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We know Poisson Integral Formula can solve the Dirichlet boundary value problem for the Laplace equation on the unit disk. But for such a problem as below:

Find the solution to the Dirichlet boundary value problem for the Laplace equation on the pie wedge $\displaystyle W=\left\{0<\theta<\frac{\pi}{4}, \quad 0<r<1\right\}$, when the nonzero boundary data $u(1, \theta)=h(\theta)$ appears only on the curved portion of its boundary.

Why can't we use Poisson Integral Formula to solve the Laplace equation on this pie wedge? I mean, it seems that the solution shouldn't change when we use a sub-region of the original problem.

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  • $\begingroup$ You can if there is a Poisson kernal for this domain. Can you find it? $\endgroup$ – Dylan Mar 7 at 12:45
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The Poisson integral formula is a particular case of Green's representation formula, which uses the Green function of a region (and/or its derivative) to represent the solution to Dirichlet/Neumann problems. Green's function is the solution of your problem with a delta-type source at some generic point in your domain and homogeneous Dirichlet data on the boundary. It depends critically on the shape of the boundary (think of the electrostatic potential created by a unit charge when you impose that certain curve (namely the boundary of your domain) is an equipotential line or surface). Thus, you get a different Poisson kernel for the upper half-plane and for the disk. In one case the $x$ axis is a zero-potential line, in the other one the unit circle is a zero-potential line, etc. In physical terms, you have different distributions of the induced charges on the boundary.

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