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

Zhang, Junli; Wang, Jialin

doi:https://doi.org/10.1155/2022/7853139

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Research Article | Open Access

Volume 2022 | Article ID 7853139 | https://doi.org/10.1155/2022/7853139

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

Junli Zhang¹and Jialin Wang²

Academic Editor: Pietro d Avenia

Received02 Sept 2021

Accepted06 Oct 2021

Published24 Jan 2022

Abstract

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 -harmonic approximation, where the coefficients in the nonlinear subelliptic system are discontinuous and satisfy the VMO condition for , ellipticity and growth condition with the growth index for the Heisenberg gradient variable, and the nonhomogeneous terms satisfy the controllable growth condition and the natural growth condition, respectively.

1. Introduction

Kohn in [1] proved estimates for the operator constructed by Hörmander’s vector fields (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 estimates and Lipschitz estimates to the operator on the Heisenberg group for suitable , where is the vertical vector field. To the nondivergence linear degenerate elliptic operator constructed by Hörmander’s vector fields, Bramanti and Zhu in [4] established estimates with and belonging to VMO spaces related to and Schauder estimates with and being in Hölder spaces for strong solutions. It is important in [4] that the difference between equations without and with was pointed out. When in (3) is basis vector fields and is the drift vector field on homogeneous groups, many scholars have obtained regularities to the operator £ with coefficients and satisfying appropriate conditions, such as [5–8]. In addition, Austin and Tyson in [9] achieved the -smoothness for the operator on the Heisenberg group by using the geometric analysis method.

Note that the equations studied in the above-cited papers are linear. In this paper, we consider the regularity to the weak solution of discontinuous subelliptic systems with drift term on where is the bounded domain in , belongs to the vanishing mean oscillation space (which is abbreviated as VMO) and satisfies the ellipticity on and polynomial growth conditions with the growth index for , and also is continuous for and differentiable for with continuous derivatives,

is the horizontal vector field and is the vertical vector field in . For more information about , see Section 2. The nonhomogeneous term satisfies the controllable growth condition or natural growth condition. We will use the -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.

More regularity for the elliptic system without drift term, one can refer to [10–12] (Euclidean space) and [13–15] (Heisenberg group).

Now, for any , and the growth index , we list the hypotheses that the system satisfies.

(H1). Let satisfy the following ellipticity and polynomial growth conditions (growth index ): where denote the usual derivative of with respect to the variable , .

(H2). Assume that is continuous for . More precisely, there exists a bounded, concave, and nondecreasing continuous modulus with such that

(H3). Let be differentiable for the variable with continuous derivatives, that is, there exists a bounded, concave, and nondecreasing continuous modulus with such that

(H4). For all , satisfies the following VMO condition: where is a bounded function and satisfies

Here, we have used in (11) and (12) the notation

(HC) (controllable growth condition). The nonhomogeneous term satisfies the following controllable growth condition where is a positive constant, and denotes the homogeneous dimension of the Heisenberg group.

Obviously, we can see that system (5) includes the system

We state the main result.

Theorem 1. Assume that and satisfy the assumptions (H1)-(H4) and (HC). If and is a weak solution to system (5), i.e., for all , then, there exists a relatively closed singular set such that for any , we have Moreover, for any , we have where is a local Morrey space. The singular set satisfies where

Corollary 2. Assume that and satisfy the assumptions (H1)-(H4) and the following assumption:
(HN). The nonhomogeneous term satisfies the following natural growth condition for where and are constants depending only on .
Then, we have for weak solution to system (5) under the assumption , where and is same as in Theorem 1.

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 -harmonic approximation method was first introduced by Duzaar and Steffen in [16] and then extended to other cases by some authors, see [17–19]. In this paper, we use the -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 -harmonic lemma, the decay estimate, and iteration relations. In Section 4, the proof of Theorem 1 is given.

2. Preliminaries

2.1. The Heisenberg Group and Some Function Spaces on

The Euclidean space with the group multiplication where leads to the Heisenberg group . The left invariant vector fields generated by commutation the Lie algebra on are and the only nontrivial commutator of such fields is

We call that are the horizontal vector fields on and the vertical vector field. Denote the horizontal gradient of a smooth function on by

The homogeneous dimension of is . The Haar measure in is equivalent to the Lebesgue measure in . We denote the Lebesgue measure of a measurable set by

The Carnot-Carathèodary metric (C-C metric) between two points in is the shortest length of the horizontal curve joining them, denoted by . The ball induced by the C-C metric is

For its Korànyi metric is denoted by

The C-C metric is equivalent to the Korànyi metric

For the horizontal Sobolev space is defined as which is a Banach space under the norm

The local horizontal Sobolev space is and the space is the closure of in .

Similar to the definition in [20], Morrey space and Campanato space on Heisenberg group are defined as follows.

Definition 3 (Morrey space). Let . For the function , if then, we say that belongs to the Morrey space denoted by , where .

Definition 4 (Campanato space). Let . For the function , if then, we say that belongs to the Campanato space denoted by , and its norm is defined as

Lemma 5 (see [21, 22]). If for any , we have , then .

Lemma 6 (Sobolev inequality, [23]). For , and for any , it holds where

Then, the following four lemmas are true.

For the proof of Lemma 6, see [24] and [25].

2.2. Horizontal Affine Function and Some Lemmas

Let , and . Denote the horizontal components of by

Let be a horizontal affine function. Following [13], if the horizontal affine function is a minimizer of the functional then, we have where stands for the matrix , and is a positive constant defined as

According to the meaning of , one has the following Poincaré inequality ([13]): where .

