Consider Dirichlet boundary value problem on unit disk : $u_{xx}+u_{yy}=0$.
Then, is there a constant $c>0$ satisfying the following?
-For every solution $u$ that $\int_0^{2\pi}\left|\frac{d^2u(e^{it})}{dt^2}\right|^p dt\le 1$, $\|u_{xx}\|_p\le c$
(as long as the all integrals are well-defined)
P.S. You may assume $u$ to be either strong or classical(or else), as you please.