Variations of generalized area functionals and p-area minimizers of bounded variation in the Heisenberg group

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. 1. Introduction and Statement