Let $f$ be a non-negative function that satisfies the following:
- $\int_0^\infty \int_M \vert \nabla f(x,t) \vert^2 \;dxdt <\infty$
- $\int_M f(x,\cdot)\;dx=0$
where $M$ is a compact subset of $\mathbb R^2$. Could we deduce that $f=0$?
What is the slightest regularity assumption that one should impose on $f$ in order for the limit $\lim\limits_{t \to \infty} \int_M \vert \nabla f(x,t) \vert^2 \;dx$ to exist?
Because if the limit exists then it has to be zero and since $\vert \nabla f(x,t) \vert^2 \ge 0$ then $\lim\limits_{t \to \infty}\int_M \vert \nabla f(x,t) \vert^2 \;dx=0$ yields that $f(x,\cdot)=const.$ Finally, by the second property we could conclude that $f=0$.
I would appreciate if somebody could help me complete this argument (or fix it in case I am mistaken). Any help is welcome.
Thanks in advance!