Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 2 / 31E / b

Testing a weak Neumann Poisson problem with the constant function one gives ${\int }_{\Omega }f=0$, proving necessity. For sufficiency, use the closed Hilbert space of mean-zero $H^1$ functions. The given Poincaré inequality makes $\Vert \nabla v{\Vert }_{L^2}$ an equivalent norm there. SetThe negative sign corresponds to $\Delta u=f$. Poincare's inequality gives boundedness of $\ell$ and coercivity of $a$. The Lax-Milgram theorem supplies a unique mean-zero solution. Every $v\in H^1$ is the sum of its mean-zero part and a constant; since $\int f=0$, the weak identity extends to every $v\in H^1$. This is precisely the weak equation with zero normal derivative as its natural boundary condition. Existence and uniqueness in the mean-zero space hold exactly when $\bar{f}=0$.