Existence and Uniqueness for P-Area Minimizers in the Heisenberg Group

Details Existence and Uniqueness for P-Area Minimizers in the Heisenberg Group Existence and Uniqueness for P-Area Minimizers in the Heisenberg Group

2006-01-10

arxiv.org/abs/math/0601208v2

Math. Ann., 337 (2007) 253-293.

Mathematics

Differential Geometry

37 pages, 5 figures

Scientific paper

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.

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