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

Abstract

Abstract. 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.

Regularity results for minimizers of (2, q)-growth functionals in the Heisenberg Group

“…is covered by (3)(4)(5)(6)(7)(8)(9)(10)(11)(12). The second one is accounted for by (3)(4)(5)(6)(7)(8)(9)(10)(11), provided that |h| < 1 400 e…”

“…In this step, we estimate the terms on the r.h.s. of (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) in a way that allows us to apply Lemma 2.8 at the end. The expression ϑ(ρ) in the lemma corresponds to the integrals on the l.h.s of (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) over the ball with radius ρ, that is,…”

“…We insert this into (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16), and obtain an estimate only involving horizontal second derivatives as well as some first-order terms:…”

“…Therefore, we also need to improve the integrability of difference quotients. We return to estimate (3)(4)(5)(6)(7)(8)(9)(10), and apply the subelliptic Sobolev embedding theorem as follows:…”

“…Hölder regularity results for the gradient of solutions of quasi-linear equations were given by Capogna [4,5], using a technique involving fractional pseudo-differential operators; this was also generalized to partial regularity for nonlinear systems by Capogna and Garofalo [7] (see also [17]). Regularity results focusing on the vertical derivative were developed by Marchi [39,40] and Domokos [8]: In particular, T u ∈ L p loc was established for solutions of equations with p-growth, under the condition that 2 ≤ p < 4, employing an iterative method based on fractional difference quotients.…”