Regularity for sub-elliptic systems with VMO-coefficients in the Heisenberg group: the sub-quadratic structure case

Jialin Wang; Maochun Zhu; Shujin Gao; Dongni Liao

doi:10.1515/anona-2020-0145

Article Open Access

Regularity for sub-elliptic systems with VMO-coefficients in the Heisenberg group: the sub-quadratic structure case

Jialin Wang , Maochun Zhu , Shujin Gao and Dongni Liao

Published/Copyright: August 7, 2020

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From the journal Advances in Nonlinear Analysis Volume 10 Issue 1

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.

Keywords: Partial Hölder continuity; Heisenberg group; sub-quadratic controllable growth; sub-quadratic natural growth; VMO-coefficient; p-Laplacian

MSC 2010: 35H20; 35B65; 32A37

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, ⋯ X_2n} with the horizontal vector fields X_i : (i = 1, ⋯, 2n) in ℍⁿ, we consider sub-elliptic systems of the type

−∑i=12nXiAiα(ξ,u,Xu)=Bα(ξ,u,Xu),inΩ,α=1,2,⋯,N, (1.1)

where the primary coefficient Aiα ∈ VMO and satisfies some standard ellipticity and growth conditions with polynomial growth rate p ∈ (1, 2), and the inhomogeneous term B^α conforms to either controllable growth conditions, or natural growth conditions under an additional smallness assumption on the weak solutions. For the precise statement of the assumptions, and more details about the Heisenberg group, we refer to (H1)-(H4)-(HC) and (HN) below, and Section 2, respectively.

The main new aspect of this paper is the fact that we are able to deal with the inhomogeneity B^α : ℝ²ⁿ⁺¹ × ℝ^N × ℝ^2n×N → ℝ^N that satisfies the sub-quadratic controllable growth conditions, as well as sub-quadratic natural growth conditions, respectively, and the primary coefficient Aiα : ℝ²ⁿ⁺¹ × ℝ^N × ℝ^2n×N → ℝ^2n×N that satisfies only a VMO-condition in ξ and is continuous in u. More precisely, we assume that the partial mapping ξ ↦ Aiα (ξ, u, P)/(1 + |P|)^p−1 is VMO uniformly in (u, P), in the sense of (1.5) below and, moreover, u ↦ Aiα (ξ, u, P)/(1 + |P|)^p−1 is continuous in the sense of (1.3) below. Our tool of choice is the use of an appropriate Sobolev-Poincaré inequality, and the harmonic approximation lemma; see Lemma 3.1, Lemma 3.3 below, respectively. The method of proof employed here will avoid the use of L^q − L^p-estimates for the horizontal gradient and reverse Hölder inequalities. Our results essentially extend those results that the coefficients are continuous with respect to variables ξ and u to the case of the coefficients being VMO in the first variable ξ. We point out that partial Hölder continuity is the best one can expect under such weak assumptions concerning regularity of the structural functions Aiα and B^α in the (ξ, u)-variables.

We now impose the precise structure assumptions for coefficients Aiα and B^α we are dealing with.

(H1). The primary coefficient Aiα satisfies following ellipticity and growth conditions for a growth exponent 1 < p < 2:

DPAiα(ξ,u,P)P0,P0≥ν(1+|P|)p−2|P0|2,|Aiα(ξ,u,P)|+(1+|P|)|DPAiα(ξ,u,P)|≤L(1+|P|)p−1, (1.2)

for any choice of ξ ∈ Ω, u, u₀ ∈ ℝ^N and P, P₀ ∈ ℝ^2n×N. Here structure constants ν ≤ 1 ≤ L < ∞.

(H2). The vector field Aiα is continuous with respect to the second variable u. More precisely, there exists a bounded, concave and non-decreasing moduli of continuity ω : [0, ∞ → [0, 1] with lims→0 ω(s) = 0 = ω(0) such that

|Aiα(ξ,u,P)−Aiα(ξ,u0,P)|≤Lω|u−u0|p(1+|P|)p−1,1<p<2. (1.3)

(H3). The vector field Aiα is differentiable in the third variable P with continuous derivatives. This infers the bounded, concave and non-decreasing modulus μ : [0, ∞) → [0, 1] such that μ(t) ≤ t, lims→0 μ(s) = 0 = μ(0), and we have

|DPAiα(ξ,u,P)−DPAiα(ξ,u,P0)|≤Lμ|P−P0|1+|P|+|P0|(1+|P|+|P0|)p−2,1<p<2. (1.4)

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 ↦ Aiα (ξ, u, P)/(1 + |P|)^p−1 satisfies the following VMO-condition uniformly in u and P:

Aiα(ξ,u,P)−Aiα(⋅,u,P)ξ0,r≤vξ0(ξ,r)(1+|P|)p−1,for allξ∈Br(ξ0),1<p<2. (1.5)

where v_ξ₀ : ℝ²ⁿ⁺¹ × [0, ρ₀] → [0, 2L] are bounded functions satisfying

limρ→0V(ρ)=0,whereV(ρ)=supξ0∈Ωsup0<r≤ρ0∫−Br(ξ0)∩Ωvξ0(ξ,r)dξ. (1.6)

Here we have used the short-hand notation

Aiα(⋅,u,P)ξ0,r:=∫−Br(ξ0)∩ΩAiα(ζ,u,P)dζ=Br(ξ0)∩Ω−1∫Br(ξ0)∩ΩAiα(ζ,u,P)dζ.

(HC) (Controllable growth condition). The inhomogeneity B^α satisfies sub-quadratic controllable growth condition

|Bα(ξ,u,P)|≤C1+|u|p∗−1+|P|p(1−1p∗),p∗=pQQ−p,1<p<Q,any constantp∗≥p,p≥Q, (1.7)

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^∗ = pQQ−p in our setting.

(HN) (Natural growth condition). For |u| ≤ M = supΩ |u|. The term B^α satisfies sub-quadratic natural growth condition

|Bα(ξ,u,P)|≤a|P|p+b,1<p<2, (1.8)

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 Aiα with Hölder continuity in the variables (ξ, u) to the assumptions of Dini continuity, and proved partial regularity result with optimal estimates for the modulus of continuity for the horizontal derivative Xu.

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 C_y and a first-order excess functional Ψ. For the case p ≥ 2, the functional Ψ is defined by

Ψ(ξ0,ρ,l)=∫−Bρ(ξ0)u−lρ(1+|Xl|)2+u−lρ(1+|Xl|)pdξ,

with the horizontal affine functions l : ℝ²ⁿ → ℝ^N 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

Ψ(ξ0,ρ,l)=∫−Bρ(ξ0)Vu−lρ2dξ, (1.9)

where V(A)=(1+|A|2)4p−2A for A ∈ ℝ^k, k ∈ ℕ₊. On the other hand, we define the Campanato type excess functional C_y by

Cy(ξ0,ρ)=ρ−py∫−Bρ(ξ0)|u−uξ0,ρ|pdξ,1<p<2,0<y<1,

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 B_ξ₀(ρ) ⊂ ⊂ U, then, for some fixed θ ∈ (0, 1), one obtain an excess improvement Ψ (ξ₀, θ r, l_ξ₀,θr) ≤ C₄θ²Ψ_*(ξ₀, r, l_ξ₀,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 C_y-excess remain bounded. Finally, the boundedness of the C_y-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 HW^1,p with 1 < p < 2. Therefore, we have to establish the following Sobolev-Poincaré inequality with the function V (see Lemma 3.1 below),

∫−Bρ(ξ0)Vu−uξ0,ρρ2QQ−pdξQ−p2Q≤CP∫−Bρ(ξ0)VXu2dξ12,

with the constant C_P 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 Aiα (ξ, u, Xu) and B^α(ξ, u, Xu) satisfy the assumptions (H1)-(H4) and (HC). Let u ∈ HW^1,p(Ω, ℝ^N) with 1 < p < 2 be weak solutions to the systems (1.1), i.e.,

∫ΩAiα(ξ,u,Xu)⋅Xφdξ=∫ΩBα(ξ,u,Xu)⋅φdξ,∀φ∈C0∞(Ω,RN). (1.10)

Then, there exists a relatively closed singular set Ω₀ ⊂ Ω such that u ∈ Cloc0,y (Ω ∖ Ω₀, ℝ^N) for every y ∈ (0, 1). Moreover, for any λ ∈ (0, Q) we have Xu ∈ Llocp,λ (Ω ∖ Ω₀, ℝ^2n×N) with the Morrey parameter λ = Q − p(1 − y). Finally, we have that the singular set satisfies Ω₀ ⊂ Σ₁ ∪ Σ₂, where

Σ1=ξ0∈Ω:limr→0⁡sup(Xu)ξ0,r=∞,Σ2=ξ0∈Ω:limr→0⁡inf∫−Br(ξ0)V(Xu)−V(Xu)ξ0,r2dξ>0

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 Aiα (ξ, u, Xu) and B^α(ξ, u, Xu) satisfy the assumptions (H1)-(H4) and (HN). Let u ∈ HW^1,p(Ω, ℝ^N) ∩ L^∞ (Ω, ℝ^N) be weak solutions to the system (1.1). Then, we have the same results that u ∈ Cloc0,y (Ω ∖ Ω₀, ℝ^N) and Xu ∈ Llocp,λ (Ω ∖ Ω₀, ℝ^2n×N) as Theorem 1.

