On Regularity and Existence of Viscosity Solutions of Nonlinear Second Order, Elliptic Equations

Abstract

In 1957 there appeared De Giorgi’s famous paper [2] on Holder estimates for weak solutions of linear elliptic equations in divergence form, $$Lu = {D_i}({a^{ij}}{D_j}u) = 0.$$ (1.1) As is well known, the results and techniques of this paper brilliantly opened the study of second order, quasilinear elliptic equations in more than two variables, including the regularity theory of extremals of multiple integrals of the form, $$\int_\Omega {F(x,u,Du)} ,$$ (1.2) and weak solutions of quasilinear, elliptic equations in divergence form, $${D_i}{A^i}(x,u,Du) + B(x,u,Du) = 0.$$ (1.3) The regularity theory was further developed by Ladyzhenskaya and Ural’tseva and Morrey in their respective books [16], [22] and the relationship between De Giorgi’s estimate and the general existence theory is also described for example in the book [4].