Regularity for a Nonlinear Discontinuous Subelliptic System with Drift on the Heisenberg Group

. In this paper, we prove the partial Hölder regularity of weak solutions and the partial Morrey regularity to horizontal gradients of weak solutions to a nonlinear discontinuous subelliptic system with drift on the Heisenberg group by the A -harmonic approximation, where the coe ﬃ cients in the nonlinear subelliptic system are discontinuous and satisfy the VMO condition for x , ellipticity and growth condition with the growth index 1 < p < 2 for the Heisenberg gradient variable, and the nonhomogeneous terms satisfy the controllable growth condition and the natural growth condition, respectively.


Introduction
Kohn in [1] proved L 2 estimates for the operator constructed by Hörmander's vector fields fX 1 , X 2 ,⋯,X q , X 0 g (see [2]) based on the energy estimate and a subelliptic estimate. Moreover, some authors also inspected the regularity of solutions to linear degenerate elliptic equations with drift term by establishing singular integral estimates. For example, Folland and Stein in [3] established L p estimates and Lipschitz estimates to the operator on the Heisenberg group for suitable α, where T is the vertical vector field. To the nondivergence linear degenerate elliptic operator constructed by Hörmander's vector fields, Bramanti and Zhu in [4] established L p estimates with a ij ðxÞ and a 0 ðxÞ belonging to VMO spaces related to fX 1 , X 2 ,⋯,X q , X 0 g and Schauder estimates with a ij ðxÞ and a 0 ðxÞ being in Hölder spaces for strong solutions. It is important in [4] that the difference between equations without X 0 and with X 0 was pointed out. When X 1 , X 2 , ⋯, X q in (3) is basis vector fields and X 0 is the drift vector field on homogeneous groups, many scholars have obtained regularities to the operator £ with coefficients a ij and a 0 satisfying appropriate conditions, such as [5][6][7][8]. In addition, Austin and Tyson in [9] achieved the C ∞ -smoothness for the operator on the Heisenberg group ℍ n weak solution of discontinuous subelliptic systems with drift term Tu on ℍ n where Ω is the bounded domain in ℍ n , A k i belongs to the vanishing mean oscillation space (which is abbreviated as VMO) and satisfies the ellipticity on ℝ 2n×N and polynomial growth conditions with the growth index 1 < p < 2 for ∇ H u, and also A k i is continuous for u and differentiable for ∇ H u with continuous derivatives, X i ði = 1, 2,⋯,2nÞ is the horizontal vector field and T is the vertical vector field in ℍ n . For more information about ℍ n , see Section 2. The nonhomogeneous term B k satisfies the controllable growth condition or natural growth condition. We will use the A-harmonic approximation method to conclude the partial Hölder regularity to the weak solutions and the partial Morrey regularity to the horizontal gradients of the weak solutions.
(H1). Let A k i satisfy the following ellipticity and polynomial growth conditions (growth index 1 < p < 2): where D P A k i denote the usual derivative of A k i with respect to the variable P, 0 < λ ≤ 1 ≤ Λ < ∞.
Its proof is direct by combining the proof of Theorem 1 in this paper with the proof of Theorem 1.2 in [15].
Let us recall that the A-harmonic approximation method was first introduced by Duzaar and Steffen in [16] and then extended to other cases by some authors, see [17][18][19]. In this paper, we use the A-harmonic approximation method described in [15] to conclude Theorem 1. Different from [15], the system considered by us has a drift term, which brings new challenges to our research. Actually, the processing of drift term are different from that the processings of other terms in the system. Moreover, Lemmas 11-14 in Section 3 used in proving Theorem 1 are different from the corresponding lemmas in [15] and will be rebuilt. This paper is organized as follows: in Section 2, we introduce the related knowledge of the Heisenberg group, some function spaces on the Heisenberg group, horizontal affine functions, and some necessary lemmas. In Section 3, we show a Caccioppoli-type inequality for weak solution to (5), the approximately A-harmonic lemma, the decay estimate, and iteration relations. In Section 4, the proof of Theorem 1 is given.

