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 $V\ge 0$, integrating the local energy over shrinking balls proves this statement: the boundary contribution is $u_t{\partial }_{\nu }u-\frac{1}{2}(u_t^2+|\nabla u{|}^2+V{u}^2)\le 0$.