= Local wave energy estimate
{title2=$E_{B_r}(T)\leq E_{B_{r+T}}(0)$}
For the unit-speed <wave equation>, the integral of $(u_t^2+|\nabla u|^2)/2$ over $B_r(x_0)$ at time $T\geq0$ is at most its initial integral over $B_{r+T}(x_0)$. On a shrinking ball the outward energy flux minus the loss from its moving boundary is $-((u_t-\partial_nu)^2+|\nabla_{\mathrm{tan}}u|^2)/2$, hence nonpositive. This estimate establishes the <domain of dependence> and <finite propagation speed> without any assumptions at spatial infinity.
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