Abstract
We consider nonlinear sub-elliptic systems with VMO-coefficients for the case 1 < p < 2 under controllable growth conditions, as well as natural growth conditions, respectively, in the Heisenberg group. On the basis of a generalization of the technique of 𝓐-harmonic approximation introduced by Duzaar-Grotowski-Kronz, and an appropriate Sobolev-Poincaré type inequality established in the Heisenberg group, we prove partial Hölder continuity results for vector-valued solutions of discontinuous sub-elliptic problems. The primary model covered by our analysis is the non-degenerate sub-elliptic p-Laplacian system with VMO-coefficients, involving sub-quadratic growth terms.
1 Introduction and statements of main results
In this paper, we consider discontinuous sub-elliptic systems with sub-quadratic growth coefficients that belong to the space of functions with vanishing mean oscillation (VMO, for short) in the Heisenberg group ℍn. We establish optimal partial Hölder continuity for vector-valued weak solutions in the sense that the solution is Hölder continuous on an open subset of its domain with full measure. More precisely, let Ω be a bounded domain, and horizontal gradient X = {X1, ⋯ X2n} with the horizontal vector fields Xi : (i = 1, ⋯, 2n) in ℍn, we consider sub-elliptic systems of the type
where the primary coefficient
The main new aspect of this paper is the fact that we are able to deal with the inhomogeneity Bα : ℝ2n+1 × ℝN × ℝ2n×N → ℝN that satisfies the sub-quadratic controllable growth conditions, as well as sub-quadratic natural growth conditions, respectively, and the primary coefficient
We now impose the precise structure assumptions for coefficients
(H1). The primary coefficient
for any choice of ξ ∈ Ω, u, u0 ∈ ℝN and P, P0 ∈ ℝ2n×N. Here structure constants ν ≤ 1 ≤ L < ∞.
(H2). The vector field
(H3). The vector field
With respect to the dependence on the first variable ξ, we do not impose a continuity condition, but we merely assume the following VMO-condition.
(H4). The mapping ξ ↦
where vξ0 : ℝ2n+1 × [0, ρ0] → [0, 2L] are bounded functions satisfying
Here we have used the short-hand notation
(HC) (Controllable growth condition). The inhomogeneity Bα satisfies sub-quadratic controllable growth condition
where C is a positive constant. We note that Q ≥ 3 is the homogeneous dimension in non-Abelian Heisenberg groups (see (2.1) below), and the exponent p ∈ (1, 2). So those infer that p < Q, and then, p∗ =
(HN) (Natural growth condition). For |u| ≤ M =
where a = a(M) and b = b(M) are constants possibly depending on M.
Now we mention some results on elliptic systems. Duzaar and Grotowski [9] prove optimal partial Hölder continuity for nonlinear elliptic systems with quadratic growth p = 2, by a new method so-called 𝓐-harmonic approximation introduced by Duzaar and Steffen [15]. Then, the method was extended to non-quadratic growth cases. Duzaar and Mingione [12, 13] consider systems of p-Laplacian type. Many partial regularity results have been established for more general nonlinear elliptic problems with Hölder, or Dini continuous coefficients; see, for example, [6, 8, 11, 28]. Furthermore, with respect to discontinuous elliptic problems, we refer to Bögelein, Duzaar, Habermann and Scheven [1], Ragusa [23], Zheng [35], Kanazawa [26], Goodrich, Ragusa and Scapellato [20], Polidoro and Ragusa [22], Scapellato [24], and Tan, Wang and Chen [27] and the references therein.
Several regularity results were focused on sub-elliptic systems in Heisenberg groups, or Hörmander vector fields; see Bramanti [2]. Xu and Zuily [34], Capogna and Garofalo [5], and Shores [25] showed partial regularity for quasi-linear sub-elliptic systems with quadratic growth p = 2. Their methods depend on generalization of classical freezing coefficient method. Then, by the generalization of the method of 𝓐-harmonic approximation, Föglein [16] treated homogeneous nonlinear sub-elliptic systems with Hölder continuous coefficients, under super-quadratic growth conditions p ≥ 2 in the Heisenberg group, and established partial Hölder continuity for the horizontal gradient Xu. Later Wang and Liao [30] considered the case of 1 < p < 2 for inhomogeneous systems in Carnot groups. Furthermore, Wang, Liao and Gao [31] weakened assumptions on coefficients
Regularity results for discontinuous sub-elliptic systems with VMO coefficients instead of continuous coefficients have been established in the work [7] by Di Fazio and Fanciullo, and [19] by Gao, Niu and Wang for the case of quadratic growth; [32] by Wang and Manfredi, [14] by Dong and Niu, [36] by Zheng and Feng, and [33] by Wang, Zhang and Yang for non-quadratic growth conditions. We note that the regularity results in [14] and [36] have a limitation of p near 2, and the result in [19] holds only under a strong smallness condition for the dimension. In contrast, our partial Hölder continuity result stated below, is valid for the full range 1 < p < 2 in any dimension.
