# Neumann Boundary Conditions from a solution of a Dirichlet problem

Given a two dimensional Dirichlet boundary problem:
$-\Delta u =0$ in $\Omega$,
$u=g_D$ on $\Gamma=\partial \Omega$.
with $\Omega=\{x \in \mathbb{R}^2:| x|<r\}$

I have a solution $u(x)=e^{2 \pi x_1}\cos(2 \pi x_2)$, $x \in \mathbb{R^2}$ of this problem, so that $g_D(x)=u(x)$ on $\Gamma$.
Now I want to obtain the Neumann boundary data which are given by $g_N=\frac{\partial u}{\partial n}=n\cdot \nabla u$. To get this data do I just have to calculate nabla of $u$ and $n$ is just the normal on $\Gamma$, the normal on the circle with radius r?

## 1 Answer

The normal vector for a circle for a point $(x_1,x_2) \in \partial \Omega$ is simply given by $\vec{n}=\frac{1}{r}(x_1,x_2)$. Using the formulas known, you can simply calculate $\frac{\partial u}{\partial \vec{n}}=<\nabla f, \vec{n}>=1/r(2 \pi e^{2 \pi x_1}cos(2\pi x_2)x_1-e^{2 \pi x_1}sin(2 \pi x_2)2 \pi x_2)=\frac{2 \pi e^{2 \pi x_1}}{r}(sin(2 \pi x_2)x_1-sin(2 \pi x_2)x_2)$