Throughout the paper, we define

Lemma 7 (see [26]). For any and , it holds (1)(2)(3)(4)(5)(6) for with

Lemma 8 (Sobolev-Poincaré-type inequality, [15]). Let and with . Then, it follows where and depends only on . In particular, we have Let be a bilinear form with constant tensorial coefficients. We recall that a map is -harmonic if and only if it holds for any testing function .

Lemma 9 (see [15]). Let be a weak solution of the constant coefficient system Then, is smooth and there exists such that for any ,

Lemma 10 (see [15]). Given , for any , there exist constants and and a bilinear form on satisfying that for , If is an approximate -harmonic map, i.e., for any , it holds then, there exists a -harmonic map satisfying

3. Some Lemmas

For convenience, we introduce some notations: where

Lemma 11 (Caccioppoli-type inequality). Let be a weak solution to (5) under the assumptions (H1)-(H4) and (HC). Then, for any and the horizontal affine function with , we have where is a positive constant depending on .

Proof. We choose a standard cut-off function with on and Taking a testing function in (17), we have Dividing the equality above by the measure of the ball, it yields Note that 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 on the right hand side of the above equality, it gets The treatments to the terms 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 , it gains By (8), , 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 . Noting is independent of and so we use to obtain Now, substituting (64)–(70) into (63), and taking small enough, it implies Then, (56) is proved.

Lemma 12 (approximately -harmonic lemma). Assume the assumptions of Theorem 1 are satisfied. For with and a horizontal affine function with , we define then, for all , it follows where . Here, we say that is an approximately -harmonic map.

Proof. A direct calculation gives The treatment of is similar to that of Lemma 4.2 in [15]. In fact, we use (10), the monotonicity of , Lemma 7, Young’s inequality, Jensen’s inequality and Hölder’s inequality to gain Now, let us estimate . Since we see Noting from (17) that so we have The treatments of and are similar to that of Lemma 4.2 in [15]. To be specific, by (11), Lemma 7, Young’s inequality and , one has We use (9), Young’s inequality, Jensen’s inequality and Lemma 7 to get It is worth noting that the treatments of and are different from that in [15]. Using the assumption (14), Hölder’s inequality and Lemma 6, we obtain Noting it implies In order to deal with , we denote so 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.

Lemma 13 (decay estimate). Assume the assumptions of Theorem 1 are satisfied and with . For constants and from the -harmonic approximation Lemma 10, we impose the following smallness conditions: (i)(ii)Then, it holds where and denote the minimizing horizontal affine functions with and , respectively, and the constant depends only on .

Proof. We divide several steps to prove (90).

Step 1. Let us take where . We first claim that satisfies the assumptions (51) and (52) Lemma 10.
In fact, for and any , we have by using Lemma 12, Lemma 7 (2), and assumptions (ii) and (i) that Now, it deduces from Lemma 7 (2) and Lemma 11 that Then, the assumptions (51) and (52) of Lemma 10 are satisfied by (92) and (93).
Using Lemma 10, it follows that there exists an -harmonic function satisfies and

Step 2. We estimate .
Let us denote and compute by (3) and (2) of Lemma 7, Lemma 8 and (94) that In order to estimate (97), we need to deal with . Noting Lemma 9, it derives and so we have Hence, for , we use Lemma 7 (1), (99), (100), Hölder’s inequality and (95) to obtain By substituting (101) into (97), we get Since in (102) is arbitrary, we take especially . Noting we use (ii) and the monotonicity of to obtain Therefore, it shows from (102) that Now, we substitute into (105) and use Lemma 7 (2) to gain Since is a minimizer of , we have

Step 3. We estimate .
To do so, let us first deal with . Since is bounded almost everywhere by (94), we denote its upper bound by . It implies by Lemma 7 (1) that (1)when , it follows(2)when , we haveHence, Now, by using Lemma 8, (101), Lemma 7 (1) and (2), (94) and the similar proof to (102), we get We substitute into the inequality above to obtain and know from that is a minimizer of that

Step 4. Combining (107) with (114), we derive (90). Lemma 13 is proved.
Before stating a new lemma, we introduce Campanato-type functions. For the fixed Hölder exponent , define a Campanato-type function by We can prove the following lemma from Lemma 13.

Lemma 14 (iteration relations). Assume the assumptions of Theorem 1 are satisfied. For any , there exist constants and , such that if for with , one has then, for any , it holds

Proof. Its proof is similar to Lemma 4.4 in [15]. Actually, has been proved in [15], so we only need to take and change the estimate of in [15] to

4. Proof of Theorem 1

Proof of Theorem 1 is finished with two steps.

Step 1. We prove . In fact, by Lebesgue’s differentiation theorem ([27]), we get , so our aim is to show that is Hölder continuous for every . For any , we use Lemma 8, (43) and Lemma 7 (1) to gain For any and , using (from it, one sees ), and Hölder’s inequality, it infers where is used in the third inequality.
By the definition of and , (117) and (119), we know that for any and , there exists a radius such that Using the continuity of integrals, it follows that there exists a neighborhood of so that for any , Then, Lemma 14 shows so i.e., . Therefore, we have from Lemma 5.

Step 2. We prove . For , it implies by Lemma 7 (1) that (1)if , then (2)if , then so Thus, i.e., where .
Therefore, Theorem 1 is proved.

Data Availability

No data is used.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 11771354 and No. 12061010) and the Natural Science Foundation of Jiangxi Province grant 20202BAB201004.

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Copyright © 2022 Junli Zhang and Jialin Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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