scholarly journals Maximum principle for second order elliptic equations and applications

1989 ◽  
Vol 138 (2) ◽  
pp. 343-348 ◽  
Author(s):  
Michelangelo Franciosi
Keyword(s):  
Maximum Principle ◽  
Elliptic Equations ◽  
Second Order ◽  
Second Order Elliptic Equations

scholarly journals On the twice differentiability of viscosity solutions of nonlinear elliptic equations

1989 ◽  
Vol 39 (3) ◽  
pp. 443-447 ◽  
Author(s):  
Neil S. Trudinger
Keyword(s):  
Maximum Principle ◽  
Viscosity Solutions ◽  
Elliptic Equations ◽  
General Structure ◽  
Main Idea ◽  
Nonlinear Elliptic Equations ◽  
Second Order ◽  
Nonlinear Elliptic ◽  
Almost Everywhere ◽  
Second Order Elliptic Equations

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.

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