Remark 1.3

It is worth noting that the choice

Aiα(ξ,u,P)=a(ξ)1+|P|2p−22Piαfori∈{1,⋯,2n},α∈{1,⋯,N}

makes the sub-elliptic p-Laplacian system with VMO-coefficients, involving sub-quadratic growth terms

−∑i=12nXiAiα(ξ)1+Xu2p−22Xiuα=Bα(ξ,u,Xu)

just as a special case of (1.1), where Aiα (ξ) ∈ VMO, and 1 < p < 2. So, combining the result for 2 ≤ p < ∞ established by Wang and Manfredi in [32], our partial Hölder continuity results covers the model case of sub-elliptic p-Laplacian system with 1 < p < ∞. It is remarkable that Zheng and Feng [36] showed everywhere regularity for weak solutions of sub-elliptic p-harmonic systems while p is very closed to 2.

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 ∈ HW^1,1 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 C_y-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:

(ξ¯,t)⋅(η¯,t~)=ξ¯+η¯,t+t~+12∑i=1nxiy~i−x~iyi,

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

Xi≡Xi(ξ)=∂∂xi−yi2∂∂t,Xn+i≡Xn+i(ξ)=∂∂yi+xi2∂∂t,T≡T(ξ)=∂∂t

for i = 1, 2, ⋯, n, and note that the special structure of the commutators:

Xi,Xi+n=−Xi+n,Xi=T,elseXi,Xj=0,andT,T=T,Xi=0,

that is, (ℍⁿ, ⋅) is a nilpotent Lie group of step 2. X = (X₁, ⋯, X_2n) 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

d(ξ~,ξ)=ξ−1⋅ξ~Hn.

The measure used on ℍⁿ is the Haar measure (Lebesgue measure in ℝ²ⁿ⁺¹), and the volume of the homogeneous ball B_R (ξ₀) = {ξ ∈ ℍⁿ : d(ξ₀, ξ) < R} is given by |B_R (ξ₀)|_ℍⁿ = R²ⁿ⁺² |B₁ (ξ₀)|_ℍⁿ =Δ ω_n R^Q, where the number

Q=2n+2 (2.1)

is called the homogeneous dimension of ℍⁿ, and the quantity ω_n is the volume of the homogeneous ball of radius 1.

Let 1 ≤ p ≤ +∞. We denote by

HW1,p(Ω)=u∈Lp(Ω)|Xiu∈Lp(Ω),i=1,⋯,k

the horizontal Sobolev space. Then, HW^1,p(Ω) is a Banach space under the norm

uHW1,p(Ω)=uLp(Ω)+∑i=1kXiuLp(Ω).

For u ∈ HW^1,q(B_R (ξ₀)), 1 < q < Q and 1 ≤ p ≤ qQQ−q , Lu [29] showed the following Poincaré type inequality associated with Hörmander vector fields, which is naturally valid for ℍⁿ:

∫−BR(ξ0)u−uξ0,Rpdξ1p≤CpR∫−BR(ξ0)Xuqdξ1q. (2.2)

The inequality (2.2) is valid for p = q (≥ 2).

Throughout the paper, we shall use the functions V, W: ℝ^k → ℝ^k defined by

V(ς)=(1+ς2)p−24ς,W(ς)=ς/(1+ς2−p)12 (2.3)

for each ς ∈ ℝ^k, k ∈ ℕ and p > 1. The functions V and W are locally bi-Lipschitz bijection on ℝ^k.

The following inequality

1+ς22−p2⩽1+ς2−p⩽2p21+ς22−p2,

immediately yields

W(ς)⩽V(ς)⩽2p4W(ς). (2.4)

The purpose of introducing W is the fact that in contrast to V2m, , the function W2m is convex. In fact, firstly by direct computation yields that W2p(t)=t2p(1+t2−p)−1p is a convex and monotone increasing function on [0, ∞) with W2p (0) = 0; secondly we have

Wς1+ς222p≤Wς1+ς222p≤Wς12p+Wς22p2,ς1,ς2∈Rn.

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 12 in the left of the first inequality (1) below, since the fact that 2^−1/2 < 2^(p−2)/4 for p > 0.

Lemma 2.1

Let p ∈ (1, 2) and V : ℝ^k → ℝ^k be the functions defined in (2.3). Then, for any ς₁, ς₂ ∈ ℝ^k and t > 0, the following inequalities hold:

12minς1,ς1p2≤2(p−2)/4minς1,ς1p2⩽Vς1⩽minς1,ς1p2;
V(tς1)⩽maxt,tp2V(ς1);
|V(ς₁ + ς₂)| ⩽ C(p)(|V(ς₁)| + |V(ς₂)|);
p2ς1−ς2⩽V(ς1)−V(ς2)/(1+ς12+ς22)p−24⩽C(p,k)ς1−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(B_ρ(ξ₀), ℝ^N), ξ₀ ∈ ℝ²ⁿ⁺¹, and consider the horizontal components

ξ¯=(x1,...,xn,y1,...,yn)andξ0¯=(x01,...,x0n,y01,...,y0n).

If the function

lξ0,ρ(ξ¯)=lξ0,ρ(ξ0¯)+Xlξ0,ρ(ξ¯−ξ0¯),

minimizes the functional

l↦∫−Bρ(ξ0)|u−l|2dξ,

among horizontal affine function l : ℝ²ⁿ → ℝ^N, Then, we have

lξ0,ρ(ξ0¯)=uξ0,ρ=∫−Bρ(ξ0)udξ,

and

Xlξ0,ρ=Q−2c0QQ+2ρ2∫−Bρ(ξ0)u⊗(ξ¯−ξ0¯)dξ, (2.5)

where the vector u ⊗ (ξ̄ − ξ0¯ ) has components uα(x1−x01,x2−x02,...,x2n−x02n) with α = 1, 2, …, N, and c₀ is a positive constant defined by

c0=∫0π(sin⁡θ)ndθ∫0π(sin⁡θ)n−1dθ=[(2k−2)!!]2(2k−1)!!(2k−3)!!2π,n=2k−1,[(2k−1)!!]2(2k)!!(2k−2)!!π2,n=2k. (2.6)

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(B_ρ(ξ₀), ℝ^N), θ ∈ (0, 1).We denote by l_ξ₀,ρ and l_ξ₀,θρ, the horizontal affine function defined as above for the radii ρ and θρ. Then, we have

|Xlξ0,ρ−Xlξ0,θρ|p≤Q−2c0QQ+2θρp∫−Bθρ(ξ0)u−lξ0,ρpdξ, (2.7)

and, more generally,

Xlξ0,ρ−Xlp≤Q−2c0QQ+2ρp∫−Bρ(ξ0)u−lpdξ, (2.8)

for every horizontal affine function l : ℝ²ⁿ → ℝ^N.

Proof

By the identity (2.5) and Hölder's inequality, we obtain

Xlξ0,ρ−Xlξ0,θρp=Q−2c0QQ+2θρpp∫−Bθρ(ξ0)u−lξ0,ρ(ξ¯0)−Xlξ0,ρ(ξ¯−ξ¯0)⊗(ξ¯−ξ¯0)dξp≤Q−2c0QQ+2θρ2p∫−Bθρ(ξ0)u−lξ0,ρ(ξ¯0)−Xlξ0,ρ(ξ¯−ξ¯0)pdξ∫−Bθρ(ξ0)ξ¯−ξ¯0pp−1dξp−1≤Q−2c0QQ+2θρp∫−Bθρ(ξ0)u−lξ0,ρpdξ, (2.9)

where we have used the fact that ∫−Bθρ(ξ0)ξ¯−ξ¯0pp−1dξp−1≤(θρ)−p.

For (2.8), it follows

Xlξ0,ρ−Xlp≤Q−2c0QQ+2ρ2p∫−Bρ(ξ0)u−l(ξ¯0)−Xl(ξ¯−ξ¯0)pdξ∫−Bρ(ξ0)ξ¯−ξ¯0pp−1dξp−1=Q−2c0QQ+2ρp∫−Bθρ(ξ0)u−lpdξ.

□

According to the definition of the function l_ξ₀,ρ, the following version of the Poincaré inequality (2.2) is true, that is,

∫−Bρ(ξ0)u−lξ0,ρ(ξ¯)pdξ1p≤Cpρ∫−Bρ(ξ0)Xu−Xlξ0,ρqdξ1q,

where 1 < q < Q, 1 ≤ p ≤ qQQ−q .

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 ∈ HW^1,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 ∈ HW^1,1, 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 ∈ HW^1,p(B_ρ(ξ₀), ℝ^N) with B_ρ (ξ₀) ⊂ Ω, Then, it follows

∫−Bρ(ξ0)Vu−uξ0,ρρ2p∗pdξp2p∗≤CP∫−Bρ(ξ0)VXu2dξ12, (3.1)

with p^∗ = pQQ−p the Sobolev critical exponent of p; here the constant C_P depends only on Q, N and p. In particular, the inequality also holds if we substitute 2 for 2p∗p .

Proof

We introduce the operator of fractional integration on Ω of order 1 as follows

I1(f)(ξ)=∫Ω|f(η)|d(ξ,η)Bξ,d(ξ,η)dη,ξ∈Bρ(ξ0).

Based on Theorem 2.7 in [3] by Capogna, Danielli and Garofalo, we deduce for 1 < p < +∞

∫−Bρ(ξ0)|I1(f)(ξ)|p∗dξ1p∗=Cρ∫−Bρ(ξ0)|f(ξ)|pdξ1p, (3.2)

where we denote by p^∗ = pQQ−p the Sobolev critical exponent of p, and the number Q the homogeneous dimension in ℍⁿ.

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

u(ξ)−uξ0,ρ≤C∫Bcρ(ξ0)|Xu(η)|d(ξ,η)Bξ,d(ξ,η)dη,ξ∈Bρ(ξ0).

