= Neumann Poisson problem
{c}
{title2=$-\Delta u=f,\quad\partial_nu=0$}
On a smooth bounded connected domain, the homogeneous Neumann problem seeks $u\in H^1(U)$ such that $\int\nabla u\cdot\nabla v=\int fv$ for every $v\in H^1(U)$. For $f\in L^2(U)$ it is solvable exactly when $\int f=0$, and the <weak solution> is unique up to constants. The <Poincare-Wirtinger inequality> and the <Riesz representation theorem> give existence on the <mean-zero Sobolev space>. The test function one gives necessity of the compatibility condition.
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