Partial Hölder continuity for nonlinear sub-elliptic systems with VMO-coefficients in the Heisenberg group

Jialin Wang; Juan J. Manfredi

doi:10.1515/anona-2015-0182

Artikel Open Access

Partial Hölder continuity for nonlinear sub-elliptic systems with VMO-coefficients in the Heisenberg group

Jialin Wang und Juan J. Manfredi

Veröffentlicht/Copyright: 2. Juli 2016

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Aus der Zeitschrift Advances in Nonlinear Analysis Band 7 Heft 1

Abstract

We consider nonlinear sub-elliptic systems with VMO-coefficients in the Heisenberg group and prove partial Hölder continuity results for weak solutions using a generalization of the technique of 𝒜-harmonic approximation. The model case is the following non-degenerate p-sub-Laplace system with super-quadratic natural growth with respect to the horizontal gradients Xu:

-∑i=12⁢nXi⁢(a⁢(ξ)⁢(1+|X⁢u|2)(p-2)/2⁢Xi⁢uα)=fα,α=1,2,…,N,

where a⁢(ξ)∈VMO and 2<p<∞.

Keywords: Partial Hölder continuity; Heisenberg group; nonlinear sub-elliptic system; VMO-coefficient; super-quadratic natural growth

MSC 2010: 35H20; 35B65

1 Introduction and statements of main results

The aim of this paper is to prove partial Hölder continuity of weak solutions for nonlinear sub-elliptic systems in divergence form with coefficients that belong to the space of functions with vanishing mean oscillation (VMO) in the Heisenberg group ℍn. The primary example covered by our analysis is the non-degenerate p-sub-Laplacian system

(1.1)-∑i=12⁢nXi⁢(a⁢(ξ)⁢(1+|X⁢u|2)(p-2)/2⁢Xi⁢uα)=fα,α=1,2,…,N,

with a⁢(ξ)∈VMO and 2<p<∞. Here, we denote by ξ=(x1,x2,…,xn,y1,y2,…,yn,t) points in ℍn, and we use the following notation:

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

and

X⁢u=[X1⁢u1X1⁢u2⋯X1⁢uNX2⁢u1X2⁢u2⋯X2⁢uN⋮⋮⋱⋮X2⁢n⁢u1X2⁢n⁢u2⋯X2⁢n⁢uN]2⁢n×N.

In Section 2 we will present additional details about the Heisenberg group.

More generally, we shall consider the following nonlinear sub-elliptic system in divergence form:

(1.2)-∑i=12⁢nXi⁢Aiα⁢(ξ,u,X⁢u)=fα⁢(ξ,u,X⁢u) in ⁢Ω,α=1,2,…,N,

where Ω is a bounded domain in ℍn,

A:=(Aiα):Ω×ℝN×ℝ2⁢n×N→ℝ2⁢n×N and f:=(fα):Ω×ℝN×ℝ2⁢n×N→ℝN.

When the coefficients Aiα is Hölder continuous or Dini continuous, partial regularity results for nonlinear sub-elliptic systems have been established by several authors; see, for example, [4, 17, 27, 29, 28]. We focus our attention on weakening the assumptions on Aiα to be discontinuous, i.e., the partial mapping ξ→(1+|P|)-(p-1)⁢Aiα⁢(ξ,u,P) is only VMO in ξ uniformly in (u,P), in the sense of (1.5) below and, moreover, u→(1+|P|)-(p-1)⁢Aiα⁢(ξ,u,P) is continuous in the sense of (1.3) below. The method of proof employed here will avoid the use of Lq, Lp-estimates for the horizontal gradient and reverse Hölder inequalities. Our tool of choice is the use of the harmonic approximation lemma, see Lemma 4.1 below. More precisely, we assume the following super-quadratic structural conditions on Aiα and fα:

