# Laplace's equation with periodic Dirichlet boundary conditions

Consider a Laplace's equation with Dirichlet boundary condition:

${\nabla ^2}\Phi = 0$ in a domain $D$ with given Dirichlet Boundary condition: $\Phi=\Phi_o$ at $\partial D$ (smooth, but not analytic boundary)

The problem is, that $\Phi_o$ is a periodic function and it changes depending on the position.

For example, $\Phi_o(x,y) = \tan^{-1}(y/x)$ for elliptic domain, ($\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$) and thus $\Phi\in [0,2\pi)$; in this case, the non-continuous boundary condition arises on $(x=a,y=0)$ that divides two distinct values ($\Phi_o(y>0) = 2\pi$ and $\Phi_o(y<0)=0$).

As a consequence, the numerical discontinuity is observed at some of the boundary and deteriorate the solution greatly. Is there any possible treatment to aid this problem? Any comment would be greatly appreciated.

## 1 Answer

This is not just a "numerical discontinuity". Are you sure that this is the problem you want to solve? $\Phi$ looks like a phase, which is defined up to $2\pi$. That may be what you refer to as "periodic", although the context makes it unclear as one thinks of space-periodicity of BCs.

If your solution $\Phi$ is up to $2\pi$, I would rather solve e.g. for $u=\cos\Phi$.

• Thanks for your comment. I haven't consider such possibility of misreading. In this context, would you please help me clearify the solution? As I have only minimal mathematical background, it would be great if the question I've raised becomes more mathematically correct and meaningful and not complexifies other readers. – Hayoung.Chung Nov 4 '15 at 0:45
• Clarifying the question you mean? I think you should post more details on where the problem comes from. – Joce Nov 5 '15 at 9:56
• I have a perfect example: a liquid crystal. Their interaction energy is governed by how they are distributed, and this is where such phase (as you mentioned) is originated. Laplace equation, therefore, is method to obtain lowest liquid crystal energy . If this answer is not enough, please look Frederiks distortion energy in wikipedia. – Hayoung.Chung Nov 5 '15 at 10:56
• There is no generic answer I believe. Please ask a question with sufficient detail. – Joce Nov 5 '15 at 12:00