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Variations of generalized area functionals and p-area minimizers of bounded variation in the Heisenberg group
by Jih-Hsin Cheng   Jenn-Fang Hwang

Vol. 5 No. 4 (2010) P.369~P.412

ABSTRACT

We prove the existence of a continuous $BV$ minimizer with $C^{0}$ boundary value for the $p$-area (pseudohermitian or horizontal area) in a parabolically convex bounded domain. We extend the domain of the area functional from $BV$ functions to vector-valued measures. Our main purpose is to study the first and second variations of such a generalized area functional including the contribution of the singular part. By giving examples in Riemannian and pseudohermitian geometries, we illustrate several known results in a unified way. We show the contribution of the singular curve in the first and second variations of the $p$-area for a surface in an arbitrary pseudohermitian $3$-manifold.

KEYWORDS
Minimizer, p-area, BV, Heisenberg group, first variation, second variation,

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 35L80, 35J70, 32V20, 53A10, 49Q10

MILESTONES