= Boundary term at infinity in Green's second identity
{title2=$\lim_{R\to\infty}\int_{|x|=R}(u\partial_nv-v\partial_nu)dS=0$}
The whole-space form of <Green's second identity> requires the displayed boundary limit, together with the existence of the relevant volume integrals. Pointwise decay of the fields does not by itself control their normal derivatives or the growing sphere area. In three dimensions, sufficient power bounds are $u=O(r^{-a})$, $\nabla u=O(r^{-a-1})$, $v=O(r^{-b})$ and $\nabla v=O(r^{-b-1})$, with $a+b>1$: the surface integral is then $O(R^{1-a-b})$.
For a smooth counterexample to pointwise decay alone, take $u=(1+r^2)^{-1/2}$ and $v=\cos r\,(1+r^2)^{-1/2}$. Both are smooth radial functions on all of $\mathbb R^3$ and tend to zero. Nevertheless $u v'-v u'=-\sin r/(1+r^2)$, giving surface integral $-4\pi R^2\sin R/(1+R^2)$, which has no zero limit. Screened <Green functions> with exponential decay avoid this particular problem.
Back to article page
