Problem set 2

1. Suppose the function $u\in C^2(\R^n)$ satisfies $\Delta u = \lambda u$ where $\lambda$ is a positive constant. Assume that $u(x)\neq0$ for some $x$
   1. Let $p\in\R^n$ be any given point. Show that $M(r) = r^{-(n-1)} \int_{\partial B_r(p)} u(x) dS(x)$ satisfies a second order linear ordinary differential equation of the form \[ M''(r) + a(r) M'(r) + b(r) M(r) = 0, \] and find the coefficients $a(r)$ and $b(r)$.
   2. Assuming still that $\lambda\gt 0$, show that there can be no bounded open subset $U\subset\R^n$ such that the solution $u$ of $\Delta u = \lambda u$ vanishes on $\partial U$, and such that $u(x)\neq0$ for at least one $x\in U$.
2. Let $U\subset\R^n$ be an unbounded domain, and suppose that we have a nonnegative function $H\in C^2(U)\cap C^0(\bar U$ that satisfies $\Delta H\lt0$ on $U$, and for which \[ \lim_{\substack{|x|\to\infty\\x\in U}} H(x) = \infty. \] Prove that if $u\in C^2(U)\cap C^0(\bar U)$ satisfies $\Delta u\geq0$, and \[ \lim_{\substack{|x|\to\infty\\ x\in U}} \frac{|u(x)|}{H(x)} = 0 \] then \[ \sup_{x\in U} u(x) \leq \sup_{x\in\partial U} u(x). \] Hint: let $\epsilon\gt0$ be arbitrary, and show that $v_\epsilon(x) = u(x) - \epsilon H(x)$ is bounded from above, and attains its supremum at some $x_\epsilon\in\bar U$. Use this to show that $\sup_{U} v_\epsilon = \sup_{\pd U} v_\epsilon$. Use \[ u(x) \leq v_\epsilon(x) + \epsilon H(x) \leq \epsilon H(x) + \sup_{\pd U} v_\epsilon \leq \epsilon H(x) + \sup_{\pd U}u \]