Assume that $u: B^n\subset R^n\to R$ is two times differentiable almost everywhere in the unit ball $B^n$ that is continuous on boundary and that solves the PDE almost everywhere $$L u(x):=\sum_{i,j} a_{ij}(x) \partial_{ij}u(x)=f\in L^p(\Omega).$$ Here $L$ is a strongly elliptic operator. Assume in addition that $v\in W^{2,p}_{B}$ for some $p>1$ is weak solution of the same differential equation $Lv=f$ having the same boundary value as $u$. Whether $v=u$ if $u\in W^{1,n}_{B}$?