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Problem set 8, PDE

Problem 1

Find the Green's function for the Laplace equation in the upper half-space $latex \R^{n+1}_+ = \{x_{n+1}>0\}$ and then derive a formal integral representation for a solution of the Dirichlet problem

$latex \begin{cases} \Delta u = 0 &\text{in } \R^{n+1}_+\\ u = \phi & \text{on }\{x_{n+1}=0\}.\end{cases}$

Problem 2

Use the Poisson integration formula to prove the Harnack inequality

$latex \displaystyle\Big(\frac{R}{R+r}\Big)^{n-2} \frac{R-r}{R+r} u(x_0) \le u(x) \le \Big(\frac{R}{R-r}\Big)^{n-2} \frac{R+r}{R-r}u(x_0),$

for a harmonic function u in $latex B_R(x_0)\subset\R^n$, where $latex r=|x-x_0|<R$.

Problem 3

Use Problem 2 to give another proof of Liuville's theorem: a harmonic function in $latex \R^n$ bounded from above, or from below, is constant.