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Volume 16, Issue 2

# Obstacle Problems and Maximal Operators

Pablo Blanc
• Departmento de Mathematica, FCEyN, Buenos Aires University, Ciudad Universitaria, Pab 1 (1428), Buenos Aires, Argentina
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• Other articles by this author:
/ Juan P. Pinasco
• Departmento de Mathematica, FCEyN, Buenos Aires University, Ciudad Universitaria, Pab 1 (1428), Buenos Aires, Argentina
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• Other articles by this author:
/ Julio D. Rossi
• Corresponding author
• Departmento de Mathematica, FCEyN, Buenos Aires University, Ciudad Universitaria, Pab 1 (1428), Buenos Aires, Argentina
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• Other articles by this author:
Published Online: 2016-04-13 | DOI: https://doi.org/10.1515/ans-2015-5044

## Abstract

Fix two differential operators ${L}_{1}$ and ${L}_{2}$, and define a sequence of functions inductively by considering ${u}_{1}$ as the solution to 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 Ω and a fixed boundary datum g on $\partial \mathrm{\Omega }$. We show that in this way we obtain an increasing sequence that converges uniformly to a viscosity solution to the minimal operator associated with ${L}_{1}$ and ${L}_{2}$, that is, the limit u verifies $\mathrm{min}\left\{{L}_{1}u,{L}_{2}u\right\}=0$ in Ω with $u=g$ on $\partial \mathrm{\Omega }$.

MSC 2010: 35J70; 49N70; 91A15; 91A24

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Revised: 2015-08-18

Accepted: 2015-08-19

Published Online: 2016-04-13

Published in Print: 2016-05-01

Citation Information: Advanced Nonlinear Studies, Volume 16, Issue 2, Pages 355–362, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365,

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