Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Hörmander type

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Sobolev Mappings between Manifolds and Metric Spaces (2009, background): “Spaces supporting Poincaré inequalities play a fundamental role in the modern theory of quasiconformal mappings [46, 47], geometric rigidity problems [7], nonlinear subelliptic equations (see, for example, [11, 22, 20, 33, 34]), and nonlinear potential theory [1, 6].”
Regularity results for quasilinear elliptic equations in the Heisenberg group (2007, background): “Regularity results for minima of general vectorial and strictly convex functionals of the type (8), including the one in (9), have been proved by Capogna & Garofalo [6], relying on explicit a priori estimates for solutions to constant coefficients systems.”
On biharmonic maps and their generalizations (2003, result): “To support this judgment, let us just mention here the work of Capogna and Garofalo [ 5 ], Jost and Xu [23], C.Y.”
Gradient regularity for elliptic equations in the Heisenberg group (2007, background): “HEISENBERG GRADIENT REGULARITY 3 fact has first been established by Capogna & Garofalo in [10]; another proof is given by Föglein [20].”
Differentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg group (2004, methodology): “[2] L. Capogna, D. Danielli and N. Garofalo, An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. P.D.E 18 (1993), 1765-1794.”