Equivalence of Viscosity Solutions between Obstacle and Gradient Constraint Problems

Abstract

It is proved that the unique viscosity solution of degenerate/singular elliptic partial differential equations (PDE for short) under gradient constraint coincides with that of the PDE with suitably selected bilateral obstacles. To this end, it is necessary to establish the Lipschitz estimates on viscosity solutions of bilateral obstacle problems. In the case of singular elliptic PDE, the notion of -solutions is adapted.