 Solution

The assumed inequality is equivalent toand equality holds at $w$. For $v\in H$ and $t\in \mathbb{R}$, expand $J(w+tv)\ge 0$ and use $J(w)=0$:Since this holds for both signs of arbitrarily small $t$, the linear coefficient vanishes:Every $v\in H^1(U)$ is a mean-zero function plus a constant. The same identity holds for constants because ${\int }_{U}w=0$, so it holds for all $v\in H^1(U)$. This is precisely the weak formulation ofwhere the Neumann boundary condition is the natural boundary condition encoded by the weak formulation.