Noting that W^2/p(t) is monotone increasing and convex, we apply W^2/p(t) to both sides of the last inequality and have by Jensen’s inequality

W2/pu(ξ)−uξ0,ρρ≤Cρ∫R2n+1W~2/pXu(η)d(ξ,η)Bξ,d(ξ,η)dη,

and

W~Xu(η)=0,η∉Bcρ(ξ0),WXu(η),η∈Bcρ(ξ0).

One can check that W(|Xu(η)|) ∈ L^p(B_ρ (ξ₀)), which implies W͠(|Xu(η)) ∈ L^p(ℝ²ⁿ⁺¹). Then, applying the inequality (3.2) yields

∫Bρ(ξ0)Wu(ξ)−uξ0,ρρ2QQ−pdξQ−ppQ=∫Bρ(ξ0)W2/pu(ξ)−uξ0,ρρpQQ−pdξQ−ppQ≤Cρ∫−Bρ(ξ0)∫R2n+1W~2/pXu(η)d(ξ,η)Bξ,d(ξ,η)dηpQQ−pdξQ−ppQ≤Cρ∫−Bρ(ξ0)I1W2/p(Xu)(ξ)pQQ−pdξQ−ppQ≤C∫−Bρ(ξ0)W2Xu(ξ)dξ1p,

∫−Bρ(ξ0)Wu(ξ)−uξ0,ρ2QQ−pdξQ−p2Q≤C∫−Bρ(ξ0)W2Xu(ξ)dξ12.

We obtain the assertion of the theorem, first for W, and Then, also for V by (2.4). □

Let 𝓐 ∈ Bil(Ω × ℝ^N × ℝ^2n×N, ℝ^2n×N) be a bilinear form with constant tensorial coefficients. We say that a map h ∈ C^∞(B_ρ(ξ₀), ℝ^N) is 𝓐-harmonic if and only if

∫−Bρ(ξ0)A(Xh,Xφ)dξ=0

holds for all testing function φ ∈ C0∞ (B_ρ(ξ₀), ℝ^N).

Shores in [25] showed that weak solutions h ∈ HW^1,2(Ω, ℝ^N) 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 ∈ HW^1,2(Ω, ℝ^N),

supBρ/2(ξ0)Xh2+X2h2≤C0ρ−2∫−Bρ(ξ0)Xh2dξ.

Therefore, we can argue as the proof of Proposition 2.10 in [4] to obtain the same estimate for any 𝓐-harmonic function h ∈ HW^1,1 (Ω, ℝ^N).

Lemma 3.2

Let h ∈ HW^1,1 (Ω, ℝ^N) be weak solutions of the constant coefficient systems. Then, h is smooth and there exists C₀ ≥ 1 such that for any B_ρ (ξ₀) ⊂ Ω

supBρ/2(ξ0)Xh2+X2h2≤C0ρ−2∫−Bρ(ξ0)Xh2dξ. (3.3)

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 ℝ^2n×N with the properties

A(P,P)≥ν|P|2andA(P,P¯)≤L|P||P¯|,P,P¯∈R2n×N.

Furthermore, let w ∈ HW^1,p(B_ρ (ξ₀), ℝ^N) be an approximate 𝓐-harmonic map in the sense that the following estimate holds

∫−Bρ(ξ0)A(Xw,Xφ)dξ≤δysupBρ(ξ0)⁡|Xφ|,∀φ∈C0∞(Bρ(ξ0),RN),

and

∫−Bρ(ξ0)|V(Xw)|2dξ≤y2.

Then, there exists an 𝓐-harmonic map h ∈ C^∞(B_ρ(ξ₀), ℝ^N) which satisfies

∫−Bρ(ξ0)Vw−yhρ2dξ≤y2εand∫−Bρ(ξ0)|V(Xh)|2dξ≤1.

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 ∈ HW^1,p(Ω, ℝ^N) 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 B_r(ξ₀) ⊂ ⊂ Ω, and any horizontal affine functions l : ℝ²ⁿ → ℝ^N with |l(ξ̄₀)| + |Xl| ≤ M₀, we have the estimate

∫−Br2(ξ0)|V(Xu−Xl)|2dξ≤Cc∫−Br(ξ0)Vu−lr2dξ+ω∫−Br(ξ0)(|u−l(ξ0¯)|p)dξ+V(r)+(r2+rp′)∫−Br(ξ0)(|Xu|p+|u|p∗+1)dξp′(p∗)′,

where C_c is a positive constants depending only on Q, p, ν, L, M₀, and the exponent p′ = pp−1 , and (p^∗)′ = p∗p∗−1 with p^∗ = pQQ−p .

Proof

We choose a standard cut-off function ϕ ∈ C0∞ (B_r(ξ₀), [0, 1]) with ϕ ≡ 1 on Br2 (ξ₀) and |Xϕ| ≤ 4r . Then, φ = ϕ²(u − l) can be taken as a testing function for sub-elliptic systems (1.1). Hence, we have

∫−Br(ξ0)Aiα(ξ,u,Xu)ϕ2(Xu−Xl)dξ=−2∫−Br(ξ0)Aiα(ξ,u,Xu)ϕ(u−l)Xϕdξ+∫−Br(ξ0)Bα(ξ,u,Xu)ϕ2(u−l)dξ,

where we have used the fact that Xφ = ϕ²(Xu − Xl) + 2ϕ(u − l)Xϕ.

In view of the identities ∫−Br(ξ0)Aiα(⋅,l(ξ0¯),Xl)ξ0,rXφdξ=0, and

−∫−Br(ξ0)Aiα(ξ,u,Xl)ϕ2(Xu−Xl)dξ=2∫−Br(ξ0)Aiα(ξ,u,Xl)ϕ(u−l)Xϕdξ−∫−Br(ξ0)Aiα(ξ,u,Xl)Xφdξ.

It follows for weak solutions u of systems (1.1) that

I0:=∫−Br(ξ0)[Aiα(ξ,u,Xu)−Aiα(ξ,u,Xl)]ϕ2(Xu−Xl)dξ=2∫−Br(ξ0)[Aiα(ξ,u,Xl)−Aiα(ξ,u,Xu)]ϕ(u−l)Xϕdξ+∫−Br(ξ0)[Aiα(ξ,l(ξ0¯),Xl)−Aiα(ξ,u,Xl)]Xφdξ+∫−Br(ξ0)[(Aiα(⋅,l(ξ0¯),Xl))ξ0,r−Aiα(ξ,l(ξ0¯),Xl)]Xφdξ+∫−Br(ξ0)Bα(ξ,u,Xu)ϕ2(u−l)dξ=:2I1+I2+I3+I4, (4.1)

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 B_r(ξ₀) into four subsets: Ω₁ := B_r(ξ₀) ∩ {|Xu − Xl| ≤ 1} ∩ u−lr≤1 , Ω₂ := B_r(ξ₀) ∩ {|Xu − Xl| ≤ 1} ∩ u−lr≥1 , Ω₃ := B_r(ξ₀) ∩ {|Xu − Xl| ≥ 1} ∩ u−lr≤1 , and Ω₄ := B_r(ξ₀) ∩ {|Xu − Xl| ≥ 1} ∩ u−lr≥1 .

Using the second inequality of (1.2), |Xϕ| ≤ 4r , Young’s inequality, and Lemma 2.1, we derive the following bound for I₁ on the subset Ω₁.

∫−Ω1[Aiα(ξ,u,Xu)−Aiα(ξ,u,Xl)]ϕ(u−l)Xϕdξ≤L∫−Ω1∫01(1+|Xl+s(Xu−Xl)|)p−2ds|Xu−Xl||u−l||Xϕ|ϕdξ≤4L∫−Ω1ϕ|2V(Xu−Xl)|u−lrdξ≤2ε∫−Ω1ϕ2|V(Xu−Xl)|2dξ+32L2ε−1∫−Ω1u−lr2dξ, (4.3)

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

∫−Ω2[Aiα(ξ,u,Xu)−Aiα(ξ,u,Xl)]ϕ(u−l)Xϕdξ≤2p2(p−1)ε∫−Ω2ϕpp−1|V(Xu−Xl)|pp−1dξ+(4L)pε1−p∫−Ω2u−lrpdξ≤2ε∫−Ω2ϕ2|V(Xu−Xl)|2dξ+2(4L)pε1−p∫−Ω2Vu−lr2dξ, (4.4)

where we have used Lemma 2.1, and the inequality |V(Xu−Xl)|pp−1 ≤ |V(Xu − Xl)|² ≤ |Xu − Xl|² ≤ 1 for 1 < p < 2 on the set Ω₂.
By Young’s inequality and Lemma 2.1, it follows that on the subset Ω₃,

∫−Ω3[Aiα(ξ,u,Xu)−Aiα(ξ,u,Xl)]ϕ(u−l)Xϕdξ≤ε∫−Ω3ϕp|Xu−Xl|pdξ+(4L)pp−1ε11−p∫−Ω3u−lr2dξ≤2ε∫−Ω3ϕ2|V(Xu−Xl)|2dξ+2(4L)pp−1ε11−p∫−Ω3Vu−lr2dξ, (4.5)

where we have used the fact that u−lrpp−1≤u−lr2 as pp−1 ≥ 2 and u−lr ≤ 1 on the subset Ω₃.
On the subset Ω₄, it holds that by the assumption |l( ξ0¯ )| + |Xl| ≤ M₀

∫−Ω4[Aiα(ξ,u,Xu)−Aiα(ξ,u,Xl)]ϕ(u−l)Xϕdξ≤22+1pL∫−Ω4ϕV(Xu−Xl)2pu−lrdξ≤∫−Ω4εϕpp−1V(Xu−Xl)2p−1dξ+22p+1Lp∫−Ω4ε1−pu−lrpdξ≤ε∫−Ω4ϕ2|V(Xu−Xl)|2dξ+C(p,L)ε1−p∫−Ω4Vu−lr2dξ, (4.6)

here, we have used the smallness assumption Φ(ξ₀, r, l) := ⨍_{B_r(ξ₀)}|V(Xu − Xl)|² dξ ≤ 1 and ϕpp−1 ≤ ϕ².

