# Solving $\Box u=0$ in an open set with $u(p)$ and $\mathrm{grad}\, u(p)$ prescribed

Consider a $C^k$, $k\ge 2$, Lorentzian manifold $(M,g)$ and let $\Box$ be the usual wave operator $\nabla^a\nabla_a$. Given $p\in M$, $s\in\Bbb R,$ and $v\in T_pM$, can we find a neighborhood $U$ of $p$ and $u\in C^k(U)$ such that $\Box u=0$, $u(p)=s$ and $\mathrm{grad}\, u(p)=v$?

• Would Mathematics be a better home for this question? – Qmechanic Sep 8 '17 at 5:07
• You want to specify data at a single point only? Seems unlikely that this problem is well-posed - the classical theory (see e.g. Hormander, Taylor, John's PDE books) says data must be specified on a noncharacteristic ("space-like") submanifold. It's been a year or two since I've read this stuff otherwise I would answer more completely. – icurays1 Sep 9 '17 at 14:33
• @icurays1 I am not looking for uniqueness, I just want some solution to exist with those properties. – Ryan Unger Sep 9 '17 at 14:40
• The standard existence theory @icurays1 mentions should specialize: pick nice coordinates near $p$ and prescribe $u, \partial_t u$ on some patch of $\{t = t(p)\}$, which in particular lets you choose $u(p), \nabla u(p).$ – Anthony Carapetis Sep 11 '17 at 11:48
• @AnthonyCarapetis Right, that was my idea. But there's some technical issues with solving the equation backwards in time and forwards, then matching up the solutions smoothly. That can probably be fixed, but I'm hoping there's a simpler solution. – Ryan Unger Sep 11 '17 at 13:25