Weak Huygens principle
= Weak Huygens principle
The <weak Huygens principle> is the domain-of-dependence statement that initial data outside a backward light cone cannot influence its tip. It permits an interior tail and is weaker than the <Strong Huygens principle>. For $u_{tt}-\Delta u+V(x)u=0$ with $V\ge0$, integrating the local <energy> over shrinking balls proves this statement: the boundary contribution is $u_t\partial_\nu u-\tfrac12(u_t^2+|\nabla u|^2+Vu^2)\le0$.