Title: Some recent results on L_p-theory for equations with singular and degenerate coefficients

Abstract: We consider classes of elliptic and parabolic equations whose coefficients are singular or degenerate of the porotype $x_d^\alpha$ on the domain $\{x_d >0\}, where $\alpha$ is a real number. Two boundary conditions on \{x_d =0\}$ are studied: the homogeneous Diritchlet boundary condition or the conormal boundary one. Depending on the boundary conditions, suitable weighted Sobolev spaces are found to establish existence, uniqueness and regularity estimates of solutions. As $\alpha$ may not be in (-1, 1)$, our weight $x_d^\alpha$ may not be in A_2 as commonly considered in literature. Moreover, the results also demonstrate that different Sobolev spaces are required for the two different boundary conditions, a phenomenon that is not seen in the case that the coefficients are uniformly elliptic. The talk is based on joint work with Hongjie Dong (Brown University).