Existence and Uniqueness for P-Area Minimizers in the Heisenberg Group
| dc.creator | Cheng, Jih-Hsin | |
| dc.creator | Hwang, Jenn-Fang | |
| dc.creator | Yang, Paul | |
| dc.date | 2006-01-10 | |
| dc.date | 2006-08-01 | |
| dc.date.accessioned | 2026-07-07T09:32:04Z | |
| dc.date.available | 2026-07-07T09:32:04Z | |
| dc.description | In \cite{CHMY04}, we studied $p$-mean curvature and the associated $p$-minimal surfaces in the Heisenberg group from the viewpoint of PDE and differential geometry. In this paper, we look into the problem through the variational formulation. We study a generalized $p$-area and associated ($p$-) minimizers in general dimensions. We prove the existence and investigate the uniqueness of minimizers. Since this is reduced to solving a degenerate elliptic equation, we need to consider the effect of the singular set and this requires a careful study. We define the notion of weak solution and prove that in a certain Sobolev space, a weak solution is a minimizer and vice versa. We also give many interesting examples in dimension 2. An intriguing point is that, in dimension 2, a $C^2$-smooth solution from the PDE viewpoint may not be a minimizer. However, this statement is true for higher dimensions due to the relative smallness of the size of the singular set. | |
| dc.description | 37 pages, 5 figures | |
| dc.identifier | https://arxiv.org/abs/math/0601208 | |
| dc.identifier | http://arxiv.org/abs/math/0601208 | |
| dc.identifier | Math. Ann., 337 (2007) 253-293. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/158682 | |
| dc.subject | Differential Geometry | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 35L80; 35J70; 32V20; 53A10; 49Q10 | |
| dc.title | Existence and Uniqueness for P-Area Minimizers in the Heisenberg Group | |
| dc.type | text |