The unique solvability of second order elliptic and parabolic equations (in either divergence form or non-divergence form) in Sobolev spaces is well known if the leading coefficients are, for example, uniformly continuous.
However, in general, it is not possible to solve equations in Sobolev spaces unless the coefficients have some regularity assumptions.
In this talk we will discuss some possible classes of discontinuous coefficients with which elliptic and parabolic equations are uniquely solvable in Sobolev spaces.
Especially, as our main results, we will focus on the unique solvability of equations with coefficients only measurable in one spatial variable and having small mean oscillations in the other variables (called partially BMO coefficients).
We will also discuss some applications of our results as well as a new approach to a priori L_p estimates.
Most of the talk is based on joint work with Nicolai Krylov and with Hongjie Dong.