Give a bound of $\|(u_{xx}+u_{yy})\|_p$ by $\|\nabla(\nabla u)\|_p$

Question

Consider twice differentiable $u$ defined on unit disk, that is constantly zero on the boundary.

Then, is there fixed $a>0$ s.t. $a\|\nabla(\nabla u)\|_p\le \|u_{xx}+u_{yy}\|_p$?

Here, you may choose any $p>1$ as you please.