The typical strategy in partial regularity depends on decay estimates for certain excess functionals, which measure the oscillations of the solution or its gradient in a suitable sense. In this paper, we are working with a combination of a zero-order excess functional Cy and a first-order excess functional Ψ. For the case p ≥ 2, the functional Ψ is defined by
with the horizontal affine functions l : ℝ2n → ℝN defined in the subsection 2.2 below. It is straightforward to adapt the standard 𝓐-harmonic approximation lemma by utilizing L2-theory combined with the standard Sobolev inequality; see Wang and Manfredi [32] for the super-quadratic natural growth case. However, in the present situation, we treat the case of sub-quadratic controllable growth, and sub-quadratic natural growth, respectively. So one should establish the decay estimate for the following excess functional
where
which provides a measure of the oscillations in the weak solutions u itself. It is remarkable that the excess functionals defined above involve only u, which simplifies the proofs of our partial regularity results. It is shown that if Ψ is small enough on a ball Bξ0(ρ) ⊂ ⊂ U, then, for some fixed θ ∈ (0, 1), one obtain an excess improvement Ψ (ξ0, θ r, lξ0,θr) ≤ C4θ2Ψ*(ξ0, r, lξ0,r) under smallness condition assumptions; see for example, Lemma 4.3. At this point, one has to assume smallness on the Ψ*-excess. Also we note that such an excess improvement estimate has two different quantities Ψ and Ψ* on the left, and the right hand side, respectively. Therefore, in contrast to the standard proof of partial regularity, the excess improvement cannot be iterated directly to yield an excess-decay estimate for Ψ-excess. In the present situation, however, iteration of the excess improvement yields that the Ψ-excess in (1.9) and also the Cy-excess remain bounded. Finally, the boundedness of the Cy-excess on any scale leads immediately to desired Hölder continuity of weak solutions u via the integral characterization of continuity by Campanato. We point out that the idea of such a combination of two excess functionals has its origin by Foss and Mingione [17] for continuous vector fields and integrands, and then, adapted to discontinuous problems with VMO coefficients for p ≥ 2 by Bögelein- Duzaar-Habermann-Scheven [1]. It is worth mentioning that we obviously do not have access to use L2-theory for functions in the horizontal Sobolev space HW1,p with 1 < p < 2. Therefore, we have to establish the following Sobolev-Poincaré inequality with the function V (see Lemma 3.1 below),
with the constant CP dependence only on N, p, Q. This inequality is an essential tool in order to get the regularity result. It is also one technique point where our case differs from the case p ≥ 2 in [32].
Under the previous assumptions(H1)-(H4) and (HC), and (H1)-(H4) and (HN), respectively, we establish the following two partial Hölder continuity results.
Theorem 1.1
Assume that coefficients
Then, there exists a relatively closed singular set Ω0 ⊂ Ω such that u ∈
with the functional V defined in (2.3), and the singular set has (2n + 1)-Lebesgue measure zero |Ω0| = 0 and its complement Ω ∖ Ω0 is a set of full measure in Ω.
