Hölder Continuity for Sub-Elliptic Systems Under the Sub-Quadratic Controllable Growth in Carnot Groups

This paper is devoted to optimal partial regularity of weak solutions to nonlinear sub-elliptic systems for the case 15m52 under the controllable growth condition in Carnot groups. We begin with establishing a Sobolev-Poincare type inequality for the function u2HW1;m(V;R ) with m2 (1; 2), and then partial regularity with optimal local HoÈlder exponent for horizontal gradients of weak solutions to the systems is established by usingA-harmonic approximation technique. MATHEMATICS SUBJECT CLASSIFICATION (2010). 35H20, 35B65. KEYWORDS. Carnot group; Nonlinear sub-elliptic system; Sub-quadratic controllable growth condition; Optimal partial regularity; A-harmonic approximation technique 1. Introduction and statements of main results In this paper, we consider nonlinear sub-elliptic systems of second order in divergence form under the sub-quadratic (15m52) controllable growth condition in Carnot groups, and settle optimal partial regularity for horizontal gradients of weak solutions. More precisely, let V G be a bounded domain in a Carnot group G with general step, and consider the following system ÿ Xk iˆ1 XiA a i (j;u;Xu) ˆ B(j;u;Xu); j 2 V; u 2 R; Xu 2 R (1:1) (*) Research supported by National Natural Science Foundation of China (No. 11126294 and No. 11201081), supported by Natural Science Foundation of Jiangxi, China (No. 2010GQS0021), and supported by Science and Technology Planning Project of Jiangxi Province, China (No. GJJ13657). (**) Indirizzo dell'A.: Key Laboratory of Numerical Simulation Technology of Jiangxi Province, Gannan Normal University, Ganzhou, 341000, Jiangxi, P. R. China. E-mail: jialinwang1025@hotmail.com ( ) Indirizzo degli Autori: School of Mathematics and Computer Science, Gannan Normal University, Ganzhou 341000, Jiangxi, P. R. China. with sub-quadratic controllable structure conditions (H1)-(H3) and (C), where (H1): Ai (j;u; p) is differentiable with respect to p, with bounded and continuous derivatives, that is, there exists a constant C such that A i;p b (j;u; p) C(1‡ p j j2) 2 ; (j;u; p) 2 V R R; 15m52; (1:2) where we denote by A i;p b ( ) ˆ @A a i ( ) @pjb ; (H2): Ai (j;u; p) satisfies the following ellipticity condition A i;p b (j;u; p)hi h b j l(1‡ p j j) mÿ2 2 h j j; 8 h 2 R; (1:3) where l is a positive constant; (H3): Ai (j;u; p) is HoÈlder continuous with exponent g 2 (0; 1) in the first and second variables, i.e., Ai (j;u; p)ÿ Ai (~j; ~ u; p) K( u j j)(dm(j; ~j)‡ uÿ ~ u ) g m(1‡ p j j2) 2 ; (1:4) where K( ) : [0;1)! [0;1) is monotonously nondecreasing. Without loss of generality, it is convenient to take K( ) 1; (C) (Controllable growth condition): Denote r ˆ mQ Qÿm and require B(j;u; p) j j4a p j jm(1ÿ r ‡b u j jrÿ1‡ c; (1:5) where Q 3 is the homogeneous dimension in Carnot groups (see (2.3) below), and a; b and c are positive constants. Furthermore, (H1) infers that there exists a continuously nonnegative and bounded function v(s; t) : [0;1) [0;1)! [0;1), where v(s; 0) ˆ 0 for all s, and v(s; t) is monotonously nondecreasing in s for fixed t; v(s; t) is concave and monotonously nondecreasing in t for fixed s, such that for all (j;u; p); (~j; ~ u; ~ p) 2 V R R, A i;p b (j;u; p)ÿ A i;p b (j;u; ~ p) C(1‡ p j j‡ ~ p j j2) 2 v( p j j; pÿ ~ p j j): (1:6) As is well known, even under reasonable assumptions on Ai ; B a in the systems, people cannot in general expect that weak solutions of nonlinear elliptic systems of equations will be classical (i.e. C-solutions) like elliptic scalar equations. This was first shown by De Giorgi [1]. Then the goal is to establish partial regularity of weak solutions for systems. Such regularity 170 Jialin Wang Dongni Liao Zefeng Yu means that for any weak solution u of a system, there exists an open subset V0 V such that VnV0 has a zero Lebesgue measure and u or its gradient Du is locally regular in V0 . We refer the reader to monographs of Giaquinta [2, 3] and Chen-Wu [4]. There are different methods to prove partial regularity: the direct approach was first carried out by Giaquinta-Modica [5]; Guisti-Miranda [6] was earlier to employ the blow-up method; furthermore, Duzaar and Grotowski in [7] generalized so called A-harmonic approximation technique in [8] and gave the remarkable proof of partial regularity for systems with quadratic growth conditions(m ˆ 2). This method has two major advantages: the first is that we only need to establish a Caccioppoli type inequality; this avoids having to prove an inverse HoÈlder inequality. The other is that one can obtain the optimal HoÈlder exponent in partial regularity. Then, Duzaar et al. [9] studied partial regularity of almost minimizers of quasi-convex variational integrals with sub-quadratic growth, and we also note that the degenerate p-Laplacian version of the method has been obtained in [10], and applied to the partial regularity in [11]. In the paper [9], Duzaar et al. actually provide a partially new proof of the original regularity result