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

By Jih-Hsin Cheng, Jenn-Fang Hwang, Paul Yang. 2006

94 citations recorded by the source corpus.

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 C2-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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/0a0099df99f6e530db862608f4dd834a34fab171

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

The Bernstein problem for intrinsic graphs in Heisenberg groups and calibrations (2006, unknown): “1”
Smoothness of Lipschitz minimal intrinsic graphs in Heisenberg groups , n > 1 (2010, methodology): “Such solutions arise as the sub-Riemannian mean curvature equation is approximated by Riemannian problems which express the mean curvature in an approximating Riemannian metrics (see [31] and [7] for the relation between Riemannian and sub-Riemannian curvature).”
Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group H1 (2006, unknown): “[CH], [RR1], [RR2], [HP], [ASV], [DGN2], [GS], [ CHY ], [BSV].”
Sub-Riemannian calculus on hypersurfaces in Carnot groups (2006, unknown): “The following is only a partial list of references [1–3,6, 14–16,26,28,43–45,48,49,56,74,84,85,90].”
Examples of area-minimizing surfaces in the subriemannian Heisenberg group H^1 with low regularity (2008, background): “These cones are a generalization of the one obtained by Cheng, Hwang and Yang [9, Ex.”
Regularity of C1 smooth surfaces with prescribed p-mean curvature in the Heisenberg group (2007, background): “3) in [7]) where S(u) denotes the singular set of u, consisting of the points (called singular points) where ∇u + ~ F = 0, and N ≡ N~ F ≡ ∇u+~ F |∇u+~ F | (called Legendrian or horizontal normal).”