We prove the hypoellipticity for systems of Hörmander type with constant coefficients in Carnot groups of step 2. This result is used to implement blow-up methods and prove partial regularity for local minimizers of non-convex functionals, and for solutions of non-linear systems which appear in the study of non-isotropic metric structures with scalings. We also establish estimates of the Hausdorff dimension of the singular set.
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Luca Capogna, Nicola Garofalo, Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Hörmander type. J. Eur. Math. Soc. 5 (2003), no. 1, pp. 1–40DOI 10.1007/S100970200043