The term Aiα⁢(ξ,u,P) is differentiable in P, and there exists a constant L such that
|DP⁢Aiα⁢(ξ,u,P)|≤L⁢(1+|P|2)(p-2)/2,(ξ,u,P)∈Ω×ℝN×ℝ2⁢n×N,p>2,
The term Aiα⁢(ξ,u,P) is uniformly elliptic, that is, for some λ>0 we have
∑i,j=12⁢n∑α,β=1NDP⁢Aiα⁢(ξ,u,P)⁢ηiα⁢ηjβ≥λ⁢(1+|P|)p-2⁢|η|2 for all ⁢η∈ℝ2⁢n×N.
The term Aiα⁢(ξ,u,P) is continuous with respect to the second variable u, while DP⁢Aiα⁢(ξ,u,P) is continuous with respect to P. More precisely, there exist two bounded, concave and non-decreasing moduli of continuity ω,μ:[0,∞)→[0,1], with lims→0⁡ω⁢(s)=0=ω⁢(0) and lims→0⁡μ⁢(s)=0=μ⁢(0), such that
(1.3)|Aiα⁢(ξ,u,P)-Aiα⁢(ξ,u0,P)|≤L⁢ω⁢(|u-u0|2)⁢(1+|P|)p-1,
(1.4)|DP⁢Aiα⁢(ξ,u,P)-DP⁢Aiα⁢(ξ,u,P0)|≤L⁢μ⁢(|P-P0|1+|P|+|P0|)⁢(1+|P|+|P0|)p-2,
whenever ξ∈Ω,u,u0∈ℝN and P,P0∈ℝ2⁢n×N.
The mapping ξ↦Aiα⁢(ξ,u,P)/(1+|P|)p-1 satisfies the following VMO-condition uniformly in u and P:
(1.5)|Aiα⁢(ξ,u,P)-(Aiα⁢(⋅,u,P))ξ0,r|≤𝐯ξ0⁢(ξ,r)⁢(1+|P|)p-1 for all ⁢ξ∈Br⁢(ξ0),
whenever ξ0∈Ω,r∈(0,ρ0],u∈ℝN and P∈ℝ2⁢n×N, where ρ0>0 and 𝐯ξ0:ℝ2⁢n+1×[0,ρ0]→[0,2⁢L] are bounded functions satisfying
limρ→0⁡𝐕⁢(ρ)=0,where 𝐕⁢(ρ)=supξ0∈Ω⁡sup0<r≤ρ⁡⨍Br⁢(ξ0)∩Ω𝐯ξ0⁢(ξ,r)⁢𝑑ξ.
Here, we have used the notation
(Aiα⁢(⋅,u,P))ξ0,r:=⨍Br⁢(ξ0)∩ΩAiα⁢(ζ,u,P)⁢𝑑ζ.
The functions fα satisfy the super-quadratic natural growth condition
(1.6)|fα⁢(ξ,u,P)|≤a⁢|P|p+b for |u|≤M,
where the positive constants a=a⁢(M) and b=b⁢(M) depend only on M.

From (H2), we have the following estimate:

(1.7)(Aiα⁢(ξ,u,P)-Aiα⁢(ξ,u,P0))⁢(P-P0)≥λ0⁢((1+|P0|)p-2⁢|P-P0|2+|P-P0|p),

with a positive constant λ0. Indeed, it is easy to verify, by the ellipticity assumption (H2), that

(Aiα⁢(ξ,u,P)-Aiα⁢(ξ,u,P0))⁢(P-P0)=(∑i=12⁢n∑α=1N∫01dd⁢t⁢Aiα⁢(ξ,u,t⁢P+(1-t)⁢P0)⁢𝑑t)⁢(P-P0)iα

=∑i,j=12⁢n∑α,β=1N∫01∂∂⁡Pjβ⁢Aiα⁢(ξ,u,t⁢P+(1-t)⁢P0)⁢(P-P0)iα⁢(P-P0)jβ⁢𝑑t

≥λ⁢∫01(1+|t⁢P+(1-t)⁢P0|2)(p-2)/2⁢|P-P0|2⁢𝑑t.

Then, using the inequality (see [20, Lemma 2.1])

2-2⁢(1+δ)≤∫01(1+|t⁢P+(1-t)⁢P0|2)δ/2⁢𝑑t(1+|P|2+|P0|2)δ/2≤2δ/2 for all ⁢δ≥0,

we obtain (1.7). Clearly, the choice

Aiα⁢(ξ,u,P)=a⁢(ξ)⁢(1+|P|2)(p-2)/2⁢Piα for ⁢i=1,…,2⁢n,α=1,…,N,

makes system (1.1) fall into the class considered in (1.2), where a⁢(ξ)∈VMO.

