Partial Differential Equations

Assignment 4

1. [5 points] (Spherical Symmetry) Let$R$be a rotational matrix, that is$RR^t = I$, and$u$be harmonic in$\mathbb{R}^n$. Define$v$by$v(x) = u(Rx)$. Show that$v$is harmonic in$\mathbb{R}^n$.

2. [5 points] (Schwarz reflection principle) Let$\Omega$be a domain in$\mathbb{R}^2$symmetric about the$x$-axis and let$\Omega^+ = \{(x, y) : y > 0\}$be the upper part. Assume$u \in C(\Omega^+)$is harmonic in$\Omega^+$with$u = 0$on$\partial\Omega^+ \cap \{y = 0\}$. Define

   $$v(x, y) = \begin{cases} u(x, y), & y \geq 0, (x, y) \in \Omega, \\ -u(x, -y), & y < 0, (x, y) \in \Omega. \end{cases}$$

   Show that$v$is harmonic.

3. [5 points] (Harnack’s inequality) Let$u \geq 0$be harmonic in a domain$\Omega$. Let$V \subset\subset \Omega$be connected, open and let$d = d(V, \partial\Omega)$be the distance from$V$to the boundary$\partial\Omega$. Use MVT in suitable open balls to prove that

   $$2^n u(y) \geq u(x) \geq 2^{-n} u(y)$$

   for all$x, y \in V$satisfying$|x - y| \leq \frac{r}{4}$. Use this estimate to prove the following: There are constants$C_1, C_2 > 0$depending on$V$such that

   $$C_1 u(y) \geq u(x) \geq C_2 u(y)$$

   for all$x, y \in V$.

4. [5 points] (Harnack’s Convergence Theorem) Let$u_n:\Omega\to\mathbb{R}$be a monotonically increasing sequence of harmonic functions. If there exists$y\in\Omega$for which the sequence$\{u_n(y)\}_{n\in\mathbb{N}}$is bounded, then$u_n$converges on any$\Omega'\subset\subset\Omega$uniformly to a harmonic function.

5. [5 points] (Eigen Values) Consider the PDE$-\Delta u = \lambda u$in$\Omega$,$u = 0$on$\partial\Omega$where$\lambda$is a scalar and$\Omega$is a bounded open set. If$\lambda \leq 0$, prove that$u \equiv 0$and there is no non-trivial solution.