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This question might sound naive but on p.34 of his book, he is considering the Poisson-dirichlet problem with $\Delta u=-f$ on $U$ with $u=g$ on $\partial U$. He then derives a formula for the general solution but says that it is not much useful since $\frac{\partial u}{\partial \nu}$ on $\partial U$ is unknown.

But this seems strange to me because isn't it that $\frac{\partial u}{\partial \nu} = \frac{\partial g}{\partial \nu}$ on $\partial U$ since $\frac{\partial u}{\partial \nu}$ on $\partial U$? Here, $\frac{\partial u}{\partial \nu} = Du\cdot \nu$ where $\nu$ is the unit normal to $U$.

If indeed I misunderstood the above, then if I had two boundary conditions $g_1$ and $g_2$ then could it be possible that $\frac{\partial u}{\partial \nu}$ on $\partial U$ is different for each boundary condition?

Thanks!

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In general $g$ is not defined off the boundary so taking the directional derivative in the normal direction is not well defined. –  Evan Sep 23 '13 at 4:28

1 Answer 1

Evan is rigth. I will try to clarify his comment with an example.

Let's assume $g=0$, $f=-1$ and $U=\mathbb{D}$. Then $u(x,y)=-0.25(x^2+y^2)+0.25$, but, easily, the normal derivative is not zero.

In general, the operator $$ T_U:g\rightarrow \partial u/\partial \nu, $$ i.e., such that, given $g$, it solves for $u$, and then return the normal derivative is called the Dirichlet-to-Neumann map for the laplacian operator and the domain $U$. These operator have a lot of importance in inverse problems and fluid dynamics.

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