Volume 53, Issue 3
Regularity to a Kohn-Laplace Equation with Bounded Coefficients on the Heisenberg Group

Junli Zhang, Pengcheng Niu & Xiuxiu Wang

J. Math. Study, 53 (2020), pp. 265-296.

Published online: 2020-05

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  • 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.

  • AMS Subject Headings

35B50, 35B65

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

jlzhang@mail.nwpu.edu.cn (Junli Zhang)

pengchengniu@nwpu.edu.cn (Pengcheng Niu)

18821781582@163.com (Xiuxiu Wang)

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@Article{JMS-53-265, author = {Zhang , JunliNiu , Pengcheng and Wang , Xiuxiu}, title = {Regularity to a Kohn-Laplace Equation with Bounded Coefficients on the Heisenberg Group}, journal = {Journal of Mathematical Study}, year = {2020}, volume = {53}, number = {3}, pages = {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.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v53n3.20.03}, url = {http://global-sci.org/intro/article_detail/jms/16920.html} }
TY - JOUR T1 - Regularity to a Kohn-Laplace Equation with Bounded Coefficients on the Heisenberg Group AU - Zhang , Junli AU - Niu , Pengcheng AU - Wang , Xiuxiu JO - Journal of Mathematical Study VL - 3 SP - 265 EP - 296 PY - 2020 DA - 2020/05 SN - 53 DO - http://doi.org/10.4208/jms.v53n3.20.03 UR - https://global-sci.org/intro/article_detail/jms/16920.html KW - Heisenberg group, Kohn-Laplace equation, local maximum principle, Hölder regularity, weak Harnack inequality. AB -

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.

Junli Zhang, Pengcheng Niu & Xiuxiu Wang. (2020). Regularity to a Kohn-Laplace Equation with Bounded Coefficients on the Heisenberg Group. Journal of Mathematical Study. 53 (3). 265-296. doi:10.4208/jms.v53n3.20.03
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