Optimal partial regularity for nonlinear sub-elliptic systems with Dini continuous coefficients in Carnot groups

We employ the method of an \(\mathcal{A}\)-harmonic approximation to investigate the sub-elliptic system (1.1) under this weaker assumption, and we establish optimal partial regularity results: Roughly speaking, assume additionally to the standard hypotheses (see precisely (H1), (H2), and (H4) below) that \((1 + \vert p \vert )^{ - \frac{m}{2}}A_{i}^{\alpha}(\xi,u,p)\) satisfies (1.2) and (1.3). Let \(u \in HW^{1,m}(\Omega,{ R}^{N})\) be a weak solution of (1.1). Then u is of class \(C^{1}\) outside a closed singular set \(\operatorname{Sing} u \subset\Omega\) of Haar measure 0. Furthermore, for \(\xi_{0} \in\Omega\backslash \operatorname{Sing}u\), the derivative Xu of u has the modulus of continuity \(r \to M(r)\) in a neighborhood of \(\xi_{0} \). Our result is optimal in the sense that when \(\mu(\rho) = \rho^{\gamma}\), \(0 < \gamma< 1\), we have \(M(r) = \gamma^{ - 1}r^{\gamma}\), and \(\Gamma ^{1,\gamma} \) regularity is known to be optimal in that case; see [1].

As is well known, one, in general, cannot expect that weak solutions of the sub-elliptic system (1.1) will be classical, i.e. \(C^{2}\)-solutions, even under reasonable assumptions on \(A_{i}^{\alpha}\) and \(B^{\alpha}\). This was first shown by De Giorgi [3] in the Euclidean space; we also refer the reader to Giaquinta [4], and Chen and Wu [5] for further discussion and additional examples. Then the goal is to establish partial regularity theory. Moreover, a new method called \(\mathcal{A}\)-harmonic approximation technique is introduced by Duzaar and Steffen in [6], and simplified by Duzaar and Grotowski in [7], to study elliptic systems with quadratic growth case. Then similar results have been proved for more general \(A_{i}^{\alpha}\) or \(B^{\alpha}\) in the Euclidean setting; see [8–11] for Hölder continuous coefficients, and [12–14] for Dini continuous coefficients.

However, turning to sub-elliptic equations and systems in Carnot groups, some new difficulties will arise due to non-commutativity of the horizontal vectors \(X_{i}\). With regard to cases of sub-elliptic equations, we refer the reader to Domokos [15–17], Capogna [18, 19], Manfredi and Mingione [20], and Mingione et al. [21] for more interesting results and details. Several results were focused on sub-elliptic systems. Capogna and Garofalo [22], and Shores [23] considered the quadratic growth case, and Föglein [24], Wang and Niu [1], and Wang and Liao [2] treated non-quadratic cases. We note that the assumption of Hölder continuity on the coefficients \(A_{i}^{\alpha}\) is required for those regularity results of sub-elliptic systems mentioned above.

When the assumption of Hölder continuity of \(A_{i}^{\alpha}\) is weakened to Dini continuity, one may ask how to establish partial regularity for the nonlinear sub-elliptic systems under superquadratic controllable growth conditions in the Carnot groups. This paper is devoted to this topic. To define weak solutions to (1.1), we assume the following structure conditions on \(A_{i}^{\alpha}\) and \(B^{\alpha}\):

We denote by \(\omega_{G}=\vert {B(\xi_{0},1)} \vert _{G} \) the volume of unit pseudo-ball. Then the volume of the pseudo-ball \(B_{R}(\xi _{0})= \{ {\eta\in{G}| {d(\xi_{0} ,\xi) < R}}\}\) is given by \(\vert {B(\xi_{0} ,R)} \vert _{G} = \omega_{G} R^{Q} \), where \(Q = \sum_{l = 1}^{r} {l m_{l} } \) is the homogeneous dimension of G.

Following [12], for technical convenience, letting \(\eta(t)=\mu^{2}(\sqrt{t})\), we have the corresponding properties for η: (η1) η is continuous, nondecreasing and \(\eta {(0 ) = 0}\); (η2) η is concave, and \(r \to r^{ - \gamma}\eta(r)\) is nonincreasing for some exponent \(\gamma\in (0,1)\); (η3) \(H(r):=4M^{2}(\sqrt{r}) = [\int_{0}^{r} {\frac{\sqrt{\eta(\rho)} }{\rho}}\,d\rho ]^{2} < \infty\) for some \(r > 0\). Changing κ by a constant, but keeping \(\kappa\ge1\), we can assume that (η4) \(\eta(1)=1\), implying \(\eta(t)\geq t\) for \(t\in[0,1]\).

In the sequel, we let \(\rho_{1} (s,t) = (1 + s+t)^{ - 1}\kappa(s + t)^{ - 1}\), and \(K_{1}(s,t)=(1+t)^{2m}\kappa^{4}(s+t)\) for \(s, t\ge0\). The constant C may vary from line to line. Note that \(\rho_{1} \le 1\) and that \(s \to\rho_{1} (s,t)\), \(t \to\rho_{1} (s,t)\) are nonincreasing functions.

In this section, we begin by introducing the \(\mathcal{A}\)-harmonic approximation lemma, and the proof is similar to [7], also see Föglein [24] with \(p=2\) in the Heisenberg group. We mainly prove the Caccioppoli type inequality for weak solutions of the systems (1.1) with controllable growth conditions.

In this section, we will complete the proof of the partial regularity results via the following lemmas. In the sequel, we always suppose that \(u\in HW^{1,m}(\Omega,{R}^{N})\) with \(m> 2\) is a weak solution to (1.1) with the assumptions of (H1)-(H4), and (μ1)-(μ3). First of all, we provide a linearization strategy for nonlinear sub-elliptic systems as in (1.1).

The following lemma is to establish the excess improvement of the functional Φ as in (1.11). 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.

By the proof method of Lemma 5.1 in [12] and conditions (4.26)-(4.31), Lemma 6 can be proved. As we know, it is sufficient to complete the proof of Theorem 1 once we obtain Lemma 6.