As is well known, we can not expect the weak solutions of the nonlinear system (1.2) to be classical, i.e., C2-solutions, even under reasonable assumptions on the coefficients Aiα and fα. This was first shown by De Giorgi [7]; we also refer the reader to [19, 6] for further discussion and additional examples. Then the goal is to establish partial regularity of solutions. The method of 𝒜-harmonic approximation was extended to include systems of p-Laplacian type by Duzaar and Mingione [15, 14], see also [16, 13]. Similar results have been proved under more general assumptions for Aiα and fα in the Euclidean setting; see [5] for Hölder continuous coefficients, [11, 12, 25] for Dini continuous coefficients and [1, 21, 26, 31] for VMO coefficients.

With respect to sub-elliptic equations and systems in the Heisenberg and general Carnot groups, new difficulties arise due to non-commutativity of the horizontal vectors Xi. For the case of sub-elliptic equations, we refer readers to [10, 8, 9, 2, 3, 23] and to [24] for more known results and details. For additional results focused on sub-elliptic systems, see [30, 4, 27] for the quadratic growth case, and [17, 29, 28] for non-quadratic cases. We note that the assumptions of continuity on the coefficients Aiα are required for those regularity results of sub-elliptic systems mentioned above. We also note that the quadratic growth case p=2 with VMO coefficients has been settled by Gao, Niu and Wang [18], who proved partial Hölder continuity of solutions. Recently, Zheng and Feng [32] considered the degenerate p-sub-Laplace system, and establish local Hölder continuity of weak solutions when p is near 2.

Under the previous assumptions(H1)–(H5), we adapt the method of 𝒜-harmonic approximation to the sub-elliptic systems (1.2) in the Heisenberg group, and establish partial Hölder continuity for weak solutions.

Theorem 1.1.

Assume that the coefficients Aiα and fα satisfy assumptions (H1)–(H5) with λ0>2⁢a⁢(M)⁢M, where λ0 is defined in (1.7) and a⁢(M) is defined in (H5). Let u∈HW1,p⁢(Ω,RN)∩L∞⁢(Ω,RN) be a weak solution to system (1.2), i.e.,

∫ΩAiα⁢(ξ,u,X⁢u)⁢Xi⁢φα⁢𝑑ξ=∫Ωfα⁢(ξ,u,X⁢u)⁢φα⁢𝑑ξ for all ⁢φ∈C0∞⁢(Ω,ℝN).

Then there exists a relatively closed set Sing⁡u⊂Ω such that u∈C0,γ⁢(Ω∖Sing⁡u,RN) for every γ∈(0,1). Furthermore, Sing⁡u⊂Σ1∪Σ2, where

Σ1={ξ0∈Ω:supr>0(|uξ0,r|+|(Xu)ξ0,r|)=∞},Σ2={ξ0∈Ω:limr→0+inf⨍Br⁢(ξ0)|Xu-(Xu)ξ0,r|pdξ>0}.

Thus, the singular set has (2⁢n+1)-Lebesgue measure zero, i.e., |Singu|=0, and its complement Ω∖Sing⁡u is a set of full measure in Ω.

Note that the Haar measure in 𝐇n is just the Lebesgue measure in ℝ2⁢n+1. The assumption 2⁢a⁢(M)⁢M<λ0 is necessary for our proof of Theorem 1.1. Whether we may assume a⁢(M)⁢M<λ0 as a replacement is still an open issue. However, the following result shows that we can avoid the condition 2⁢a⁢(M)⁢M<λ0 by slightly strengthening (1.6).

Theorem 1.2.

If we substitute condition (1.6) with |fα⁢(ξ,u,P)|≤a⁢|P|p-ϵ+b, with ϵ>0, then the conclusion of Theorem 1.1 is still valid without the requirement 2⁢a⁢(M)⁢M<λ0.