Theorem 1.2
Assume that coefficients
Remark 1.3
It is worth noting that the choice
makes the sub-elliptic p-Laplacian system with VMO-coefficients, involving sub-quadratic growth terms
just as a special case of (1.1), where
The organization of this paper is as follows. In Section 2, we collect some basic notions and facts associated to Heisenberg groups, involving quasi-distance, horizontal Sobolev spaces, and horizontal affine function and some estimates. In Section 3, firstly an appropriate Sobolev-poincaré inequality which plays an important part on proving Hölder regularity is established. Then, an 𝓐-harmonic approximation lemma, and a prior estimate for weak solution h ∈ HW1,1 to the constant coefficient homogeneous sub-elliptic systems are given. In Section 4, we prove partial regularity results of Theorem 1.1 under sub-quadratic controllable structure assumptions (H1)-(H4) and (HC) by several steps. Step 1 is to gain a suitable Caccioppoli-type inequality which is an essential tool to get partial regularity. An appropriate linearization strategy is given in the second step. Then, one can achieve that solutions are approximately 𝓐-harmonic by the linearization procedure, and an excess improvement estimate for the functional Ψ is obtained under two smallness condition assumptions, by combining with 𝓐-harmonic approximation lemma in the third steps. Once the excess improvement is established, the iteration for the Ψ-excess and the Cy-excess can be acquired in Step 4. Finally, we show boundedness of the Campanato-type excess which leads immediately to desired Hölder continuity and Morrey regularity of Theorem 1.1. The last section shows the results of Theorem 1.2 under sub-quadratic natural structure assumptions (H1)-(H4) and (HN). In such a case, we establish appropriate estimates just for the natural growth term, and the rest procedure is similar to the proof of Theorem 1.1.
2 Preliminaries
In this section, we will give introduction of the Heisenberg group ℍn and definitions of several function spaces, and some elementary estimates which will be used later.
2.1 Introduction of the Heisenberg group ℍn
The Heisenberg group ℍn is defined as ℝ2n+1 endowed with the following group multiplication:
for all ξ = (ξ̄, t) = (x1, x2, ⋯, xn, y1, y2, ⋯, yn, t), ξ̃ = (η̄, t̃) = (x̃1, x̃2, ⋯, x̃n, ỹ1, ỹ2, ⋯, ỹn, t̃). Its neutral element is 0, and its inverse to (ξ̄, t) is given by (− ξ̄, − t).
The basic vector fields corresponding to its Lie algebra can be explicitly calculated, and are given by
for i = 1, 2, ⋯, n, and note that the special structure of the commutators:
that is, (ℍn, ⋅) is a nilpotent Lie group of step 2. X = (X1, ⋯, X2n) is said to be the horizontal gradient, and T vertical derivative.
The homogeneous norm is defined by ∥(ξ̄, t)∥ℍn = (∥ξ̄∥4 + 16 t2)1/4, and the metric induced by this homogeneous norm is given by
The measure used on ℍn is the Haar measure (Lebesgue measure in ℝ2n+1), and the volume of the homogeneous ball BR (ξ0) = {ξ ∈ ℍn : d(ξ0, ξ) < R} is given by |BR (ξ0)|ℍn = R2n+2 |B1 (ξ0)|ℍn
is called the homogeneous dimension of ℍn, and the quantity ωn is the volume of the homogeneous ball of radius 1.
Let 1 ≤ p ≤ +∞. We denote by
the horizontal Sobolev space. Then, HW1,p(Ω) is a Banach space under the norm
For u ∈ HW1,q(BR (ξ0)), 1 < q < Q and 1 ≤ p ≤
The inequality (2.2) is valid for p = q (≥ 2).
Throughout the paper, we shall use the functions V, W: ℝk → ℝk defined by
for each ς ∈ ℝk, k ∈ ℕ and p > 1. The functions V and W are locally bi-Lipschitz bijection on ℝk.
The following inequality
immediately yields
The purpose of introducing W is the fact that in contrast to
The following lemma includes some useful properties of the function V. The proof can be found in Lemma 2.1 of [4]. For simplicity, here, we replace the coefficient 2(p−2)/4 with
Lemma 2.1
Let p ∈ (1, 2) and V : ℝk → ℝk be the functions defined in (2.3). Then, for any ς1, ς2 ∈ ℝk and t > 0, the following inequalities hold:
|V(ς1 + ς2)| ⩽ C(p)(|V(ς1)| + |V(ς2)|);
|V(ς1) − V(ς2)| ⩽ C(p, k)|V(ς1 − ς2)|;
|V(ς1 − ς2)| ⩽ C(p, M)|V(ς1) − V(ς2)| for all ς2 with |ς2| ≤ M.
2.2 Horizontal affine function and estimates in ℍn
Let u ∈ L2(Bρ(ξ0), ℝN), ξ0 ∈ ℝ2n+1, and consider the horizontal components
If the function
minimizes the functional
among horizontal affine function l : ℝ2n → ℝN, Then, we have
and
where the vector u ⊗ (ξ̄ −
Here, we use the notation (2k − 2)!! = (2k − 2)(2k − 4)⋅⋅⋅4 × 2 and (2k − 1)!! = (2k − 1)(2k − 3)⋅⋅⋅3 × 1. The proof of the results above can be found in [32] by Wang and Manfredi. On the basis of this formula, elementary calculations yield the following estimates.
