Guest Lecture in Geometric PDE

A well-known example by N. N. Ural'tseva suggests that for fixed p > 2 there is no unique $W^2_p$-solvability of elliptic equations under p > the condition that the leading coefficients are measurable in two spatial variables. We will present a recent result which gives the unique $W^2_p$-solvability of parabolic and elliptic equations under this condition, when p\ge 2 is close to 2 depending on the ellipticity constant. This is joint work with N. V. Krylov.