I will be talking about equations with discontinuous coefficients and about a progress in studying them which occurred quite recently since 2007.
For linear equations with VMO coefficients a Sobolev space theory was developed by Chiarenza, Frasca, and Longo (elliptic case) in 1991 and (parabolic case) by Bramanty and Cerutti in 1993. These authors used explicit representations of solutions and the Coifman-Rochberg-Weiss commutator theorem from the theory of singular integrals. Since 2007 a new approach was developed based on various versions of the Fefferman-Stein theorem and pointwise estimates of the sharp functions of the highest order derivatives. This approach works equally well for linear equations and systems and fully nonlinear equations when no explicit formulas for solutions are available.
Speaker: Nicolai Krylov, University of Minnesota