Obstacle problem and maximal operators

JUAN PABLO PINASCO; PABLO BLANC; J. D. ROSSI

ADVANCED NONLINEAR STUDIES

ADVANCED NONLINEAR STUDIES, INC

Lugar: San Antonio, Texas, Estados Unidos; Año: 2016 vol. 16 p. 355 - 362

1536-1365

Fix two differential operators, $L_1$ and $L_2$and define a sequence of functions inductively by considering $u_1$ as the solution for the Dirichlet problem for an operator $L_1$ and then $u_n$ as the solution to the obstacle problem for an operator $L_i$ ($i=1,2$ alternating them) with obstacle given by the previous term $u_{n-1}$ in a domain $Omega$ and a fixed boundary datum $g$ on $partial Omega$. We show that in this way we obtain an increasing sequence that converge uniformly to a viscosity solution to the minimal operator associated with $L_1$ and $L_2$, that is, the limit $u$ verifies $min { L_1 u, L_2 u } =0$ in $Omega$ with $u=g$ on $partial Omega$.