We follow the strategy used by Bögelein, Duzaar, Habermann and Scheven in the Euclidean case in [1], with the necessary modification to handle the subelliptic case. First, we define a class of horizontal affine functions, and compute the minimizer of the functional via explicit polar coordinates formulas (see Lemma 3.1 below), and establish some estimates for affine functions in Heisenberg groups. The method of using Taylor’s expansion in [11] cannot be easily adapted to our situation. Instead, we choose different auxiliary functions, and use a Poincaré type inequality from [22] to obtain the desired excess improvement estimates. Our partial Hölder continuity results are valid for the full range 2<p<∞.

2 Preliminaries

The Heisenberg group ℍn is defined as ℝ2⁢n+1 endowed with the following group multiplication:

((ξ1,t),(ξ~1,t~))↦(ξ1+ξ~1,t+t~+12⁢∑i=1n(xi⁢y~i-x~i⁢yi)),

for all

ξ=(ξ1,t)=(x1,x2,…,xn,y1,y2,…,yn,t),ξ~=(ξ~1,t~)=(x~1,x~2,…,x~n,y~1,y~2,…,y~n,t~).

Its neutral element is 0, and its inverse to (ξ1,t) is given by (-ξ1,-t). Particularly, the mapping (ξ,ξ~)↦ξ⋅ξ~-1 is smooth, so (ℍn,⋅) is a Lie group.

The basic vector fields corresponding to its Lie algebra can be explicitly calculated, and are given by

(2.1)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 the special structure of the commutators:

[Xi,Xi+n]=-[Xi+n,Xi]=T,else [Xi,Xj]=0 and [T,T]=[T,Xi]=0,

that is, (ℍn,⋅) is a nilpotent Lie group of step 2. The vector X=(X1,…,X2⁢n) is called horizontal gradient and T vertical derivative.

The homogeneous norm is defined by

∥(ξ1,t)∥=(|ξ1|4+16⁢t2)1/4,

and the metric induced by this homogeneous norm is given by

d⁢(ξ~,ξ)=∥ξ-1⋅ξ~∥.

The measure used on ℍn is the Haar measure (Lebesgue measure in ℝ2⁢n+1), and the volume of the homogeneous ball BR⁢(ξ0)={ξ∈ℍn:d⁢(ξ0,ξ)<R} is given by

|BR⁢(ξ0)|ℍn=R2⁢n+2⁢|B1⁢(ξ0)|ℍn=Δωn⁢RQ,

where the number Q=2⁢n+2 is called the homogeneous dimension of ℍn, and the quantity ωn is the volume of the homogeneous ball of radius 1.

The horizontal Sobolev spaces HW1,p⁢(Ω) (1≤p<∞) are defined as follows:

HW1,p⁢(Ω)={u∈Lp⁢(Ω):Xi⁢u∈Lp⁢(Ω),i=1,2,…,2⁢n}.

Then HW1,p⁢(Ω) is a Banach space with the norm

∥u∥HW1,p⁢(Ω)=∥u∥Lp⁢(Ω)+∑i=12⁢n∥Xiu∥Lp⁢(Ω).

Lu [22] showed the following Poincaré type inequality related to Hörmander’s vector fields for u∈HW1,q⁢(BR⁢(ξ0)), 1<q<Q, 1≤p≤q⁢Q/(Q-q):

(2.2)(⨍BR⁢(ξ0)|u-uξ0,R|pdξ)1/p≤CpR(⨍BR⁢(ξ0)|Xu|qdξ)1/q,

where

uξ0,R=⨍Br⁢(ξ0)u⁢𝑑ξ=|Br⁢(ξ0)|ℍn-1⁢∫Br⁢(ξ0)u⁢𝑑ξ.

Note the fact that the horizontal vectors Xi defined in (2.1) fits Hörmander’s vector fields, and that (2.2) is valid for p=q (≥2).

3 Horizontal affine function and estimates

Lemma 3.1.

Let u∈L2⁢(Bρ⁢(ξ0),RN), ξ0∈R2⁢n+1, and consider the horizontal components

ξ¯:=(x1,x2,…,xn,y1,y2,…,yn),ξ¯0:=(x01,x02,…,x0n,y01,y02,…,y0n).

