In the research of elliptic and parabolic equationequations, the Schauder estimate is the one one of the most important issues for them. In thesethis topic, we always bound the norm of higher regularity in the small ball by a bigger one. That is, for the elliptic equation $ \operatorname{div}(A(x)\nabla u)=0 $, we have the estimetesestimates like $ \left\|u\right\|_{C^{0,\alpha}(B_1)}\leq C\left\|u\right\|_{L^2(B_2)} $, where $ B_r=B(0,r) $ is the ball with center $ 0 $ and radius $ r $. I want to ask why we do not study such estimates for hyperbolic equations.
Became Hot Network Question