Lemma 2.2
Let u ∈ L2(Bρ(ξ0), ℝN), θ ∈ (0, 1).We denote by lξ0,ρ and lξ0,θρ, the horizontal affine function defined as above for the radii ρ and θρ. Then, we have
and, more generally,
for every horizontal affine function l : ℝ2n → ℝN.
Proof
By the identity (2.5) and Hölder's inequality, we obtain
where we have used the fact that
For (2.8), it follows
□
According to the definition of the function lξ0,ρ, the following version of the Poincaré inequality (2.2) is true, that is,
where 1 < q < Q, 1 ≤ p ≤
3 Sobolev-Poincaré type inequality and 𝓐-harmonic approximation
We know that L2-theory cannot be directly used to obtain appropriate estimates for solutions u ∈ HW1,p with 1 < p < 2, so in this section, we first establish a suitable version of Sobolev-Poincaré type inequality with functions V. This inequality is an essential tool in proving partial regularity. Then, we give a prior estimate for 𝓐-harmonic functions h ∈ HW1,1, and introduce an 𝓐-harmonic approximation lemma which plays an important part in getting excess improvement estimates.
Lemma 3.1
(Sobolev-Poincaré type inequality). Let p ∈ (1, 2) and u ∈ HW1,p(Bρ(ξ0), ℝN) with Bρ (ξ0) ⊂ Ω, Then, it follows
with p∗ =
Proof
We introduce the operator of fractional integration on Ω of order 1 as follows
Based on Theorem 2.7 in [3] by Capogna, Danielli and Garofalo, we deduce for 1 < p < +∞
where we denote by p∗ =
Lu [21] gave a representation formula for a function on graded nilpotent Lie groups for the left invariant vector fields; see Lemma 3.1 there. One form of the representation states that there exist constants c > 1 and C > 1 such that
Noting that W2/p(t) is monotone increasing and convex, we apply W2/p(t) to both sides of the last inequality and have by Jensen’s inequality
and
One can check that W(|Xu(η)|) ∈ Lp(Bρ (ξ0)), which implies W͠(|Xu(η)) ∈ Lp(ℝ2n+1). Then, applying the inequality (3.2) yields
or
We obtain the assertion of the theorem, first for W, and Then, also for V by (2.4). □
Let 𝓐 ∈ Bil(Ω × ℝN × ℝ2n×N, ℝ2n×N) be a bilinear form with constant tensorial coefficients. We say that a map h ∈ C∞(Bρ(ξ0), ℝN) is 𝓐-harmonic if and only if
holds for all testing function φ ∈
Shores in [25] showed that weak solutions h ∈ HW1,2(Ω, ℝN) of the constant coefficient homogeneous sub-elliptic systems belongs to C∞ in the subset Ω0 ⊂ Ω. Then, the following estimate holds for the solution h ∈ HW1,2(Ω, ℝN),
Therefore, we can argue as the proof of Proposition 2.10 in [4] to obtain the same estimate for any 𝓐-harmonic function h ∈ HW1,1 (Ω, ℝN).
Lemma 3.2
Let h ∈ HW1,1 (Ω, ℝN) be weak solutions of the constant coefficient systems. Then, h is smooth and there exists C0 ≥ 1 such that for any Bρ (ξ0) ⊂ Ω
Similarly to [10], one can establish the following version of 𝓐-harmonic approximation for the case 1 < p < 2 in Heisenberg groups.
Lemma 3.3
Let 0 < ν ≤ L and 1 < p < 2 be given. For every ε > 0, there is a constant δ = δ(Q, N, p, ν, L, ε) ∈ (0, 1] assume that y ∈ [0, 1] and that 𝓐 is a bilinear form on ℝ2n×N with the properties
Furthermore, let w ∈ HW1,p(Bρ (ξ0), ℝN) be an approximate 𝓐-harmonic map in the sense that the following estimate holds
and
Then, there exists an 𝓐-harmonic map h ∈ C∞(Bρ(ξ0), ℝN) which satisfies
4 Partial Hölder continuity for sub-quadratic controllable growth
In this section, we prove the partial regularity result of Theorem 1 under the assumptions of sub-quadratic controllable structure conditions. Now we begin with the following.