of Carozza et al. [12]. Later, Chen and Tan in [13, 14] extended Duzaar and Grotowski's results [7] to more general nonlinear elliptic systems under the super-quadratic growth (m > 2) and subquadratic growth (15m52), respectively. Several regularity results were focused on systems constructed by basic vector fields in Carnot groups. Capogna and Garofalo in [15] showed the partial HoÈlder regularity for quasi-linear sub-elliptic systems under the quadratic structure conditions in Carnot groups of step two. Shores in [16] considered a homogeneous quasi-linear system under the quadratic growth condition on the Carnot group with general step. She first established higher differentiability and smoothness for weak solutions of the system with constant coefficients, and then deduced the partial regularity. Their methods depend mainly on generalization of classical direct method in the Euclidean space. Later, by the method of A-harmonic approximation, FoÈglein in [17] treated the homogeneous nonlinear system ÿ X2n iˆ1 XiA a i (j;Xu) ˆ 0; a ˆ 1; ;N on the Heisenberg group under super-quadratic structure conditions. She got partial regularity for the horizontal gradient of weak solutions to the initial system. Then the first author and Niu in [18] considered more genHoÈlder Continuity for Sub-Elliptic Systems Under etc. 171 eral nonlinear sub-elliptic systems (see (1.1)) in Carnot groups under superquadratic growth conditions, and established the optimal partial regularity of the horizontal gradient of weak solutions. In this paper, we will apply the method of A-harmonic approximation adapted to the setting of Carnot groups to study partial regularity for the system (1.1) under the controllable growth conditions with sub-quadratic case (15m52). The key point is to establish a certain excess-decay estimate for the excess function F. In the case m 2, this function is given by F(j0; r; p0) ˆ Br(j0) ÿ1 G Z Br(j0) Xuÿ p0 j j‡ Xuÿ p0 j j dj; whereas in the case 15m52, one uses F(j0; r; p0) ˆ Br(j0) ÿ1 G Z Br(j0) V (Xu)ÿ V (p0) j jdj; (1:7) where V (A) ˆ (1‡ jAj) 4 mÿ2 for A 2 Hom(R;R). It is shown that if F(j0; r; p0) is small enough on a ball Br(j0) V, then for some fixed u 2 (0; 1) one has the excess improvement F(j0; ur; p0) CuF(j0; r; p0). Iteration of this result yields the excess-decay estimate which implies the regularity result. Although the underlying philosophy in this paper is encouraged by that in [14, 9], some different treatment are necessary. Since basic vector fields (see (2.1) below) of Lie algebras corresponding to the Carnot group are more complicated than gradient vector fields in the Euclidean space, we have to find a different auxiliary function in proving Caccioppoli type inequality. Inspired by [17], we choose horizontal variables to construct such a suitable function (see Remark 1 below); Besides, the non-horizontal derivatives of weak solutions will happen in the Taylor type formula on the Carnot group and cannot been effectively controlled in the present hypotheses. So the method employing Taylor's formula in [14] is not appropriate in our setting. In order to obtain the desired decay estimate, we need establish and use the Sobolev-Poincare type inequality (3.1) instead. The main result in this paper is as follows. THEOREM 1. Assume that coefficients Ai and B a satisfy conditions (H1)-(H3) and (C), and u 2 HW1;m(V;R) be a weak solution to the system (1.1) with V G, i.e., Z V Ai (j;u;Xu)XiW dj ˆ Z V B(j;u;Xu)Wdj 8 W 2 C1 0 (V;R): (1:8) 172 Jialin Wang Dongni Liao Zefeng Yu Then there exists an open subset V0 V; such that u 2 G(V0;R), with g in (1.4). Furthermore, VnV0 ˆ S1 [ S2 and meas (VnV0) ˆ 0, where S1 ˆ n j0 2 V : lim r!0‡ sup (Xu)j0;r ÿ ˆ 1o; S2 ˆ ( j0 2 V : lim r!0‡ inf Br(j0) j jÿ1 Z Br(j0) Xuÿ (Xu)j0;r dj > 0): To the best of our konwledge, in the Heisenberg group, G regularity of weak solutions to sub-elliptic p-Laplacian equations is valid for 15p51 and was proved by several people during the 90's in [19, 20, 21], but when turning to G regularity, it is worthy of pointing out that the remarkable contribution of HoÈlder continuity for the gradient of weak solutions to subelliptic p-Laplacian equations is due to Capogna [22, 23], Marchi [24, 25], Domokos [26, 27], Manfredi and Mingione et al. [28, 29], but the exponent p should be near 2, and the limitation 2 p54 appears in the most recent of the cited works. Recently, Garofalo in [30] obtained the G regularity of weak solution which possess some special symmetries for 2 p. When turning to partial continuity of weak solutions, our result shows that the G continuity is also valid for exponent 15p52. The plan of this paper is organized as follows: In Section 2, we introduce two functions and their some useful properties, and collect some basic notions and facts associated to Carnot groups. Since in the subquadratic case we are dealing with functions belonging to the horizontal Sobolev space HW with 15m52, in the proof of the main results, a Sobolev-Poincare type inequality (see (3.1) bel