Oblique derivative problem for elliptic equations in non-divergence form with $VMO$ coefficients

summary:A priori estimates and strong solvability results in Sobolev space $W^{2,p}(\Omega)$, $1<p<\infty$ are proved for the regular oblique derivative problem $$ \begin{cases} \sum_{i,j=1}^n a^{ij}(x)\frac{\partial^2u}{\partial x_i\partial x_j} =f(x) \text{ a.e. } \Omega \\ \frac{\partial u}{\partial \ell}+\sigma(x)u =\varphi(x) \text{ on } \partial \Omega \end{cases} $$ when the principal coefficients $a^{ij}$ are $V\kern -1.2pt MO\cap L^\infty$ functions