4.1 Caccioppoli-type inequality
We know that Caccioppoli-type inequality is a preliminary tool to prove partial regularity for systems. So in this subsection, we shall prove a Caccioppoli-type inequality for weak solutions to the sub-elliptic systems (1.1) with sub-quadratic controllable growth conditions.
Lemma 4.1
(Caccioppoli-type inequality). Let u ∈ HW1,p(Ω, ℝN) be weak solutions of the nonlinear sub-elliptic systems (1.1) under the assumptions (H1)-(H4)-(HC). Then, for any ξ0 = (x1, ⋯, xn, y1, ⋯, yn, t) ∈ Ω with Br(ξ0) ⊂ ⊂ Ω, and any horizontal affine functions l : ℝ2n → ℝN with |l(ξ̄0)| + |Xl| ≤ M0, we have the estimate
where Cc is a positive constants depending only on Q, p, ν, L, M0, and the exponent p′ =
Proof
We choose a standard cut-off function ϕ ∈
where we have used the fact that Xφ = ϕ2(Xu − Xl) + 2ϕ(u − l)Xϕ.
In view of the identities
It follows for weak solutions u of systems (1.1) that
with the obvious labelling for I0 − I4.
We first estimate the left-hand side of (4.1). By the first inequality of (1.2), Young’s inequality and definition of the function V (2.3), we have
where we have used the elementary inequality 1 + |a|2 + |b − a|2 ⩽ 3(1 + |a|2 + |b|2), and 1 < p < 2.
Now, we are going to estimate the terms I1 − I4 on the right-hand side of (4.1). For small positive ϵ < 1 appearing in lines, it will be fixed later.
Estimate for I1. We shall decompose the ball Br(ξ0) into four subsets: Ω1 := Br(ξ0) ∩ {|Xu − Xl| ≤ 1} ∩
Using the second inequality of (1.2), |Xϕ| ≤
where we have used the inequality (1 + |Xl + s(Xu - Xl)|)p−2 ≤ 1 for 1 < p < 2.
Similarly to the case 1, there is
where we have used Lemma 2.1, and the inequality
By Young’s inequality and Lemma 2.1, it follows that on the subset Ω3,
where we have used the fact that
On the subset Ω4, it holds that by the assumption |l(
here, we have used the smallness assumption Φ(ξ0, r, l) := ⨍Br(ξ0)|V(Xu − Xl)|2 dξ ≤ 1 and
From (4.3), (4.4), (4.5) and (4.6), we have the estimate for the term I1 as follows
where we have used the inequality
Estimate for I2. By the first inequality of (1.3), we get
To estimate the term I21, we divide the domain of integration into two parts Ω5 := Br(ξ0) ∩ |Xu − Xl| ≤ 1 and Ω6 := Br(ξ0) ∩ {|Xu − Xl| > 1}.
On the set Ω5 where |Xu − Xl| ≤ 1, it holds
where we have used in turn Young’s inequality, ω2 ≤ ω, the concavity of ω and Jensen’s inequality.
On the part Ω6 where |Xu − Xl| > 1, we find
where we have used the inequality
Combining (4.9) with (4.10) leads to
where we have use the fact
The term I22 can be estimated similarly as I21 above. Here, we split the ball Br(ξ0) into two subsets Ω7 := Br(ξ0) ∩
On the subset Ω7, it yields
We deduce on Ω8
From (4.12) and (4.13), it follows
Joining (4.8), (4.11) and (4.14), we obtain
We are now in the position to handle the term I3. By VMO-condition (1.5), the term I3 can be estimated as follows
We can argue the terms I31 and I32 as the same way treating the terms I21 and I22.
On the set Ω5 where |Xu − Xl| ≤ 1, we use vξ0 ≤ 2L and (1.6) to infer the following estimate
On the part Ω6 where |Xu − Xl| > 1, we use (1.6) and the fact that
Using (4.17) and (4.18), we get
Similarly, the term I32 can be estimated as follows
Joining (4.16), (4.19) with (4.20), we have
Estimate for I4. Using Hölder inequality, one has
To obtain an appropriate estimate for I4, we take the domain Br(ξ0) into four parts as the same way of I1.
For the case of Ω1 = Br(ξ0) ∩ {|Xu − Xl| ≤ 1} ∩
On the part Ω2 = Br(ξ0) ∩ {|Xu − Xl| ≤ 1} ∩
where we have used the fact that ⨍Br(ξ0) (|Xu|p + |u|p∗ + 1)dξ ≥ 1, and ϕ ≤ 1.
