Bulletin of the Australian Mathematical Society

Research Article

On the twice differentiability of viscosity solutions of nonlinear elliptic equations

Neil S. Trudingera1

a1 Centre for Mathematical Analysis, Australian National University, G.P.O. Box 4, Canberra, A.C.T. 2601.


We prove, under very general structure conditions, that continuous viscosity subsolutions of nonlinear second-order elliptic equations possess second order superdifferentials almost everywhere. Consequently we deduce the twice differentiability almost everywhere of viscosity solutions. The main idea of the proof is the backwards use of the Aleksandrov maximum principle as invoked in a previous work of Nadirashvili on sequences of solutions of linear elliptic equations.

(Received July 20 1988)