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Consider Poisson's equation $\nabla^2 u = 1$ on a square of side-length 1 centered at the origin. Cut out a circle of radius 1/3 at the center of this square.

Impose a von Neumann boundary condition $\frac{\partial u}{\partial n} = 0$ at the outer (squared) boundary, and a Dirichlet boundary condition $u = 0$ at the inner (circular) boundary.

I would like to find either of:

  1. An analytic solution (of course!).
  2. Asymptotics at any of the corners.

Thank you.

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  • $\begingroup$ this is the Sinai billiard --- no analytic solutions; asymptotics at the origin? there is a hole at the origin.... $\endgroup$ Oct 25, 2013 at 14:25
  • $\begingroup$ @CarloBeenakker Thank you, I'll look up more on the Sinai billiard. I meant asymptotics at any of the corners (I have corrected this in the question). $\endgroup$
    – rodms
    Oct 25, 2013 at 17:29

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