Existence and uniqueness for P-area minimizers in the Heisenberg group

Jih Hsin Cheng, Jenn Fang Hwang, Paul Yang

Research output: Contribution to journalArticle

43 Scopus citations

Abstract

In [3] 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.

Original languageEnglish (US)
Pages (from-to)253-293
Number of pages41
JournalMathematische Annalen
Volume337
Issue number2
DOIs
StatePublished - Feb 1 2007

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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