From (4.3), (4.4), (4.5) and (4.6), we have the estimate for the term I₁ as follows

I1≤2ε∫−Br(ξ0)ϕ2|V(Xu−Xl)|2dξ+C(p,L,M0)ε11−p∫−Br(ξ0)Vu−lr2dξ, (4.7)

where we have used the inequality ε11−p≥ε−1≥ε1−p for small positive constant ε < 1.

Estimate for I₂. By the first inequality of (1.3), we get

I2≤L∫−Br(ξ0)ω|u−l(ξ0¯)|p(1+|Xl|)p−1|Xφ|dξ≤C(p,L,M0)∫−Br(ξ0)ω|u−l(ξ0¯)|pϕ2|Xu−Xl|dξ+C(p,L,M0)∫−Br(ξ0)ω|u−l(ξ0¯)|pϕ|u−l||Xϕ|dξ=:I21+I22. (4.8)

To estimate the term I₂₁, we divide the domain of integration into two parts Ω₅ := B_r(ξ₀) ∩ |Xu − Xl| ≤ 1 and Ω₆ := B_r(ξ₀) ∩ {|Xu − Xl| > 1}.
1. On the set Ω₅ where |Xu − Xl| ≤ 1, it holds
  
  I21(onΩ5)≤2ε∫−Ω5ϕ4|V(Xu−Xl)|2dξ+C(p,L,M0)ε−1∫−Ω5ω2|u−l(ξ0¯)|pdξ≤2ε∫−Ω5ϕ2|V(Xu−Xl)|2dξ+C(p,L,M0)ε−1ω∫−Ω5|u−l(ξ0¯)|pdξ, (4.9)
  
  where we have used in turn Young’s inequality, ω² ≤ ω, the concavity of ω and Jensen’s inequality.
2. On the part Ω₆ where |Xu − Xl| > 1, we find
  
  I21(onΩ6)≤ε∫−Ω6ϕ2p|Xu−Xl|pdξ+C(p,L,M0)ε11−p∫−Ω6ωpp−1|u−l(ξ0¯)|pdξ≤2ε∫−Ω6ϕ2|V(Xu−Xl)|2dξ+C(p,L,M0)ε11−pω∫−Ω6|u−l(ξ0¯)|pdξ, (4.10)
  
  where we have used the inequality ωpp−1 ≤ ω.
Combining (4.9) with (4.10) leads to

I21≤2ε∫−Br(ξ0)ϕ2|V(Xu−Xl)|2dξ+C(p,L,M0)ε11−pω∫−Br(ξ0)|u−l(ξ0¯)|pdξ, (4.11)

where we have use the fact ε11−p ≥ ε⁻¹ for 0 < ε < 1.

The term I₂₂ can be estimated similarly as I₂₁ above. Here, we split the ball B_r(ξ₀) into two subsets Ω₇ := B_r(ξ₀) ∩ {|u−lr|≤1} and Ω₈ := B_r(ξ₀) ∩ {u−lr|>1} .

On the subset Ω₇, it yields

I22(onΩ7)≤2ε∫−Ω7ϕ2Vu−lr2dξ+C(p,L,M0)ε−1ω∫−Ω7|u−l(ξ0¯)|pdξ. (4.12)
We deduce on Ω₈

I22(onΩ8)≤2ε∫−Ω8ϕpVu−lr2dξ+C(p,L,M0)ε11−pω∫−Ω8|u−l(ξ0¯)|pdξ. (4.13)

From (4.12) and (4.13), it follows

I22≤2ε∫−Br(ξ0)Vu−lr2dξ+C(p,L,M0)ε11−pω∫−Br(ξ0)|u−l(ξ0¯)|pdξ. (4.14)

Joining (4.8), (4.11) and (4.14), we obtain

I2≤2ε∫−Br(ξ0)ϕ2|V(Xu−Xl)|2dξ+2ε∫−Br(ξ0)Vu−lr2dξ+C(p,L,M0)ε11−pω∫−Br(ξ0)|u−l(ξ0¯)|pdξ. (4.15)

We are now in the position to handle the term I₃. By VMO-condition (1.5), the term I₃ can be estimated as follows

I3≤∫−Br(ξ0)vξ0(1+|Xl|)p−1|Xφ|dξ≤C(p,M0)∫−Br(ξ0)vξ0ϕ2|Xu−Xl|dξ+C(p,M0)∫−Br(ξ0)vξ0|u−l|ϕ|Xϕ|dξ=:I31+I32. (4.16)

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 v_ξ₀ ≤ 2L and (1.6) to infer the following estimate

I31(onΩ5)≤∫−Ω5εϕ42V(Xu−Xl)2dξ+C(p,M0)∫−Ω5ε−1vξ02dξ≤2ε∫−Ω5ϕ2|V(Xu−Xl)|2dξ+C(p,L,M0)ε−1V(r). (4.17)
On the part Ω₆ where |Xu − Xl| > 1, we use (1.6) and the fact that vξ0pp−1=vξ0⋅vξ01p−1,vξ0≤2L, to deduce

I31(onΩ6)≤∫−Ω6εϕ2pXu−Xlpdξ+C(p,M0)∫−Ω6ε11−pvξ0p1−pdξ≤2ε∫−Ω6ϕ2|V(Xu−Xl)|2dξ+C(p,L,M0)ε11−pV(r). (4.18)

Using (4.17) and (4.18), we get

I31≤2ε∫−Br(ξ0)ϕ2|V(Xu−Xl)|2dξ+C(p,L,M0)ε11−pV(r). (4.19)

Similarly, the term I₃₂ can be estimated as follows

I32≤2ε∫−Br(ξ0)Vu−lr2dξ+C(p,L,M0)ε11−pV(r). (4.20)

Joining (4.16), (4.19) with (4.20), we have

I3≤2ε∫−Br(ξ0)ϕ2|V(Xu−Xl)|2dξ+2ε∫−Br(ξ0)Vu−lr2dξ+C(p,L,M0)ε11−pV(r). (4.21)

Estimate for I₄. Using Hölder inequality, one has

I4≤C∫−Br(ξ0)Xup+up∗+11−1p∗ϕ2(u−l)dξ≤C∫−Br(ξ0)Xup+up∗+1dξ1−1p∗∫−Br(ξ0)ϕ2(u−l)p∗dξ1p∗. (4.22)

To obtain an appropriate estimate for I₄, we take the domain B_r(ξ₀) into four parts as the same way of I₁.

For the case of Ω₁ = B_r(ξ₀) ∩ {|Xu − Xl| ≤ 1} ∩ u−lr≤1 , by Sobolev type inequality, Hölder’s inequality, Young’s inequality and Lemma 2.1, it follows that

∫−Ω1Xup+up∗+1dξ1−1p∗∫−Ω1ϕ2(u−l)p∗dξ1p∗≤Cp∫−Ω1Xup+up∗+1dξ1−1p∗r∫−Ω1u−lr+ϕ22V(Xu−Xl)pdξ1p≤C(Cp,p,ε)r2∫−Ω1Xup+up∗+1dξ21−1p∗+Cpε∫−Ω1u−lr2+2Cpε∫−Ω1ϕ4V(Xu−Xl)2dξ≤C(Cp,p,ε)r2∫−Ω1Xup+up∗+1dξ21−1p∗+2Cpε∫−Ω1Vu−lr2+2Cpε∫−Ω1ϕ2VXu−Xl2dξ. (4.23)
On the part Ω₂ = B_r(ξ₀) ∩ {|Xu − Xl| ≤ 1} ∩ u−lr≥1 , the following estimate holds

∫−Ω2Xup+up∗+1dξ1−1p∗∫−Ω2ϕ2(u−l)p∗dξ1p∗≤C(Cp,p,ε)rpp−1∫−Ω2Xup+up∗+1dξ1−1p∗pp−1+Cpε∫−Ω2u−lrp+C(Cp,p,ε)r2∫−Ω2Xup+up∗+1dξ21−1p∗+Cpε∫−Ω2ϕ4V(Xu−Xl)2dξ≤C(Cp,p,ε)rpp−1+r2∫−Ω2Xup+up∗+1dξ1−1p∗pp−1+2Cpε∫−Ω2Vu−lr2+2Cpε∫−Ω2ϕ2VXu−Xl2dξ, (4.24)

where we have used the fact that ⨍_{B_r(ξ₀)} (|Xu|^p + |u|^{p^∗} + 1)dξ ≥ 1, and ϕ ≤ 1.
On the part Ω₃ = B_r(ξ₀) ∩ {|Xu − Xl| ≥ 1} ∩ u−lr≤1 , it yields

∫−Ω3Xup+up∗+1dξ1−1p∗∫−Ω3ϕ2(u−l)p∗dξ1p∗≤Cp∫−Ω3Xup+up∗+1dξ1−1p∗r∫−Ω3u−lr+ϕ2Xu−Xlpdξ1p≤C(Cp,p)∫−Ω3Xup+up∗+1dξ1−1p∗r∫−Ω3u−lr2dξ12+r∫−Ω3ϕ2pXu−Xlp1pdξ≤C(Cp,p,ε)rpp−1+r2∫−Ω3Xup+up∗+1dξ1−1p∗pp−1+2Cpε∫−Ω3Vu−lr2+2Cpε∫−Ω3ϕ2VXu−Xl2dξ, (4.25)
For the last case of Ω₄ = B_r(ξ₀) ∩ {|Xu − Xl| ≥ 1} ∩ u−lr≥1 , we get