On the part Ω3 = Br(ξ0) ∩ {|Xu − Xl| ≥ 1} ∩
For the last case of Ω4 = Br(ξ0) ∩ {|Xu − Xl| ≥ 1} ∩
Combining the estimates (4.22)-(4.26), we find that
Joining the estimates(4.2), (4.7), (4.15), (4.21), (4.27) with (4.1), we arrive at
Here, choosing
with a constant Cc = Cc(Q, p, ν, L, M0), p′ =
For sake of simplicity, we motivated the form of the Caccioppoli inequalities in Lemma 4.1. We set
In the sequel, when the choice of ξ0 or l is clear, we frequently write Φ(r, l) or Φ(r) respectively, as a replacement of Φ(ξ0, r, l).
4.2 Approximate 𝓐-harmonicity of weak solutions
To apply 𝓐-harmonic approximation lemma, we need to establish the following lemma, which provides a linearization strategy for non-linear sub-elliptic systems (1.1).
Lemma 4.2
Under the assumptions of Theorem 1.1 are satisfied, B2ρ(ξ0) ⊆ Ω with ρ ≤ ρ0 and an arbitrary horizontal function l : ℝ2n → ℝN, we define
Then, w is approximately 𝓐-harmonic in the sense that
for all φ ∈
Proof
Without loss of generality, we assume that
with obvious labelling of J1 and J2.
In order to get the bound for the first term J1, we first use the inequality (1.4) to obtain
By the monotonicity of μ and the inequality above, it yields
Here, we decompose the ball Bρ(ξ0) into two parts Ω5 and Ω6.
On the domain Ω5 where |Xu − Xl| ≤ 1, it follows by Lemma 2.1 Young’s inequality, Jensen’s inequality, and Hölder’s inequality in turn
where we have used the inequality μ2 ≤ μ.
On the set Ω6 where |Xu − Xl| > 1, we have the following bound
where we have used
Based on the following facts
the integral J2 can be rewritten as
with the obvious meaning of J21 + J22 + J23.
Using the assumption of |l(
where we have used the inequality 0 < p − 1 < 1 in the second line.
Now, we discuss it on the domain Ω5 and Ω6, respectively.
On the set Ω5 where |Xu − Xl| ≤ 1, the following estimate holds
where we have used
On the part Ω6 where |Xu − Xl| > 1, it yields
where we have used the assumption vξ0 ≤ 2L and Lemma 2.1.
Then, we get the following estimate for J21
By first inequality of (1.3), the term J22 can be estimated as follows
Similarly, for the case of |Xu − Xl| ≤ 1 on Ω5, applying Young’s inequality, Jensen’s inequality and Lemma 2.1, we deduce that
where we have used
For the other case of |Xu − Xl| > 1 on Ω6, it follows
where we have used ωp ≤ ω ≤ 1, Jensen’s inequality and Lemma 2.1.
Thus, we get the following estimate for J22
Finally, we handle the term J23 by the same as the way for I4 to obtain
Joining the estimates (4.31)-(4.33) with (4.30), we have
Plugging (4.29) and (4.34) into (4.28), we finally arrive at
where we have employed the Caccioppoli-type inequality from Lemma 4.1, Ψ∗(ρ) ≤ C(n, p)Ψ∗(2ρ) in the last step. This yields the claim. □
4.3 Excess improvement
The strategy of our proof is to approximate the given solution in the sense of L2 by 𝓐-harmonic functions. Now we are in the position to establish the excess improvement.
Lemma 4.3
Suppose that the assumptions of Theorem 1.1 are satisfied and consider a ball Br(ξ0) ⊆ Ω with r ≤ ρ0. For the constants δ = δ (Q, N, p, L, ν, θQ+4) ∈ (0, 1] and y ∈ (0, 1] from the 𝓐-harmonic approximation Lemma 3.3, we let 0 < θ ≤
Then, there holds an excess improvement estimate
with some constants C4 that depend only on n, N, p, ν, δ and L. Here, lξ0,θr and lξ0,r denote the minimizing affine functions introduced in Lemma 2.2.
