A study of comparison, existence and regularity of viscosity and weak solutions for quasilinear equations in the Heisenberg group. (English) Zbl 1409.35202
Summary: 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.
MSC:
| 35R03 | PDEs on Heisenberg groups, Lie groups, Carnot groups, etc. |
| 35J62 | Quasilinear elliptic equations |
| 35B51 | Comparison principles in context of PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35D40 | Viscosity solutions to PDEs |
| 35D30 | Weak solutions to PDEs |
| 49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |
Keywords:
partial differential equations on the Heisenberg group; viscosity solutions; comparison principle; weak solutions; Radó type theoremReferences:
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