∫−Ω4Xup+up∗+1dξ1−1p∗∫−Ω4ϕ2(u−l)p∗dξ1p∗≤2p−1pCp∫−Ω4Xup+up∗+1dξ1−1p∗r∫−Ω4u−lrpdξ1p+r∫−Ω4ϕ2pXu−Xlp1pdξ≤C(Cp,p,ε)rpp−1∫−Ω4Xup+up∗+1dξ1−1p∗pp−1+2Cpε∫−Ω4u−lrp+2Cpε∫−Ω4ϕ2pXu−Xlpdξ≤C(Cp,p,ε)rpp−1∫−Ω4Xup+up∗+1dξ1−1p∗pp−1+4Cpε∫−Ω4Vu−lr2+4Cpε∫−Ω4ϕ2VXu−Xl2dξ. (4.26)

Combining the estimates (4.22)-(4.26), we find that

I4≤4Cpε∫−Br(ξ0)ϕ2VXu−Xl2dξ+C(Cp,p,ε)r2+rpp−1∫−Br(ξ0)Xup+up∗+1dξ1−1p∗pp−1+4Cpε∫−Br(ξ0)Vu−lr2. (4.27)

Joining the estimates(4.2), (4.7), (4.15), (4.21), (4.27) with (4.1), we arrive at

Here, choosing ε<3(1+M02)p−226+4Cp, we can absorb the first integral of the right-hand side into the left. Keeping in mind the properties of ϕ, we have thus shown

∫−Br2(ξ0)|V(Xu−Xl)|2dξ≤2Q∫−Br(ξ0)|V(Xu−Xl)|2ϕ2dξ≤Cc∫−Br(ξ0)Vu−lr2dξ+ω∫−Br(ξ0)(|u−l(ξ0¯)|p)dξ+V(r)+(r2+rp′)∫−Br(ξ0)(|Xu|p+|u|p∗+1)dξp′(p∗)′,

with a constant C_c = C_c(Q, p, ν, L, M₀), p′ = pp−1 , and (p^∗)′ = p∗p∗−1 . This proves the claim. □

For sake of simplicity, we motivated the form of the Caccioppoli inequalities in Lemma 4.1. We set

Φ(ξ0,r,l):=∫−Br(ξ0)|V(Xu−Xl)|2dξ,Ψ(ξ0,r,l):=∫−Br(ξ0)Vu−lr2dξ,

Ψ∗(ξ0,r,l):=Ψ(ξ0,r,l)+ω∫−Br(ξ0)|u−l(ξ0¯)|pdξ+V(r)+r2+rp′∫−Br(ξ0)(|Xu|p+|u|p∗+1)dξp′(p∗)′.

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, B_2ρ(ξ₀) ⊆ Ω with ρ ≤ ρ₀ and an arbitrary horizontal function l : ℝ²ⁿ → ℝ^N, we define

A=DPAiα(⋅,l(ξ0¯),Xl)ξ0,ρandw=u−l,

Then, w is approximately 𝓐-harmonic in the sense that

∫−Bρ(ξ0)A(Xw,Xφ)dξ≤C1Ψ∗(2ρ)+μΨ∗12(2ρ)+μΨ∗1p(2ρ)supBρ(ξ0)⁡|Xφ|

for all φ ∈ C0∞ (B_ρ(ξ₀), ℝ^N), and the positive constant C₁ = C(p, M₀, L, C_c).

Proof

Without loss of generality, we assume that supBρ(ξ0)⁡|Xφ|≤1. Noting that w = u − l, we compute

∫−Bρ(ξ0)A(Xw,Xφ)dξ=∫−Bρ(ξ0)∫01DPAiα(⋅,l(ξ0¯),Xl)ξ0,ρ−DPAiα(⋅,l(ξ0¯),Xl+sXw)ξ0,ρXwdsdξsupBρ(ξ0)⁡|Xφ|+∫−Bρ(ξ0)∫01DPAiα(⋅,l(ξ0¯),Xl+sXw)ξ0,ρXwdsdξsupBρ(ξ0)⁡|Xφ|=:(J1+J2)supBρ(ξ0)⁡|Xφ|, (4.28)

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

∫01DPAiα(⋅,l(ξ0¯),Xl)ξ0,ρ−DPAiα(⋅,l(ξ0¯),Xl+sXw)ξ0,ρds=∫01∫−Bρ(ξ0)DPAiα(⋅,l(ξ0¯),Xl)−DPAiα(⋅,l(ξ0¯),Xl+sXw)dξds≤∫01∫−Bρ(ξ0)DPAiα(⋅,l(ξ0¯),Xl)−DPAiα(⋅,l(ξ0¯),Xl+sXw)dξds≤L∫−Bρ(ξ0)μ|Xu−Xl|1+|Xl|(1+2|Xl|)p−2dξ.

By the monotonicity of μ and the inequality above, it yields

J1≤L∫−Bρ(ξ0)μ|Xu−Xl|1+|Xl|(1+2|Xl|)p−2|Xu−Xl|dξ≤C(p,L,M0)∫−Bρ(ξ0)μ(|Xu−Xl|)|Xu−Xl|dξ.

Here, we decompose the ball B_ρ(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

J1(onΩ5)≤C(p,L,M0)∫−Ω5μ|V(Xu−Xl)||V(Xu−Xl)|dξ≤C(p,L,M0)μ2∫−Ω5|V(Xu−Xl)|dξ+∫−Ω5|V(Xu−Xl)|2dξ≤C(p,L,M0)μ∫−Ω5|V(Xu−Xl)|2dξ12+∫−Ω5|V(Xu−Xl)|2dξ,

where we have used the inequality μ² ≤ μ.
On the set Ω₆ where |Xu − Xl| > 1, we have the following bound

J1(onΩ6)≤C(p,L,M0)∫−Ω6μ(|Xu−Xl|)|Xu−Xl|dξ≤C(p,L,M0)μpp−1∫−Ω6|Xu−Xl|dξ+∫−Ω6|Xu−Xl|pdξ≤C(p,L,M0)μ∫−Ω6|Xu−Xl|pdξ1p+∫−Ω6|V(Xu−Xl)|2dξ≤C(p,L,M0)μ∫−Ω6|V(Xu−Xl)|2dξ1p+∫−Ω6|V(Xu−Xl)|2dξ,

where we have used μpp−1 ≤ μ and Lemma 2.1. Then, we get the following estimate

J1≤C(p,L,M0)μ(Φ12(ρ))+μ(Φ1p(ρ))+Φ(ρ). (4.29)

Based on the following facts

∫−Bρ(ξ0)〈Aiα(ξ,u,Xu),Xφ〉dξ−∫−Bρ(ξ0)〈Bα(ξ,u,Xu),φ〉dξ=0and∫−Bρ(ξ0)〈Aiα(⋅,l(ξ0¯),Xl)ξ0,ρ,Xφ〉dξ=0,

the integral J₂ can be rewritten as

J2=∫−Bρ(ξ0)Aiα(⋅,l(ξ0¯),Xu)ξ0,ρ−Aiα(⋅,l(ξ0¯),Xl)ξ0,ρXφdξ=∫−Bρ(ξ0)Aiα(⋅,l(ξ0¯),Xu)ξ0,ρ−Aiα(ξ,l(ξ0¯),Xu)Xφdξ+∫−Bρ(ξ0)Aiα(ξ,l(ξ0¯),Xu)−Aiα(ξ,u,Xu)Xφdξ+∫−Bρ(ξ0)Bα(ξ,u,Xu)φdξ=:J21+J22+J23, (4.30)

with the obvious meaning of J₂₁ + J₂₂ + J₂₃.

Using the assumption of |l( ξ0¯ )| + |Xl| ≤ M₀ and VMO-condition (1.5), We find that

J21≤∫−Bρ(ξ0)vξ0(1+|Xu|)p−1dξ≤∫−Bρ(ξ0)vξ0(1+|Xl|p−1+|Xu−Xl|p−1)dξ≤(1+M0p−1)∫−Bρ(ξ0)vξ0+vξ0|Xu−Xl|p−1dξ,

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

J21(onΩ5)≤(1+M0p−1)∫−Ω5vξ0+vξ0|2V(Xu−Xl)|p−1dξ≤(1+M0p−1)∫−Ω5vξ0dξ+∫−Ω5vξ022−pdξ+2∫−Ω5|V(Xu−Xl)|2dξ≤C(p,L,M0)∫−Ω5vξ0dξ+∫−Ω5|V(Xu−Xl)|2dξ,

where we have used vξ02p−2=vξ0⋅vξ0p2−p,vξ0≤2L and Lemma 2.1.
On the part Ω₆ where |Xu − Xl| > 1, it yields

J21(onΩ6)≤(1+M0)∫−Ω6vξ0dξ+∫−Ω6vξ0pdξ+∫−Ω6|Xu−Xl|pdξ≤C(p,L,M0)∫−Ω6vξ0dξ+∫−Ω6|V(Xu−Xl)|2dξ,

where we have used the assumption v_ξ₀ ≤ 2L and Lemma 2.1.

Then, we get the following estimate for J₂₁

J21≤C(p,L,M0)V(ρ)+Φ(ρ). (4.31)

By first inequality of (1.3), the term J₂₂ can be estimated as follows

J22≤L∫−Bρ(ξ0)ω|u−l(ξ0¯)|p(1+|Xu|)p−1dξ≤L(1+M0p−1)∫−Bρ(ξ0)ω|u−l(ξ0¯)|p+ω|u−l(ξ0¯)|p|Xu−Xl|p−1dξ.

