Neumann Problem Solutions

Question

I want to prove that two solutions to a Neumann problem differ at most by a constant. $$\bigtriangledown^2=f \ in \ D$$ $$\frac{\partial{u}}{\partial{n}}=g \ on\ B$$ I don't know how to approach the problem as the interior Neumann problem in the book was solved in a very confusing way.

Answer 1

Let $u_1$ and $u_2$ be two solutions to the problem, and consider their difference $w=u_2-u_1$. We see that

$$\begin{cases} \nabla^2w=\nabla^2u_2-\nabla^2u_1=f-f=0, \\ \frac{\partial{w}}{\partial{n}}=\frac{\partial{u_2}}{\partial{n}}-\frac{\partial{u_1}}{\partial{n}}=g-g=0. \end{cases}$$

Now, we use Green's first identity on the product $w\nabla^2w$:

$$\int_Dw\nabla^2w\,\mathrm{d}\mathbf{x}=\int_Bw\frac{\partial{w}}{\partial{n}}\,\mathrm{d}S-\int_D|\nabla{w}|^2\,\mathrm{d}\mathbf{x}$$

But $\nabla^2w=0$ and $\frac{\partial{w}}{\partial{n}}=0$, so

$$\int_D|\nabla{w}|^2\,\mathrm{d}\mathbf{x}=0.$$

It follows that $\nabla{w}\equiv\mathbf{0}$, so $w\equiv\text{constant}$ and $u_2=u_1+\text{constant}$, which is what we wanted to prove.