$C^{1,\alpha}$-Regularity for variational problems in the Heisenberg group

TL;DR

This paper proves that the horizontal gradient of minima for certain variational integrals in the Heisenberg group is H"older continuous, advancing understanding of regularity in sub-Riemannian geometric analysis.

Contribution

It establishes $C^{1,eta}$ regularity for minima of scalar variational integrals with $p$-growth in the Heisenberg group, a novel result in sub-Riemannian calculus of variations.

Findings

Horizontal gradient of minima is H"older continuous.

Regularity results extend to the Heisenberg group setting.

Advances understanding of variational problems in sub-Riemannian geometry.

Abstract

We study the regularity of minima of scalar variational integrals of $p$-growth, $1<p<\infty$, in the Heisenberg group and prove the H\"older continuity of horizontal gradient of minima.