Similarly, for the case of |Xu − Xl| ≤ 1 on Ω₅, applying Young’s inequality, Jensen’s inequality and Lemma 2.1, we deduce that

J22(onΩ5)≤C(L,p,M0)ω∫−Ω5|u−l(ξ0¯)|pdξ+∫−Ω5|V(Xu−Xl)|2dξ,

where we have used ω22−p ≤ ω ≤ 1.

For the other case of |Xu − Xl| > 1 on Ω₆, it follows

J22(onΩ6)≤C(L,p,M0)ω∫−Ω6|u−l(ξ0¯)|pdξ+∫−Ω6|V(Xu−Xl)|2dξ,

where we have used ω^p ≤ ω ≤ 1, Jensen’s inequality and Lemma 2.1.

Thus, we get the following estimate for J₂₂

J22≤C(L,p,M0)ω∫−Bρ(ξ0)|u−l(ξ0¯)|pdξ+Φ(ρ). (4.32)

Finally, we handle the term J₂₃ by the same as the way for I₄ to obtain

J23≤4Cpε∫−Bρ(ξ0)ϕ2VXu−Xl2dξ+4Cpε∫−Bρ(ξ0)Vu−lρ2.+C(Cp,ε)ρ2+ρpp−1∫−Bρ(ξ0)Xup+up∗+1dξ1−1p∗pp−1≤C(Cp,ε)Φ(ρ)+Ψ(ρ)+ρ2+ρpp−1∫−Bρ(ξ0)Xup+up∗+1dξ1−1p∗pp−1 (4.33)

Joining the estimates (4.31)-(4.33) with (4.30), we have

J2≤C(p,L,M0,Cp)Φ(ρ)+Ψ(ρ)+ω∫−Bρ(ξ0)|u−l(ξ0¯)|pdξ+V(ρ)+ρ2+ρpp−1∫−Bρ(ξ0)Xup+up∗+1dξ1−1p∗pp−1=C(p,L,M0,Cp)Φ(ρ)+Ψ∗(ρ). (4.34)

Plugging (4.29) and (4.34) into (4.28), we finally arrive at

∫−Bρ(ξ0)A(Xw,Xφ)dξ≤C(p,L,M0,Cp)μ(Φ12(ρ))+μ(Φ1p(ρ))+Φ(ρ)+Ψ∗(ρ)supBρ(ξ0)⁡|Xφ|≤C(p,L,M0,Cc,Cp)μ(Ψ∗12(2ρ))+μ(Ψ∗1p(2ρ))+Ψ∗(2ρ)supBρ(ξ0)⁡|Xφ|,

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 B_r(ξ₀) ⊆ Ω with r ≤ ρ₀. For the constants δ = δ (Q, N, p, L, ν, θ^Q+4) ∈ (0, 1] and y ∈ (0, 1] from the 𝓐-harmonic approximation Lemma 3.3, we let 0 < θ ≤ 14 be arbitrary and also impose the following smallness conditions:

Ψ∗12(r)<δ2;
y:=Ψ∗(r)+δ2−2μΨ∗12(r)+μΨ∗1p(r)2≤1.

Then, there holds an excess improvement estimate

Ψ(ξ0,θr,lξ0,θr)≤C4θ2Ψ∗(ξ0,r,lξ0,r)

with some constants C₄ that depend only on n, N, p, ν, δ and L. Here, l_ξ₀,θr and l_ξ₀,r denote the minimizing affine functions introduced in Lemma 2.2.

Proof

We denote Ψ_*(r) = Ψ_*(ξ₀, r, l_ξ₀,r), and take

w~=u−lξ0,rC2

with l_ξ₀,r = u_ξ₀,r+Xl_ξ₀,r(ξ̄ − ξ̄₀) and C₂ = max{C₁, Cc }. We claim that w̃ satisfies the assumptions of 𝓐-harmonic approximation Lemma 3.3.

First note that, for our choice of the bilinear form

A=DPAiα(⋅,lξ0,r(ξ0¯),Xlξ0,r)ξ0,ρ.

Next by Lemma 4.2 with ρ = r2 and l = l_ξ₀,r, and the assumptions (i) and (ii), we find the map w̃ is approximately 𝓐-harmonic in the sense that

∫−Br2(ξ0)A(Xw~,Xφ)dξ≤yC1C2Ψ∗(r)+μΨ∗12(r)+μΨ∗1p(r)ysupBr2(ξ0)⁡|Xφ|≤yΨ∗12(r)+δ2supBr2(ξ0)⁡|Xφ|≤yδsupBr2(ξ0)⁡|Xφ| (4.35)

for all φ∈C0∞(Br2(ξ0),RN), and

∫−Br2(ξ0)|V(Xw~)|2dξ=1C22∫−Br2(ξ0)|V(Xu−Xlξ0,r)|2dξ≤Φ(r/2,lξ0,r)C22≤CcC22Ψ∗(r)≤y2. (4.36)

The estimates (4.35) and (5.2) tell us that the conditions of Lemma 3.3 are satisfied. So, there exists an 𝓐-harmonic h∈C0∞(Br2(ξ0),RN) such that

∫−Br2(ξ0)Vw~−yhr2dξ≤y2ε,and∫−Br2(ξ0)|V(Xh)|2dξ≤1.

In order to estimate excess functional

Ψ(ξ0,θr,lξ0,θr)=∫−Bθr(ξ0)Vu−lξ0,θrθr2dξ,

we now have to handle the integral ⨍_{B_θr(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 Br2(ξ0) ⊂ ⊂ Ω, and using Hölder’s inequality, we have the estimate for θ ∈ (0, 14 )

∫−Bθr(ξ0)V(X2h(ξ))2dξ≤supBr4(ξ0)X2h(ξ)2≤C0r−2∫−Br2(ξ0)Xh(ξ)2dξ≤C0Mr−2∫−Br2(ξ0)Xh(ξ)2dξ12+∫−Br2(ξ0)Xh(ξ)pdξ1p≤2C0Mr−2∫−Br2(ξ0)V(Xh(ξ))2dξ12+∫−Br2(ξ0)V(Xh(ξ))2dξ1p≤Cr−2, (4.37)

where we have used the estimate (3.3) and Lemma 2.1.

We write l^h(ξ) = h_ξ₀,θr + (Xh)_ξ₀,θr(ξ̄ − ξ0¯ ). Based on (3.1) and (4.37), it follows that

∫−Bθr(ξ0)Vw~−ylhθr2dξ≤C∫−Bθr(ξ0)Vw~−yhθr2dξ+∫−Bθr(ξ0)yVh−hξ0,θr−(Xh)ξ0,θr(ξ¯−ξ0¯)θr2dξ≤Cθ−2(2θ)−Q∫−Br2(ξ0)Vw~−yhr2dξ+CCPy2∫−Bθr(ξ0)VXh−(Xh)ξ0,θr2dξ≤Cθ−2(2θ)−Qy2ε+CCP2(θr)2y2∫−Bθr(ξ0)V(X2h)2dξ≤C(CP,C0)y2θ−2−Qε+θ2≤C(CP,C0)(1+16δ−2)θ2Ψ∗(r),

where we have taken ε = θ^Q+4.

Scaling back to u, we infer

∫−Bθr(ξ0)Vu−lξ0,r−C2ylhθr2dξ≤C22C(CP,C0)(1+16δ−2)θ2Ψ∗(r).

In view of the defining property of l_ξ₀,θr, we arrive at

∫−Bθr(ξ0)Vu−lξ0,θrθr2dξ≤C22C(CP,C0)(1+16δ−2)θ2Ψ∗(r)≤C4θ2Ψ∗(r),

here, we have denoted C4=C22C(CP,C0)(1+16δ−2). Then, it implies excess improvement estimate

Ψ(ξ0,θr,lξ0,θr)≤C4θ2Ψ∗(ξ0,r,lξ0,r).

□

4.4 Iteration

First, let y ∈ (0, 1) be an fixed Hölder exponent. We define the Campanato-type excess

Cy(ξ0,ρ)=ρ−py∫−Bρ(ξ0)|u−uξ0,ρ|pdξ,1<p<2.

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, 18 ], if

Ψ(ξ0,r,lξ0,r)<ε∗andCy(ξ0,r)<κ∗, (A₀)

for r ∈ (0, ρ_*) with B_r(ξ₀) ⊂ Ω, Then,

Ψ(ξ0,θkr,lξ0,θkr)<ε∗andCy(ξ0,θkr)<κ∗, (A_k)

respectively, for every k ∈ ℕ.

Proof

We begin by choosing the constants. First, we let

θ=minc0Q2p(Q−2)(Q+2)11−y,12C4≤18, (4.38)

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, θ^Q+4) from Lemma 3.3. Next, we fix an ε_* small sufficiently to ensure

ε∗≤minθQ+py82p,δ23δ2+48andμ(ε∗)≤ε∗. (4.39)

Then, we choose κ_* > 0 so small that

ω(κ∗)≤ε∗.

Finally, we fix ρ_* > 0 small enough to guarantee

ρ∗≤minρ0,κ∗1p(1−y),1,V(ρ∗)≤ε∗,andF(ρ∗)≤ε∗, (4.40)

here we have abbreviated F(r)=:(r2+rp′)∫−Br(ξ0)(|Xu|p+|u|p∗+1)dξp′(p∗)′.