Proof
We denote Ψ*(r) = Ψ*(ξ0, r, lξ0,r), and take
with lξ0,r = uξ0,r+Xlξ0,r(ξ̄ − ξ̄0) and C2 = max{C1,
First note that, for our choice of the bilinear form
Next by Lemma 4.2 with ρ =
for all
The estimates (4.35) and (5.2) tell us that the conditions of Lemma 3.3 are satisfied. So, there exists an 𝓐-harmonic
In order to estimate excess functional
we now have to handle the integral ⨍Bθr(ξ0) |V(X2h(ξ))|2dξ. Since the function h(ξ) is 𝓐-harmonic, we know that h(ξ) ∈ C∞(Ω) by Lemma 3.2. Noting that the boundedness |Xh(ξ)| ≤ M in the ball
where we have used the estimate (3.3) and Lemma 2.1.
We write lh(ξ) = hξ0,θr + (Xh)ξ0,θr(ξ̄ −
where we have taken ε = θQ+4.
Scaling back to u, we infer
In view of the defining property of lξ0,θr, we arrive at
here, we have denoted
□
4.4 Iteration
First, let y ∈ (0, 1) be an fixed Hölder exponent. We define the Campanato-type excess
Next, we iterate the excess improvement estimate from Lemma 4.3.
Lemma 4.4
Suppose that the assumptions of Theorem 1.1 are satisfied. For every y ∈ (0, 1), there are constant ε*, κ*, ρ* and θ ∈ (0,
for r ∈ (0, ρ*) with Br(ξ0) ⊂ Ω, Then,
respectively, for every k ∈ ℕ.
Proof
We begin by choosing the constants. First, we let
where c0 is defined in (2.6), and C4 is determined in Lemma 4.3, respectively. We note that the choice of θ fixes the constant δ = δ(Q, N, p, ν, L, θQ+4) from Lemma 3.3. Next, we fix an ε* small sufficiently to ensure
Then, we choose κ* > 0 so small that
Finally, we fix ρ* > 0 small enough to guarantee
here we have abbreviated
Now, we are in the position to prove the assertion (Ak) by induction. We assume that (Ak) is true for up to some k ∈ ℕ. Then, we prove the first part of the assertion (Ak+1), that is, the one concerning Ψ (θk+1r, lξ0,θk+1r). For this we are going to prove that the small assumptions for the excess improvement in Lemma 4.3 are satisfied. Firstly, by the assumptions of (Ak) and the choices of ε*, κ* and ρ*, we deduce
Now it is easy to check that our choice of ε* implies that the smallness condition assumptions (i)-(ii) in Lemma 4.3 are satisfied on the level θkr, that is, we have
where we have used
Consequently, we apply Lemma 4.3 with the radius θk r instead of r, this leads to
by the choice of θ in (4.38). This is the result for the first part of (Ak+1).
Now it remains to show the second part, that is, the one concerning Cy(ξ0, θk+1 r). Since lξ0,θkr = uξ0,θkr + Xlξ0,θkr(ξ̄ −
Now we are going to estimate the term
On the subset Ω9, we get
For the case of
Therefore, we deduce the following estimate
By means of (2.8) with the choice of l ≡ uξ0,θkr, and the assumption Ak, we obtain
Recalling that r ∈ (0, ρ*), we deduce that
where we have used the choice of ε* in (4.39), the choice of ρ* from (4.40) and θ in (4.38).
This proves the second part of the assertion (Ak+1) and finally complete the proof of the lemma. □
4.5 Proof of Theorem 1
Proof
By Lebesgue’s differentiation theorem, we obtain |Σ1 ∪ Σ2| = 0. So our aim is to show that every ξ0 ∈ Ω ∖ (Σ1 ∪ Σ2) is a regular point. For every 0 < ρ < dist(ξ0, ∂Ω), by Sobolev-Poincaré type inequality, it follows
Moreover, for any y ∈ (0, 1) and ρ ≤ 1, the following estimate holds
If
If
So, it yields
Keeping in mind the definitions of Σ1, Σ2, from the estimates (4.41) and (4.42), we know that there exists a radius ρ:0 < ρ < min{ρ*, dist(ξ0, ∂Ω)} such that
Using the absolute continuity of the integral, there exists a neighborhood U ⊆ Ω of ξ0 with
Applying Lemma 4.4 in any point ξ ∈ U, Then, we get
This together with Campanato’s characterization of Hölder continuous functions imply that
Hence u ∈
Furthermore, it holds for |Xu − Xlξ,σ| > 1
and we have if |Xu − Xlξ,σ| ≤ 1
Combining (4.44) and (4.45) with (4.43) and (1.6), we get for y ∈ (0, 1)
with abbreviation of
In view of the well known equivalence of Campanato and Morrey spaces for parameters λ ∈ (0, Q), it yields Xu ∈ Lp,λ(U, ℝ2n×N) with λ = Q − p(1 − y). In particular, the parameter λ can be chosen arbitrary chose to Q. This concludes the proof of Theorem 1.1. □
5 Partial Hölder continuity for sub-quadratic Natural growth
In this section, we prove the partial regularity result of Theorem 1.2 under the assumptions of sub-quadratic natural structure conditions (H1)-(H4) and (HN). In this case, we will need to restrict ourselves to bounded solution of (1.1), where the bound M =
in our present situation with a(M) defined in (1.8). Such a similar smallness condition is necessary for a partial regularity result even in the elliptic case with quadratic growth (p = 2); for example, see [18].
