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Green's function for $-\Delta$ on the plane is $-\frac{1}{2\pi}\ln\lVert x-y \rVert$

a) Using the method of reflection, compute the green's function for Laplace's equation $-\Delta u=0$

b) Compute the green's function for the quadrant with the following other boundary condition Neuwann on the x-axis and dirichlet on the y-axis

My answers

a)$U(x)= V\lvert x\rvert$ where, $x=(x,x^{2},x^{3},...x^{n})$

$-\Delta u = -\frac{\partial^{2}u}{\partial x^{2}} - \frac{\partial^{2}u}{\partial y^{2}}=0$

taking derivative of $U$

$Ux_i = V'\frac{x_i}{\lvert x\rvert}$

$Ux_ix_i = V''\frac{x_i^{2}}{\lvert x\rvert^{2}}$ + $V'\frac{\lvert x\rvert -\frac{x_i^{2}}{\lvert x\rvert}}{\lvert x\rvert^{2}}$

therefore

$-\Delta u= \sum Ux_ix_i= \sum V''\frac{x_i^{2}}{\lvert x\rvert^{2}} + V'\frac{\lvert x\rvert-\frac{x_i^{2}}{\lvert x\rvert}}{\lvert x\rvert^{2}} $

I'm stuck here.

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  • $\begingroup$ Note that $\sum_i x_i^2 \equiv |x|^2$ so you arrive at the ODE $V''(r) + \frac{V'(r)}{r}(n-1) = 0$ where $r = |x|$ and $n=2$ is the dimension. $\endgroup$
    – Winther
    Dec 18, 2018 at 15:41
  • $\begingroup$ Thanks for the clarification here $\endgroup$ Feb 8, 2019 at 9:16

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