Now, we are in the position to prove the assertion (A_k) by induction. We assume that (A_k) is true for up to some k ∈ ℕ. Then, we prove the first part of the assertion (A_k+1), that is, the one concerning Ψ (θ^k+1r, l_{ξ₀,θ^k+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 (A_k) and the choices of ε_*, κ_* and ρ_*, we deduce

Ψ∗(ξ0,θkr,lξ0,θkr)≤Ψ(ξ0,θkr,lξ0,θkr)+ω(Cy(ξ0,θkr))+V(θkr)+F(θkr)≤ε∗+ω(κ∗)+V(ρ∗)+F(ρ∗)≤4ε∗.

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 θ^kr, that is, we have

Ψ∗12(ξ0,θkr,lξ0,θkr)≤2ε∗≤δ2,

where we have used ε∗≤δ216 due to the choice of ε∗≤δ23δ2+48 in (4.39), and

y(θkr):=Ψ∗(θkr)+δ2−2μΨ∗(θkr)+μΨ∗(θkr)p2≤3ε∗+δ2−23ε∗+3ε∗p2≤3ε∗+4δ223ε∗2=ε∗3δ2+48δ2≤1.

Consequently, we apply Lemma 4.3 with the radius θ^k r instead of r, this leads to

Ψ(ξ0,θk+1r,lξ0,θk+1r)≤C4θ2Ψ∗(ξ0,θkr,lξ0,θkr)≤4C4θ2ε∗<ε∗,

by the choice of θ in (4.38). This is the result for the first part of (A_k+1).

Now it remains to show the second part, that is, the one concerning C_y(ξ₀, θ^k+1 r). Since l_ξ₀,θ^kr = u_ξ₀,θ^kr + Xl_ξ₀,θ^kr(ξ̄ − ξ0¯ ), we can estimate

Now we are going to estimate the term ∫−Bθkr(ξ0)u−lξ0,θkrθkrpdξ. Similarly, we divide the domain of integration into two subsets Ω9:=Bθkr(ξ0)∩u−lξ0,θkrθkr>1 and Ω10:=Bθkr(ξ0)∩u−lξ0,θkrθkr≤1.

On the subset Ω₉, we get

∫−Ω9u−lξ0,θkrθkrpdξ≤2∫−Ω9Vu−lξ0,θkrθkr2dξ.

For the case of u−lξ0,θkrθkr≤1 on Ω₁₀, noting the fact of 1 < p < 2, we find

∫−Ω10u−lξ0,θkrθkrpdξ≤∫−Ω10u−lξ0,θkrθkr2dξp2≤2∫−Ω10Vu−lξ0,θkrθkr2dξp2.

Therefore, we deduce the following estimate

∫−Bθkr(ξ0)u−lξ0,θkrθkrpdξ≤2∫−Bθkr(ξ0)Vu−lξ0,θkrθkr2dξp2≤2Ψp2(ξ0,θkr,lξ0,θkr)≤2ε∗p2.

By means of (2.8) with the choice of l ≡ u_ξ₀,θ^kr, and the assumption A_k, we obtain

|Xlξ0,θkr|p≤Q−2c0QQ+2θkrp∫−Bθkr(ξ0)|u−uξ0,θkr|pdξ≤(Q−2)(Q+2)c0Qp(θkr)p(y−1)Cy(ξ0,θkr)≤(Q−2)(Q+2)c0Qp(θkr)p(y−1)κ∗.

Recalling that r ∈ (0, ρ_*), we deduce that

Cy(ξ0,θk+1r≤4ρ∗p(1−y)θ−Q−pyε∗p2+(Q−2)(Q+2)c0Qpθp(1−y)κ∗≤κ∗2+κ∗2≤κ∗,

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 (A_k+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

Ψ(ξ0,ρ,lξ0,ρ)=∫−Bρ(ξ0)Vu−lξ0,ρρ2dξ≤CP2∫−Bρ(ξ0)VXu−(Xu)ξ0,ρ2dξ≤CP2C(p,M)∫−Bρ(ξ0)V(Xu)−V(Xu)ξ0,ρ2dξ. (4.41)

Moreover, for any y ∈ (0, 1) and ρ ≤ 1, the following estimate holds

Cy(ξ0,ρ)=ρ−py∫−Bρ(ξ0)|u−uξ0,ρ|pdξ≤ρp−py∫−Bρ(ξ0)u−lξ0,ρρpdξ.

If u−lξ0,ρρ>1, we have

∫−Bρ(ξ0)u−lξ0,ρρpdξ≤2∫−Bρ(ξ0)Vu−lξ0,ρρ2dξ.

If u−lξ0,ρρ≤1, we obtain u−lξ0,ρρp≤u−lξ0,ρρ2+1. Then, it implies

∫−Bρ(ξ0)u−lξ0,ρρpdξ≤2∫−Bρ(ξ0)Vu−lξ0,ρρ2dξ+1.

So, it yields

Cy(ξ0,ρ)≤2ρp−py∫−Bρ(ξ0)Vu−lξ0,ρρ2dξ+ρp−py≤ρp−py2Ψ(ξ0,ρ,lξ0,ρ)+1. (4.42)

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

Ψ(ξ0,ρ,lξ0,ρ)<ε∗andCy(ξ0,ρ)<κ∗.

Using the absolute continuity of the integral, there exists a neighborhood U ⊆ Ω of ξ₀ with

Ψ(ξ,ρ,lξ0,ρ)<ε∗andCy(ξ,ρ)<κ∗,∀ξ∈U.

Applying Lemma 4.4 in any point ξ ∈ U, Then, we get

Ψ(ξ,θkρ,lξ0,θkρ)<ε∗andCy(ξ,θkρ)<κ∗,∀ξ∈U,k∈N. (4.43)

This together with Campanato’s characterization of Hölder continuous functions imply that

supξ∈U,σ∈(0,ρ)⁡σ−py∫−Bσ(ξ)|u−uξ,σ|pdη=supξ∈U,σ∈(0,ρ)⁡Cy(ξ,σ)<κ∗<∞.

Hence u ∈ Cloc0,y (U, ℝ^N).

Furthermore, it holds for |Xu − Xl_ξ,σ| > 1

∫−Bσ(ξ)|Xu−Xlξ,σ|pdξ≤2∫−Bσ(ξ)|V(Xu−Xlξ,σ)|2dξ, (4.44)

and we have if |Xu − Xl_ξ,σ| ≤ 1

∫−Bσ(ξ)|Xu−Xlξ,σ|pdξ≤2∫−Bσ(ξ)|V(Xu−Xlξ,σ)|2dξ+1. (4.45)

Combining (4.44) and (4.45) with (4.43) and (1.6), we get for y ∈ (0, 1)

supξ∈U,σ∈(0,ρ)⁡σp(1−y)∫−Bσ(ξ)|Xu−Xlξ,σ|pdξ≤supξ∈U,σ∈(0,ρ)⁡σp(1−y)2∫−Bσ(ξ)|V(Xu−Xlξ,σ)|2dξ+1≤supξ∈U,σ∈(0,ρ)⁡σp(1−y)2CcΨ(ξ,σ,lξ,σ)+ω(Cy(ξ,σ))+V(σ)+F(σ)+1≤∞,

with abbreviation of F(r)=:(r2+rp′)∫−Br(ξ0)(|Xu|p+|u|p∗+1)dξp′(p∗)′.

In view of the well known equivalence of Campanato and Morrey spaces for parameters λ ∈ (0, Q), it yields Xu ∈ L^p,λ(U, ℝ^2n×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 = supΩ |u| satisfies the smallness assumption

2a(M)(M+M0)3(1+M02)2−p2<ν

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

(a+b)p≤(1+ε)ap+Kbp. (5.1)

Lemma 5.2

(Caccioppoli-type inequality). Let u ∈ HW^1,p(Ω, ℝ^N) ∩ L^∞ (Ω, ℝ^N) be weak solutions of the systems (1.1) under the assumptions (H1)-(H4)-(HN) with ν > 2a(M)(M + M₀) 3(1+M02)2−p2. Then, for any ξ₀ = (x¹, ⋯, xⁿ, y¹, ⋯, yⁿ, t) ∈ Ω and r ≤ 1 with B_r(ξ₀) ⊂ ⊂ Ω, and any horizontal affine functions l : ℝ²ⁿ → ℝ^N with |l( ξ0¯ )| + |Xl| ≤ M₀, we have the estimate

∫−Br2(ξ0)|V(Xu−Xl)|2dξ≤Cc∫−Br(ξ0)Vu−lr2dξ+ω∫−Br(ξ0)(|u−l(ξ0¯)|p)dξ+V(r)+r2+rp′,

where C_c is some positive constants depending only on Q, N, p, a, b, L, ν, M, M₀, and the exponent p′ = pp−1 .

Proof

Let φ = ϕ²(u − l) be a testing function for sub-elliptic systems (1.1), where the standard cut-off function ϕ ∈ C0∞ (B_r(ξ₀), [0, 1]) with ϕ ≡ 1 on Br2(ξ0) and |Xϕ| ≤ 4r . By the same way as the case of controllable growth, we have for weak solutions u of the systems (1.1)

I0′:=∫−Br(ξ0)[Aiα(ξ,u,Xu)−Aiα(ξ,u,Xl)]ϕ2(Xu−Xl)dξ=2∫−Br(ξ0)[Aiα(ξ,u,Xl)−Aiα(ξ,u,Xu)]ϕ(u−l)Xϕdξ+∫−Br(ξ0)[Aiα(ξ,l(ξ0¯),Xl)−Aiα(ξ,u,Xl)]Xφdξ+∫−Br(ξ0)[(Aiα(⋅,l(ξ0¯),Xl))ξ0,r−Aiα(ξ,l(ξ0¯),Xl)]Xφdξ+∫−Br(ξ0)Bα(ξ,u,Xu)ϕ2(u−l)dξ=:2I1′+I2′+I3′+I4′, (5.2)

with the obvious meaning for I0′−I4′ .

