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May  2019, 18(3): 1091-1115. doi: 10.3934/cpaa.2019053

## A study of comparison, existence and regularity of viscosity and weak solutions for quasilinear equations in the Heisenberg group

 Universidad Nacional de Cuyo-CONICET, Parque Gral. San Martin. M5502JMA Mendoza. Argentina

* Corresponding author

Received  December 2017 Revised  July 2018 Published  November 2018

Fund Project: The authors are supported by grants PICT 2015-1701 AGENCIA, and SECTyP UNCuyo B051

In this manuscript, we are interested in the study of existence, uniqueness and comparison of viscosity and weak solutions for quasilinear equations in the Heisenberg group. In particular, we highlight the limitation of applying the Euclidean theory of viscosity solutions to get comparison of solutions of sub-elliptic equations in the Heisenberg group. Moreover, we will be concerned with the equivalence of different notions of weak solutions under appropriate assumptions for the operators under analysis. We end the paper with an application to a Radó property.

Citation: Pablo Ochoa, Julio Alejo Ruiz. A study of comparison, existence and regularity of viscosity and weak solutions for quasilinear equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1091-1115. doi: 10.3934/cpaa.2019053
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