So this came up as I was thinking about the uniqueness of solutions to the wave equation. I have seen proofs for uniqueness on all of $\mathbb{R}$ or on bounded subsets of $\mathbb{R}^n$, but never $\textit{explicitly}$ on all of $\mathbb{R}^n$. I tried to use the energy approach method to see if I could show uniqueness on all of $\mathbb{R}^n$, however, I ran into trouble when I started doing the integration by parts. i.e., I realized that I don't know the validity of integration by parts on an unbounded domain in dimensions greater than 1. So is this valid? It seems like a proof would look something like below
$\lim_{R \rightarrow \infty} \int_{B(0,R)} u\cdot \nabla v d\bar{x} = \lim_{R \rightarrow \infty} [\int_{\partial B(0,R)} vu\cdot \hat{n} dS\bar{x}- \int_{B(0,R)} v \nabla \cdot u d\bar{x}]$
However, is $\lim_{R \rightarrow \infty} \int_{\partial B(0,R)} vu\cdot \hat{n} dS\bar{x}$ a well defined expression? Part of my worry is I looked up multidimensional integration by parts on wikipedia but it explicitly stated the expression for bounded domains and didn't mention unbounded. From experience past experience there isn't necessarily problems with taking integrals over infinite domains, but I haven't encountered many infinite integrals over a surface. Can anybody offer some illumination?
Best Answer
Yes, integration by parts on $\mathbb R^n$ means integrate by parts on $B(0,R)$ and then let $R\to\infty$. Often this is done without writing anything down, when the fact that the boundary term, the integral over $\partial B(0,R)$, tends to zero is taken as "obvious". (The unfortunate fact is that this is often more obvious to the writer than to the readers. Even more unfortunate, "obvious" claims are sometimes wrong.)
Concerning $$\lim_{R \rightarrow \infty} \int_{\partial B(0,R)} v(u\cdot \hat{n})\, dS \tag{1}$$ there are no
here. The surface of integration is $\partial B(0,R)$, which is a nice bounded surface. Once the integral is evaluated, we get a real number which depends on $R$. Then we study what happens to this number as $R\to\infty$. We are not taking "limit of $\partial B(0,R)$ as $R\to\infty$" here.
It is necessary to know something about $u$ and $v$ to conclude that the limit (1) exists and is zero. E.g., if one of $u,v$ is compactly supported, we are in great shape. Or, if one is bounded and the other decays sufficiently rapidly. Precisely: since the area of $\partial B(0,R)$ is proportional to $R^{n-1}$, it suffices to have $$ |u(x)| |v(x)| = o(R^{1-n}) $$ in order to conclude that (1) is zero.
Some elementary textbook skip over these boring parts, and as a result the statements they make are not universally true. (There are "non-physical" solutions of the heat equation, which are not unique for given initial data.)