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Suppose the values of $a$, $b$, $V_1$ and $V_2$ is given. I want to find the solution of the Laplace equation, $$\frac{\partial^2 \phi}{\partial x^2}+\frac{\partial^2 \phi}{\partial y^2}=0$$ in the orange region of the figure (with the boundary conditions given in the figure).

Can we use the separation of variable method for solving this problem? I have tried to find the solution by separation of variables, but it seems the the boundary conditions can not be satisfied. Do you have any idea for solving this problem, theoretically?

figure

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  • $\begingroup$ An naive (but useful) advise is: Use superposition principle! $\endgroup$ – Dox Jul 18 '14 at 16:15
  • $\begingroup$ For using superposition principle, we must choose complete eigen functions. Which eigenfunctions can I use? $\endgroup$ – Mojtaba Golshani Jul 18 '14 at 17:06
  • $\begingroup$ If you want analytic solution, perhaps the method of complex functions will help you: feynmanlectures.caltech.edu/II_07.html $\endgroup$ – Ján Lalinský Jul 18 '14 at 17:49
  • $\begingroup$ Well, you could try solving it numerically. The interior is obviously going to be a pain, but it should be doable. $\endgroup$ – Kyle Kanos Jul 18 '14 at 19:55
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If we ignore the inner section, we have a box with 3 sides held at V = 0 and the top edge at V = V1. I'm pretty sure this is easily solvable by separation of variables using an oscillatory solution in x with a decaying solution along y. Using superposition we can then treat the inner box as a separate problem of similar geometry/boundary conditions. The exact solution would be a bit tedious with the origin being held at a corner of the larger box but I don't see how it's not solvable in principle.

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OK, I will try to say something a bit more useful than my other response. I am still on the skeptical side of getting a closed-form, simple solution to this problem, specially using separation of variables. I think that the problem is that the potential outside of a square with that border conditions cannot be attacked using separation of variables. My reasoning is that, whatever the solution, very far from the square it should behave as $$\ln \frac{1}{\sqrt{x^2+y^2}},$$ and it is not hard to prove that this is inconsistent with a potential expressible as the product of a function depending on $x$ and a function depending on $y$. Since the potential inside a square can be solved by separation of variables, if the problem of this question could be solved by separation, then by superposition we could solve the problem of the external field to a square by separation of variables. Therefore, the original problem cannot be solve by separation of variables.

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