a^b|\nabla f|^2\ dx\ dy\\ |\nabla f|^2&=\bigg(\underbrace{\dfrac{\partial f}{\partial x}}_g\bigg)^2+\bigg(\underbrace{\dfrac{\partial f}{\partial y}}_h\bigg)^2\\ \dfrac{\partial}{dt}\left(|\nabla f|^2\right)&=\left(2\cdot\dfrac{\partial f}{\partial x}\cdot\dfrac{\partial\left(\dfrac{\partial f}{\partial x}\right)}{\partial t}\right)+\left(2\cdot\dfrac{\partial f}{\partial y}\cdot\dfrac{\partial\left(\dfrac{\partial f}{\partial y}\right)}{\partial t}\right)\\ 2|\nabla f|&=\left(2\cdot\dfrac{\partial f}{\partial x}\cdot\dfrac{\partial^2f}{\partial x\,\partial t}\right)+\left(2\cdot\dfrac{\partial f}{\partial y}\cdot\dfrac{\partial^2f}{\partial y\,\partial t}\right)\\ &=(2. \end{align}
Any tips on how to finish the problem? Thanks!