With respect to the terms I0′−I3′ , here, we choose the same estimates as (4.2), (4.7), (4.15) and (4.21), that is,

I0′≥ν3(1+M02)p−22∫−Br(ξ0)ϕ2|V(Xu−Xl)|2dξ, (5.3)

I1′≤2ε∫−Br(ξ0)ϕ2|V(Xu−Xl)|2dξ+C(p,L,M0)ε11−p∫−Br(ξ0)Vu−lr2dξ, (5.4)

I2′≤2ε∫−Br(ξ0)ϕ2|V(Xu−Xl)|2dξ+2ε∫−Br(ξ0)Vu−lr2dξ+C(p,L,M0)ε11−pω∫−Br(ξ0)|u−l(ξ0¯)|pdξ, (5.5)

I3′≤2ε∫−Br(ξ0)ϕ2|V(Xu−Xl)|2dξ+2ε∫−Br(ξ0)Vu−lr2dξ+C(p,L,M0)ε11−pV(r). (5.6)

Now we are in the position to get an appropriate bound for the term I4′ . By (H4), elementary inequality (5.1) and Young’s inequality, it yields

I4′≤∫−Br(ξ0)(a|Xu|p+b)ϕ2|u−l|dξ≤a∫−Br(ξ0)|Xu−Xl|+|Xl|pϕ2|u−l|dξ+b∫−Br(ξ0)ϕ2|u−l|dξ≤a∫−Br(ξ0)(1+ϵ)|Xu−Xl|p+1+K|Xl|pϕ2|u−l|dξ+b∫−Br(ξ0)rϕ2u−lrdξ. (5.7)

We denote by I41′ the first term of the right-hand side of (5.7). If |Xu − Xl| ≥ 1, the following estimate holds

I41′≤2a(M0+M)(1+ϵ)∫−Br(ξ0)|V(Xu−Xl)|2ϕ2dξ+a1+K|Xl|p∫−Br(ξ0)ϕru−lrdξ.

I41′≤2a(M0+M)(1+ϵ)∫−Br(ξ0)|V(Xu−Xl)|2ϕ2dξ+a1+K(1+|Xl|p)∫−Br(ξ0)ϕru−lrdξ.

Combining these estimates above, we have

I4′≤2a(M0+M)(1+ϵ)∫−Br(ξ0)|V(Xu−Xl)|2ϕ2dξ+a1+K(1+|Xl|p)+b∫−Br(ξ0)ϕru−lrdξ. (5.8)

We denote by I₄₂ the second term of the right-hand side of (5.8). If u−lr≥1, it leads to

I42′≤a1+K(1+|Xl|p)+bp∫−Br(ξ0)ϕpu−lrpdξ+rp′≤a1+K(1+|Xl|p)+bp∫−Br(ξ0)ϕpVu−lr2dξ+rp′.

If u−lr≤1, it yields

I42′≤a1+K(1+|Xl|p)+b2∫−Br(ξ0)ϕ2u−lr2dξ+r2≤a1+K(1+|Xl|p)+b2∫−Br(ξ0)ϕ2Vu−lr2dξ+r2.

So, we finally arrive at

I4′≤2a(M0+M)(1+ϵ)∫−Br(ξ0)|V(Xu−Xl)|2ϕ2dξ+a1+K(1+M0p)+b2∫−Br(ξ0)Vu−lr2dξ+r2+rp′. (5.9)

Combining (5.3)-(5.6), (5.9) and (5.2), we have

ν3(1+M02)p−22−6ε−2a(M0+M)(1+ϵ)∫−Br(ξ0)|V(Xu−Xl)|2ϕ2dξ≤C(p,L,M0)ε11−p+4ε+a1+K(1+M0p)+b2∫−Br(ξ0)Vu−lr2dξ+C(p,L,M0)ε11−pω∫−Br(ξ0)|u−l(ξ0¯)|pdξ+C(p,L,M0)ε11−pV(r)+r2+rp′. (5.10)

Noting that the smallness condition 2a(M + M₀) 3(1+M02)2−p2 < ν, we fix the constant ε > 0 small sufficiently such that the coefficient ν3(1+M02)p−22−6ε−2a(M+M0)(1+ϵ)>0. Dividing the inequality (5.10) by the positive constant, finally we deduce

∫−Br2(ξ0)|V(Xu−Xl)|2dξ≤Cc∫−Br(ξ0)Vu−lr2dξ+ω∫−Br(ξ0)|u−l(ξ0¯)|pdξ+V(r)+r2+rp′,

where C_c = 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

Φ¯(ξ0,r,l):=∫−Br(ξ0)|V(Xu−Xl)|2dξ,Ψ¯(ξ0,r,l):=∫−Br(ξ0)Vu−lr2dξ,Ψ¯∗(ξ0,r,l):=Ψ(ξ0,r,l)+ω∫−Br(ξ0)|u−l(ξ0¯)|pdξ+V(r)+r2+rp′.

Lemma 5.3

Under the assumptions of Theorem 1.1 are satisfied, B_2ρ(ξ₀) ⊆ Ω with ρ ≤ ρ₀ and an arbitrary horizontal function l : ℝ²ⁿ → ℝ^N, we define

A=DPAiα(⋅,l(ξ0¯),Xl)ξ0,ρandw=u−l,

then, w is approximately 𝓐-harmonic in the sense that

∫−Bρ(ξ0)A(Xw,Xφ)dξ≤C1′Ψ¯∗(2ρ)+μΨ¯∗12(2ρ)+μΨ¯∗1p(2ρ)supBρ(ξ0)⁡|Xφ|

for all φ ∈ C0∞ (B_ρ(ξ₀), ℝ^N), and the positive constant C1′ = C(p, a, b, M₀, L, C_c).

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,

J23≤2a(M0+M)(1+ϵ)∫−Bρ(ξ0)|V(Xu−Xl)|2ϕ2dξ+a1+K(1+M0p)+b2∫−Bρ(ξ0)Vu−lρ2dξ+ρ2+ρp′≤C(p,a,b,M0,M)Φ¯(ρ)+Ψ¯(ρ)+ρ2+ρp′≤C(p,a,b,M0,M,Cc)Φ¯(ρ)+Ψ¯∗(ρ),

where we have used the bound for the natural growth term I4′ in (5.9). The rest procedure is very similar as the proof in Lemma 4.2, and we omit them. So we obtain the claim. □

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 B_r(ξ₀) ⊆ Ω with r ≤ ρ₀. For the constants δ = δ (Q, N, p, L, ν, θ^Q+4) ∈ (0, 1] and y ∈ (0, 1] from the 𝓐-harmonic approximation Lemma 3.3, we let 0 < θ ≤ 14 be arbitrary and also impose the following smallness conditions:

Ψ¯∗12(r)<δ2;
y:=Ψ¯∗(r)+δ2−2μΨ¯∗12(r)+μΨ¯∗1p(r)2≤1.

Then, the following excess improvement estimate holds

Ψ¯(ξ0,θr,lξ0,θr)≤C4′θ2Ψ¯∗(ξ0,r,lξ0,r),

where constants C4′ depend only on Q, N, p, a, b, ν, δ and L.

By Lemma 5.4, the iteration for the Ψ-excess and the C_y-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, 18 ], if

Ψ¯(ξ0,r,lξ0,r)<ε∗andCy(ξ0,r)<κ∗, (A₀)

for r ∈ (0, ρ_*) with B_r(ξ₀) ⊂ Ω, then,

Ψ¯(ξ0,θkr,lξ0,θkr)<ε∗andCy(ξ0,θkr)<κ∗, (A_k)

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.

Tel.:+86 0797 8393663; fax: +86 0797 8395933

Acknowledgement

The authors wish to thank the referees for their careful reading of my manuscript and valuable suggestions.

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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Received: 2020-03-19

Accepted: 2020-06-29

Published Online: 2020-08-07

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/anona-2020-0145

Keywords for this article

Partial Hölder continuity; Heisenberg group; sub-quadratic controllable growth; sub-quadratic natural growth; VMO-coefficient; p-Laplacian

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Regularity for sub-elliptic systems with VMO-coefficients in the Heisenberg group: the sub-quadratic structure case

Article

Abstract

1 Introduction and statements of main results

Theorem 1.1

Theorem 1.2

Remark 1.3

2 Preliminaries

2.1 Introduction of the Heisenberg group ℍn

Lemma 2.1

2.2 Horizontal affine function and estimates in ℍn

Lemma 2.2

Proof

3 Sobolev-Poincaré type inequality and 𝓐-harmonic approximation

Lemma 3.1

Proof

Lemma 3.2

Lemma 3.3

4 Partial Hölder continuity for sub-quadratic controllable growth

4.1 Caccioppoli-type inequality

Lemma 4.1

Proof

4.2 Approximate 𝓐-harmonicity of weak solutions

Lemma 4.2

Proof

4.3 Excess improvement

Lemma 4.3

Proof

4.4 Iteration

Lemma 4.4

Proof

4.5 Proof of Theorem 1

Proof

5 Partial Hölder continuity for sub-quadratic Natural growth

Lemma 5.1

Lemma 5.2

Proof

Lemma 5.3

Proof

Lemma 5.4

Lemma 5.5

Proof of Theorem 1.2

Acknowledgement

References

Articles in the same Issue

Articles in the same Issue

Articles in the same Issue

2.1 Introduction of the Heisenberg group ℍⁿ

2.2 Horizontal affine function and estimates in ℍⁿ