ABSTRACT
This chapter discusses some qualitative properties of solutions to elliptic, parabolic and ultraparabolic partial differential equations with discontinuous coefficients, in nondivergence and divergence form. It shows that some regularity results of solutions to equations whose coefficients belong to the Sarason class of functions with vanishing mean oscillation. The chapter presents some kind of spaces that historically play an important role in the study of the boundedness of certain type of singular integral operators. It describes some important applications of VMO class to the theory of partial differential equations. The most interesting aspect of the exposition is that the hypothesis of vanishing mean oscillation allows many authors to study several type of equations, taking into account both equations in nondivergence and divergence form and systems. The chapter examines some regularity results about elliptic equations in divergence form with discontinuous coefficients.
