Abstract and Applied Analysis

Volume 2013 (2013), Article ID 950134, 12 pages

http://dx.doi.org/10.1155/2013/950134

Research Article

Partial Regularity for Nonlinear Subelliptic Systems with Dini Continuous Coefficients in Heisenberg Groups

Jialin Wang, Pingzhou Hong, Dongni Liao, and Zefeng Yu

School of Mathematics and Computer Science, Gannan Normal University, Ganzhou, Jiangxi 341000, China

Received 17 June 2013; Accepted 8 August 2013

Academic Editor: Pekka Koskela

Copyright © 2013 Jialin Wang et al. 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.

Abstract.

This paper is concerned with partial regularity to nonlinear subelliptic systems with Dini continuous coefficients under quadratic controllable growth conditions in the Heisenberg group . Based on a generalization of the technique of -harmonic approximation introduced by Duzaar and Steffen, partial regularity to the sub-elliptic system is established in the Heisenberg group. Our result is optimal in the sense that in the case of Hölder continuous coefficients we establish the optimal Hölder exponent for the horizontal gradients of the weak solution on its regular set.

1. Introduction and Statements of Main Results

In this paper, we are concerned with partial regularity of weak solutions to nonlinear sub-elliptic systems of equations of second order in the Heisenberg group in divergence form, and more precisely, we consider the following systems:

where is a bounded domain in , , the definition of is to be seen in the next section (11), , , and .

Under the coefficients assumed to be Dini continuous, the aim of this paper is to establish optimal partial regularity to the sub-elliptic system (1) in the Heisenberg group . Comparing Hölder continuous coefficients (see [1, 2] for the case of sub-elliptic systems), such assumption is weaker. More precisely, we assume for the continuity of with respect to the variables that

for all , , and , where is monotone nondecreasing and is monotone nondecreasing and concave with . We also required that be nonincreasing for some and that

We adopt the method of -harmonic approximation to the case of sub-elliptic systems in the Heisenberg groups and establish the optimal partial regularity result. Roughly speaking, assume additionally to the standard hypotheses (see precisely (H1), (H2), and (H4)) that satisfies (2) and (3). Let be a weak solution of the sub-elliptic system (1). Then, is of class outside a closed singular set Sing of the Haar measure . Furthermore, for , the derivative of has the modulus of continuity in a neighborhood of . Our result is optimal in the sense that in the case , , we have Hölder continuity to be optimal in that case.

As is well known, even under reasonable assumptions on and of the systems of equations, one cannot, in general, expect that weak solutions of (1) will be classical, that is, -solutions. This was first shown by de Giorgi [3]; 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 -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 or in the Euclidean setting; see [8–11] for Hölder continuous coefficients and [12–15] for Dini continuous coefficients.

However, turning to sub-elliptic equations and systems in the Heisenberg groups , some new difficulties will arise. Already in the first step, trying to apply the standard difference quotient method, the main difference between Euclidean and Heisenberg groups becomes clear. Any time we use horizontal difference quotients (i.e., in the directions ), extra terms with derivatives in the direction will arise due to noncommutativity (see (12)), but these cannot be controlled by using the initial assumptions on the weak solution. Several results were focused on those equations which have a bearing on basic vector fields on the Heisenberg group or, more generally, the Carnot group. Capogna [16, 17] studied the regularities for weak solutions to quasi-linear equations. Concretely, by a technique combining fractional difference quotients and fractional derivatives defined by Fourier transform, differentiability in the nonhorizontal direction, estimate, and continuity of weak solutions are obtained; see [16] for the case of Heisenberg groups and [17] for Carnot groups. To sub-elliptic -Laplace equations in Heisenberg groups, Marchi in [18–20] showed that and for by using theories of Besov space and Bessel potential space. Domokos in [21, 22] improved these results for employing the A. Zygmund theory related to vector fields. Recently, by meticulous arguments, Manfredi and Mingione in [23] and Mingione et al. in [24] proved Hölder regularity with regard to full Euclidean gradient for weak solutions and further continuity under the coefficients assumed to be smooth.

