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Given the PDE, $\Delta u=x$ in the region $x^2+y^2<1$. And $\frac{\partial u}{\partial r}=y$ on $x^2+y^2=1$. I am supposed to find all solutions.

The only machinery I know for finding solutions on bounded domains is separation of variables. But I only know how to do separation of variables for homogeneous problems. So I found the solution to $\Delta u=0$ is $u(x,y)=y$. But it is unclear how to extend this to inhomogeneous case. (It amounts to solving $\Delta u=x$ with $\frac{\partial u}{\partial r}=0$)

Maybe there some transform that turns $\Delta u=x$ into Laplace's equation? Thank-you.

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Procedure is the same – find general solution of homogeneous problem, particular solution of inhomogeneous problem, and then add them up to get general solution of inhomogeneous problem. – Kaster Mar 5 '13 at 2:47

One approach goes as follows. First, you make the boundary condition homogeneous. This can be done by introducing $v=u-g$, where $g$ is some function that satisfies $\partial_r g = y$ at the boundary of the disk. The function $v$ must then satisfy $\Delta v = x-\Delta g$. Now you use separation of variables. The idea is to expand the right hand side $f(x,y)=x-\Delta g$ in terms of the eigenfunctions of the Neumann Laplacian on the disk. So let $\phi_n$ be those eigenfunctions, and let $$ f = \sum_n b_n\phi_n. $$ We look for the solution in the form $$ v = \sum_n a_n\phi_n. $$ Then the equation is $$ \Delta v = \sum_n a_n\Delta \phi_n = \sum_n a_n\lambda_n\phi_n=\sum_n b_n\phi_n, $$ and so $$ a_n=b_n/\lambda_n, $$ that is $$ v = \sum_n \frac{b_n}{\lambda_n} \phi_n. $$ Note that we need $b_0=0$ because $\lambda_0=0$ for the Neumann Laplacian, and by a related reason the solution $v$ is unique up to a constant.

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I think we can use a green's function approach. The Neumann green representation is $$u(x) = -\int_U G(x,y)\Delta u(y) + \int_{\partial U} G(x,y)\frac{\partial u}{\partial \nu}$$ But I am still trying to think of the neumann green's function on disk. For your solution, how can I calculate the eigenfunctions of the neumann laplacian? When I try seperation of variables on $\Delta u =\lambda u$, I get a 2nd order ODE with non-constant coefficients, so I don't know how to solve. Thank-you – StuartHa Mar 6 '13 at 0:44
@Stuart: The eigenfunctions are epxressed in terms of the Bessel functions. – timur Mar 6 '13 at 2:14
Thank-you. But I am looking for a more explicit solution. I should have said this, sorry. – StuartHa Mar 13 '13 at 20:06
up vote 1 down vote accepted

I'll start by decomposing problem as follows, \begin{cases} \Delta u=x\\ \partial u/\partial r=0 \end{cases}

\begin{cases} \Delta v=0\\ \partial v/\partial r=y \end{cases}

The second problem is easy. You can do it with seperation of variables or just observe that the solution is $v=y$. For the first problem we can start by converting to polar coordinates,


Now it is natural to guess the solution is like $r^{3}\cos\theta$. Plugging this in yields that $\frac{1}{8}r^{3}\cos\theta$ works, and then you need to fix the BC so we have $u=\frac{1}{8}r^{3}\cos\theta-\frac{3}{8}r\cos\theta$. Adding $u$ and $v$ yields the solution,

$$\frac{x}{8}\left(x^{2}+y^{2}-3\right)+y + C$$

Now since I added the constant $C$, we have all solutions. This is because solution to Poisson with Neumann data is unique up to a constant. (Easy energy method to show this)

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