Partial Regularity for Nonlinear Subelliptic Systems with Dini Continuous Coefficients in Heisenberg Groups
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 ; we also refer the reader to Giaquinta  and Chen and Wu  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  and simplified by Duzaar and Grotowski in  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  for the case of Heisenberg groups and  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  and Mingione et al. in  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  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  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  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  and Wang and Liao  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  is not appropriate in our setting. In order to obtain the desired decay estimate, we use the Poincaré type inequality in  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 .
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  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 , 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  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  established a priori estimate for the weak solution to homogeneous sub-elliptic systems with constant coefficients in the Heisenberg group (also see  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),
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
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
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