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

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

2011-02-14

arxiv.org/abs/1102.2691v1

Bulletin of the Institute of Mathematics, Academia Sinica, New Series, 5 (2010) 369-412

Mathematics

Analysis of PDEs

34 pages

Scientific paper

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.

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