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

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

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.

• 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. Dec 18, 2018 at 15:41
• Thanks for the clarification here Feb 8, 2019 at 9:16