While regularities for weak solutions to sub-elliptic systems concerning vector fields are more complicated, Capogna and Garofalo in [25] showed the partial Hölder regularity for the horizontal gradient of weak solutions to quasilinear sub-elliptic systems with , being horizontal vector fields in Carnot groups of step two, where and satisfy the quadratic structure conditions. Their way relies mainly on generalization of classical direct method in the Euclidean setting. Shores in [26] considered a homogeneous quasi-linear system in the Carnot group with general step, where also satisfies the quadratic growth condition. She established higher differentiability and smoothness for weak solutions of the system with constant coefficients and deduced partial regularity for weak solutions of the original system. With respect to the case of nonquadratic growth, Föglein in [27] treated the homogeneous nonlinear system in the Heisenberg group under superquadratic structure conditions. She got a priori estimates for weak solutions of the system with constant coefficients and partial regularity for the horizontal gradient of weak solutions to the initial system. Later, Wang and Niu [1] and Wang and Liao [2] treated more general nonlinear sub-elliptic system in the Carnot groups under superquadratic growth conditions and subquadratic growth conditions, respectively.

The regularity results for sub-elliptic systems mentioned above require Hölder continuity with respect to the coefficients . When the assumption of Hölder continuity on is weakened to Dini continuity, how to establish partial regularity of weak solutions to nonlinear sub-elliptic systems in the Heisenberg group. This paper is devoted to this topic. To define weak solution to (1), we assume the following structure conditions on and .

(H1) is differentiable in , and there exist some constants such that

 Here, we write down . (H2) is uniformly elliptic; that is, for some , we have

(H3) There exist a modulus of continuity and a nondecreasing function such that

(H4) satisfies quadratic controllable growth condition

 where because ; see (16).

Without loss of generality, we can assume that and the following.

() is nondecreasing with .() is concave; in the proof of the regularity theorem, we have to require that is nonincreasing for some exponent . We also require Dini's condition (2) which was already mentioned in the introduction.() for some .

In the present paper, we will apply the method of -harmonic approximation adapting to the setting of Heisenberg groups to study partial regularity for the system (1). Since basic vector fields of Lie algebras corresponding to the Heisenberg group are more complicated than gradient vector fields in the Euclidean setting, we have to find a different auxiliary function in proving Caccioppoli type inequality. Besides, the nonhorizontal derivatives of weak solutions will happen in the Taylor type formula in the Heisenberg group and cannot be effectively controlled in the present hypotheses. So, the method employing Taylor's formula in [12] is not appropriate in our setting. In order to obtain the desired decay estimate, we use the Poincaré type inequality in [28] as a replacement. And we obtain the following main result.

Theorem 1. Assume that coefficients and satisfy (H1)–(H4), (μ1)–(μ3) and that is a weak solution to the system (1); that is,

Then, there exists a relatively closed set such that . Furthermore, and Haar meas , where

In addition, for and , the derivative has the modulus of continuity in a neighborhood of .

2. Preliminaries

The Heisenberg group is defined as endowed with the following group multiplication:

for all  , . This multiplication corresponds to addition in Euclidean . Its neutral element is , and its inverse to is given by . Particularly, the mapping is smooth, so is a Lie group.

The basic vector corresponding to its Lie algebra can be explicitly calculated by the exponential map and is given by

for , and note that the special structure of the commutators:

that is, , is a nilpotent Lie group of step . is called the horizontal gradient and the vertical derivative.

The pseudonorm is defined by

and the metric induced by this pseudonorm is given by

The measure used on is Haar measure, and the volume of the pseudoball is given by

The number

is called the homogeneous dimension of .

The horizontal Sobolev spaces are defined as

Then, is a Banach space with the norm

is the completion of under norm (18).

Lu [28] showed the following Poincaré type inequality related to Hörmander's vector fields for , , :

where we write down here and there. Note the fact that the horizontal vectors defined in (11) fit Hörmander's vector fields and that (19) is valid for .

Following [12], for technical convenience, letting , we have the corresponding properties for : () is continuous, nondecreasing and ; () is concave, and is nonincreasing for some exponent ; () for some . Changing by a constant, but keeping , we may assume the following: () , implying for . Also note that it implies that from () and (), for all .

Furthermore, the following inequality holds:

The condition (H3) becomes

Moreover, we deduce the existence of a nonnegative modulus of continuity with for all such that is nondecreasing with respect to for fixed and is concave and nondecreasing with respect to for fixed . Also, we have for ,

Using (H1) and (H2), we see that

In the sequel, the constant may vary from line to line.