We first introduce an elementary inequality showed by Kanazawa in [26]. It is useful to get suitable estimates for the natural growth term in proving Caccioppoli-type inequality.
Lemma 5.1
Consider fixed a, b ≥ 0, p ≥ 1. Then, for any ε > 0, there exists K = K(p, ε) ≥ 0 satisfying
Lemma 5.2
(Caccioppoli-type inequality). Let u ∈ HW1,p(Ω, ℝN) ∩ L∞ (Ω, ℝN) be weak solutions of the systems (1.1) under the assumptions (H1)-(H4)-(HN) with ν > 2a(M)(M + M0)
where Cc is some positive constants depending only on Q, N, p, a, b, L, ν, M, M0, and the exponent p′ =
Proof
Let φ = ϕ2(u − l) be a testing function for sub-elliptic systems (1.1), where the standard cut-off function ϕ ∈
with the obvious meaning for
With respect to the terms
Now we are in the position to get an appropriate bound for the term
We denote by
If |Xu − Xl| ≤ 1, we have |Xu − Xl|p ≤ |Xu − Xl|2 + 1. Then, it follows
Combining these estimates above, we have
We denote by I42 the second term of the right-hand side of (5.8). If
If
So, we finally arrive at
Combining (5.3)-(5.6), (5.9) and (5.2), we have
Noting that the smallness condition 2a(M + M0)
where Cc = C(Q, p, a, b, L, ν, M0, M). This yields the claim. □
For sake of simplicity, we motivated the form of the Caccioppoli inequalities in Lemma 5.2. We write
Lemma 5.3
Under the assumptions of Theorem 1.1 are satisfied, B2ρ(ξ0) ⊆ Ω with ρ ≤ ρ0 and an arbitrary horizontal function l : ℝ2n → ℝN, we define
then, w is approximately 𝓐-harmonic in the sense that
for all φ ∈
Proof
The proof is similar as the case of controllable growth. Here, we just give the different estimate for the natural growth term, that is,
where we have used the bound for the natural growth term
Applying Lemma 5.2 and Lemma 5.3, we can establish the improvement estimate for Excess functional Ψ with the same form as Lemma 4.3, that is,
Lemma 5.4
Suppose that the assumptions of Theorem 1.2 are satisfied and consider a ball Br(ξ0) ⊆ Ω with r ≤ ρ0. For the constants δ = δ (Q, N, p, L, ν, θQ+4) ∈ (0, 1] and y ∈ (0, 1] from the 𝓐-harmonic approximation Lemma 3.3, we let 0 < θ ≤
Then, the following excess improvement estimate holds
where constants
By Lemma 5.4, the iteration for the Ψ-excess and the Cy-excess can be obtained as follows,
Lemma 5.5
Suppose that the assumptions of Theorem 1.2 are satisfied. For every y ∈ (0, 1), there are constant ε*, κ*, ρ* and θ ∈ (0,
for r ∈ (0, ρ*) with Br(ξ0) ⊂ Ω, then,
respectively, for every k ∈ ℕ.
Proof of Theorem 1.2
It is enough to use Lemma 5.5, and repeat the procedure for the proof of Theorem 1.1 in the previous Subsection 4.5.
Acknowledgement
The authors wish to thank the referees for their careful reading of my manuscript and valuable suggestions.
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Funding: The research is supported by the National Natural Science Foundation of China (No.11661006), and the Science and Technology Planning Project of Jiangxi Province, China (No. GJJ190741).
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