Obstacle Problems and Maximal Operators

Abstract

Abstract Fix two differential operators L 1 ${L_{1}}$ and L 2 ${L_{2}}$ , and define a sequence of functions inductively by considering u 1 ${u_{1}}$ as the solution to the Dirichlet problem for an operator L 1 ${L_{1}}$ and then u n ${u_{n}}$ as the solution to the obstacle problem for an operator L i ${L_{i}}$ ( i = 1 , 2 ${i=1,2}$ alternating them) with obstacle given by the previous term u n - 1 ${u_{n-1}}$ in a domain Ω and a fixed boundary datum g on ∂ ⁡ Ω ${\partial\Omega}$ . We show that in this way we obtain an increasing sequence that converges uniformly to a viscosity solution to the minimal operator associated with L 1 ${L_{1}}$ and L 2 ${L_{2}}$ , that is, the limit u verifies min ⁡ { L 1 ⁢ u , L 2 ⁢ u } = 0 ${\min\{L_{1}u,L_{2}u\}=0}$ in Ω with u = g ${u=g}$ on ∂ ⁡ Ω ${\partial\Omega}$ .