Abstract
This paper is concerned with the interior regularity for nonlinear sub-elliptic systems with Dini continuous coefficients under superquadratic controllable growth conditions in Carnot groups. We adapt the technique of the $\mathcal{A}$ -harmonic approximation to the case of sub-elliptic systems in divergence form, and we show a partial regularity result for weak solutions. In particular, our result is optimal in the sense that in the case of Holder continuous coefficients we obtain directly the optimal Holder exponent for the horizontal gradient of weak solutions on its regular set.
Highlights
Under the coefficients Aαi assumed to be Dini continuous, the purpose of this paper is to establish optimal partial regularity to the system ( . ) under the superquadratic controllable growth conditions. Such an assumption is much weaker than the assumption of Hölder continuity; see [, ] for the case of sub-elliptic systems
We mainly prove the Caccioppoli type inequality for weak solutions of the systems ( . ) with controllable growth conditions
We provide a linearization strategy for nonlinear sub-elliptic systems as in ( . )
Summary
=: I + II + III + IV + V , with the obvious labeling for I-V. The left hand side of ( . ) can be estimated via the uniformly elliptic condition (H ). Using the Dini continuity condition (H ), Young’s inequality, and The term V can be estimated by using the controllable growth condition (H ), Hölder’s inequality, and Young’s inequality. The term II can be estimated using the Dini continuity condition (ρ ), which follows from the nondecreasing property of the function η(t), (η ), and our assumption ρ ≤ ρ ≤. The following lemma is to establish the excess improvement of the functional as in ). The strategy of the proof is to approximate the given solution by A-harmonic functions, for which suitable decay estimates are available from the classical theory.
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