Obstacle problem and maximal operators

Abstract

Fix two differential operators L1 and L2, and define a sequence of functions inductively by considering u1 as the solution to the Dirichlet problem for an operator L1 and then un as the solution to the obstacle problem for an operator Li (i=1,2 alternating them) with obstacle given by the previous term un−1 in a domain Ω and a fixed boundary datum g on ∂Ω. We show that in this way we obtain an increasing sequence that converges uniformly to a viscosity solution to the minimal operator associated with L1 and L2, that is, the limit u verifies min{L1u,L2u}=0 in Ω with u=g on ∂Ω.