Regularity resultado for elliptic operator

Question

Let $\Omega$ a smooth bounded domain in $\mathbb{R}^N$. Consider the following operator in the divergent form $$ L(u) = \sum_{i,j=1}^{n}( a_{i,j} u_{x_i})_{x_j} + \sum_{i=1}^{n} b_i u_{x_i} + cu. $$ We say that $L$ is elliptic if, for each $x \in \Omega$ there's a number $\lambda(x)$ such that $$ (1) \quad \quad \quad \sum_{i,j=1}^{n} a_{i,j}(x) \xi_{i}\xi_j \geq \lambda(x)|\xi|^{2}, \quad \forall \xi \in \mathbb{R}^n\backslash\{0\}. $$ The operator is uniformly elliptic if $\lambda(x)$ is constant. If $L$ is uniformly elliptic and $a_{i,j},b,c$ are regular enough, and $u \in H^1_0(\Omega)$ is a weak solution of $$ \begin{cases} L(u) = f, \Omega \\ \,\,\,\,\,\,\,\,u = 0, \partial \Omega \end{cases} $$ then $u \in W^{2,p}(\Omega)$. I would like to know if there is a analogous result for elliptic operator, which satisfies the inequality $(1)$ almost every in $\Omega$. A reference with this theory would be very nice.