3. Caccioppoli Type Inequality

In this section, we present the following -harmonic approximation lemma in the Heisenberg group introduced by Föglein [27] with as a special case and prove a Caccioppoli type inequality in our setting.

Lemma 2. Let and be fixed positive numbers and with . If for any given , there exists with the following properties:

(I)for any satisfying

(II) for any satisfying

then, there exists an -harmonic function such that

Föglein [27] established a priori estimate for the weak solution to homogeneous sub-elliptic systems with constant coefficients in the Heisenberg group (also see [25] for Carnot groups of step ). We list it as follows:

In what follows, we let and for . Note that and that , are nonincreasing functions.

Lemma 3. Let be a weak solution to the system (1) under the conditions (H1)–(H4), (μ1)–(μ3). Then, for every , , , and such that , the inequality

holds, where is the horizontal component of and

Proof. Let . Take a test function in (8) with satisfying , , and on . Then, we have , , and

Adding this to the equations

It follows that by using the hypotheses (H1), (H3) (i.e., (23), (21), resp.), and (H4),

where

Applying (H2), the left hand side of (33) can be estimated as

For to be fixed later, we have, using Young's inequality,

Using Jensen's inequality, (20), and the fact that for , we arrive at

where is an abbreviation of the function . Also, note that the application of (20) in the second last inequality is possible by our choice .

Using Young's inequality and (37) in , we obtain

And similarly, we see

Here we have used in the last inequality.

By Hölder's inequality, (19), and Young's inequality, one gets

where we have used the fact that .

Applying these estimates to (37), we obtain

Choosing , we obtain the desired inequality (29).

4. Proof of the Main Theorem

In this section, we will complete the proof of the partial regularity results via the following lemmas. In the sequel, we always suppose that is a weak solution to (1) with the assumptions of (H1)–(H4) and ()–().

Lemma 4. Let with and satisfying and . Then, there exists a constant such that

Proof. Using the fact that and the weak form (8), we deduce

It yields

Using (22), Hölder's inequality, the fact that is concave, and Jensen's inequality, we have

Similarly, using (21) and the fact that for , we obtain

where we have used the fact that which follows from the nondecreasing property of the function , , and our assumption .

In the same way, it follows that by using (21), (37), and (19),

Using Hölder's inequality, (19), and Young's inequality, we have

where we have used the assumption and the fact that and . Combining these estimates, we obtain the conclusion with .

Lemma 5. Assume that the conditions of Lemma 2 and the following smallness conditions hold:

with , together with

Then, the following growth condition holds for

where one abbreviates and with .

Proof. We define , where

Then, we have . Now, we consider such that . Applying Lemma 4 on to , we have for any ,

In consideration of the small condition (49), we see that (54) and (55) imply conditions (26) in Lemma 2. Also note that (H1) and (H3) imply condition (25). So, there exists an -harmonic function such that

Taking and replacing by , we use Lemma 3 to obtain

where

Using the fact that has mean value on the ball , the definition of , and (19), we have

where . Note that in the second last inequality we have used the fact that

In consideration of the fact that , and the assumptions and , it follows that

Let with . Combining these estimates (57)–(61) and considering the small condition (51) (it implies ; see (64) and (65)), we deduce that

We now specify , such that . Note that the small condition (50) implies with , and it yields

where we have used the a priori estimate (28) for the -harmonic function . Furthermore, using (19) and recalling the definition of and , we have

Combining these estimates with (62), we have

Then, the proof of Lemma 5 is complete.

For , we find (depending on , , , , , and such that

With from (67), we choose (depending on , , , , , , , and such that

where .

By the proof method of of Lemma 5.1 in [12] and conditions (67)-(68), Lemma 6 can be proved. As is well known, it is sufficient to complete the proof of Theorem 1 once we obtain Lemma 6.

Lemma 6. Assume that for some and one has

(1) , (2), (3).

Then, the small conditions (49)–(51) are satisfied on the balls for . Moreover, the limit exists, and the inequality

is valid for with a constant .

Proof. The proof is very similar to the proof of Lemma 5.1 in [12]. We omit it here.

Acknowledgments

The project was supported by the National Natural Science Foundation of China (no. 11201081 and no. 11126294) and by the Science and Technology Planning Project of Jiangxi Province, China, no. GJJ13657.

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