Preliminaries
2.1. The Heisenberg Group ℍ n and Some Function Spaces on ℍ n . The Euclidean space ℝ 2n+1 , n ≥ 1 with the group multiplication where x = ðx 1 , x 2 ,⋯,x 2n , tÞ, y = ðy 1 , y 2 ,⋯,y 2n , sÞ ∈ ℝ 2n+1 leads to the Heisenberg group ℍ n . The left invariant vector fields generated by commutation the Lie algebra on ℍ n are and the only nontrivial commutator of such fields is We call that X 1 , X 2 , ⋯, X 2n are the horizontal vector fields on ℍ n and T the vertical vector field. Denote the horizontal gradient of a smooth function u on ℍ n by The homogeneous dimension of ℍ n is ℘ = 2n + 2. The Haar measure in ℍ n is equivalent to the Lebesgue measure in ℝ 2n+1 . We denote the Lebesgue measure of a measurable set E ⊂ ℍ n by jEj: The Carnot-Carathèodary metric (C-C metric) between two points in ℍ n is the shortest length of the horizontal curve joining them, denoted by d. The ball induced by the C-C metric is The C-C metric d is equivalent to the Korànyi metric For 1 ≤ p < ∞, Ω ⊂ ℍ n , the horizontal Sobolev space HW k,p ðΩÞ is defined as 3

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which is a Banach space under the norm The local horizontal Sobolev space HW k,p loc ðΩÞ is and the space HW Similar to the definition in [20], Morrey space and Campanato space on Heisenberg group are defined as follows.
then, we say that g belongs to the Morrey space denoted by L p,β ðΩÞ, where Ωðx, rÞ = Ω ∩ B r ðxÞ.

Some Lemmas
For convenience, we introduce some notations:
Proof. We choose a standard cut-off function η ∈ C ∞ 0 ðB r ðx 0 Þ, ½0, 1Þ with η ≡ 1 on B r/2 ðx 0 Þ and Taking a testing function φ = η 2 ðu − lÞ in (17), we have ð Dividing the equality above by the measure of the ball, it yields Note that ðA k i ð·, lðx ′ 0 Þ, ∇ H lÞÞ x 0 ,r is a constant, so it infers by using the integration by parts that Owing to we substitute the left and right hand sides of (60) and (61) into the left and right hand sides of (59), respectively, to obtain Then, adding and subtracting the same term ð1/jB r ðx 0 ÞjÞ Ð B r ðx 0 Þ A k i ðx, lðx ′ 0 Þ, ∇ H lÞ∇ H φdx on the right hand side of the above equality, it gets The treatments to the terms I 0 , I 1 , ⋯, I 4 in (63) are similar to that of Lemma 4.1 in [15], and we simply write the processes of proofs. By (7), (44) and the known inequality ð1 + jaj + jb − ajÞ 2 ≤ 3ð1 + jaj 2 + jb − aj 2 Þ, it gains 6 Advances in Mathematical Physics By (8), j∇ H ηj ≤ 4/r, Young's inequality and Lemma 7, we have It implies from (9), Young's inequality, Jensen's inequality and Lemma 7 that Using (11), Young's inequality and Lemma 7, it follows We have by using (14), Hölder's inequality, Lemma 6, Young's inequality and Lemma 7 that The remaining task is to deal with I 5 . Noting l is independent of t and so we use jTηj ≤ c/r 2 to obtain Now, substituting (64)-(70) into (63), and taking ε small enough, it implies Then, (56) is proved. A-harmonic lemma). Assume the assumptions of Theorem 1 are satisfied. For B 2r ðx 0 Þ ⊂ Ω with r ≤ r 0 and a horizontal affine function l :

Proof. A direct calculation gives
The treatment of J 1 is similar to that of Lemma 4.2 in [15]. In fact, we use (10), the monotonicity of ϑ, Lemma 7, Young's inequality, Jensen's inequality and Hölder's inequality to gain Now, let us estimate J 2 . Since we see Noting from (17) that so we have The treatments of J 21 and J 22 are similar to that of Lemma 4.2 in [15]. To be specific, by (11), Lemma 7, Young's inequality and jlðx′ 0 Þj + j∇ H lj ≤ M 0 , one has We use (9), Young's inequality, Jensen's inequality and Lemma 7 to get It is worth noting that the treatments of J 23 and J 24 are different from that in [15]. Using the assumption (14), Hölder's inequality and Lemma 6, we obtain Noting it implies 8 Advances in Mathematical Physics In order to deal with ð1/jB r ðx 0 ÞjÞ Then, Now, we replace (80)-(87) in (79) to see Finally, we substitute (75) and (88) into (74) and then use Lemma 11 to get i.e., (73) holds.
Proof. We divide several steps to prove (90).