Regularity to a Kohn-Laplace Equation with Bounded Coefficients on the Heisenberg Group

Regularity to a Kohn-Laplace Equation with Bounded Coefficients on the Heisenberg Group

Year:    2020

Author:    Junli Zhang, Pengcheng Niu, Xiuxiu Wang

Journal of Mathematical Study, Vol. 53 (2020), Iss. 3 : pp. 265–296

Abstract

In this paper, we concern the divergence Kohn-Laplace equation

$$\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\left( {X_j^*({a^{ij}}{X_i}u) + Y_j^*({b^{ij}}{Y_i}u)} \right)} }  + Tu = f - \sum\limits_{i = 1}^n {\left( {X_i^*{f^i} + Y_i^*{g^i}} \right)}$$ with bounded coefficients on the Heisenberg group ${{\mathbb{H}}^n}$, where ${X_1}, \cdots, {X_n},{Y_1}, \cdots, {Y_n}$ and $T$ are real smooth vector fields defined in a bounded region $\Omega  \subset {\mathbb{H}^n}$. The local maximum principle of weak solutions to the equation is established. The oscillation properties of the weak solutions are studied and then the Hölder regularity and weak Harnack inequality of the weak solutions are proved.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jms.v53n3.20.03

Journal of Mathematical Study, Vol. 53 (2020), Iss. 3 : pp. 265–296

Published online:    2020-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    32

Keywords:    Heisenberg group Kohn-Laplace equation local maximum principle Hölder regularity weak Harnack inequality.

Author Details

Junli Zhang

Pengcheng Niu

Xiuxiu Wang

  1. C1,-regularity for quasilinear degenerate elliptic equation with a drift term on the Heisenberg group

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    https://doi.org/10.1016/j.bulsci.2022.103097 [Citations: 2]