If the function

lξ0,ρ⁢(ξ¯)=lξ0,ρ⁢(ξ¯0)+X⁢lξ0,ρ⁢(ξ¯-ξ¯0)

minimizes the functional

(3.1)l↦⨍Bρ⁢(ξ0)|u-l|2⁢𝑑ξ,

among horizontal affine functions l:R2⁢n→RN, then we have

lξ0,ρ⁢(ξ¯0)=uξ0,ρ=⨍Bρ⁢(ξ0)u⁢𝑑ξ

and

X⁢lξ0,ρ=Q-2c0⁢Q⁢Q+2ρ2⁢⨍Bρ⁢(ξ0)u⊗(ξ¯-ξ¯0)⁢𝑑ξ,

where the vector u⊗(ξ¯-ξ¯0) has components uα⁢(x1-x01,x2-x02,…,x2⁢n-x02⁢n) with α=1,2,…,N, and c0 is a positive constant defined by

(3.2)c0=∫0π(sin⁡θ)n⁢𝑑θ∫0π(sin⁡θ)n-1⁢𝑑θ={[(2⁢k-2)!!]2(2⁢k-1)!!⁢(2⁢k-3)!!⁢2π,n=2⁢k-1,[(2⁢k-1)!!]2(2⁢k)!!⁢(2⁢k-2)!!⁢π2,n=2⁢k.

Here, we use the notation (2⁢k-2)!!=(2⁢k-2)⁢(2⁢k-4)⁢⋯⁢4×2 and (2⁢k-1)!!=(2⁢k-1)⁢(2⁢k-3)⁢⋯⁢3×1.

Proof.

For simplicity and without loss of generality, it is enough to prove the case where N=1. We introduce the horizontal affine function l⁢(ξ¯)=a+〈b,ξ¯-ξ¯0〉, a∈ℝ, b∈ℝ2⁢n, and the function

P⁢(a,b):=⨍Bρ⁢(ξ0)(uα-a-〈b,ξ¯-ξ¯0〉)2⁢𝑑ξ,α=1,…,N.

We first note that the function P⁢(a,b) is differentiable with respect to the variables a and bi, i=1,2,…,2⁢n. A straightforward computation gives

∂⁡P∂⁡a=2⁢⨍Bρ⁢(ξ0)(a-uα+〈b,ξ¯-ξ¯0〉)⁢𝑑ξ

and

∂⁡P∂⁡bi=2⁢⨍Bρ⁢(ξ0)(〈b,ξ¯-ξ¯0〉⁢(ξ¯i-ξ¯0i)-uα⁢(ξ¯i-ξ¯0i)+a⁢(ξ¯i-ξ¯0i))⁢𝑑ξ.

In view of the symmetry of the quasi-ball Bρ⁢(ξ0) in the Heisenberg group, we have

⨍Bρ⁢(ξ0)(〈b,ξ¯-ξ¯0〉)⁢𝑑ξ=0=⨍Bρ⁢(ξ0)a⁢(ξ¯i-ξ¯0i)⁢𝑑ξ

and

⨍Bρ⁢(ξ0)bj⁢(ξ¯j-ξ¯0j)⁢(ξ¯i-ξ¯0i)⁢𝑑ξ=0 for ⁢i≠j.

So letting ∂⁡P/∂⁡a=0 and ∂⁡P/∂⁡bi=0, we deduce

a=⨍Bρ⁢(ξ0)uα⁢𝑑ξ and bi=⨍Bρ⁢(ξ0)uα⁢(ξ¯i-ξ¯0i)⁢𝑑ξ⨍Bρ⁢(ξ0)(ξ¯i-ξ¯0i)2⁢𝑑ξ.

Without loss of generality we calculate b1 in the sequel, so we are in the position of computing the average integral ⨍Bρ⁢(ξ0)(x1-x01)2⁢𝑑ξ. We take the following transformation:

x1-x01=r⁢(sin⁡θ)1/2⁢cos⁡θ1,

y1-y01=r⁢(sin⁡θ)1/2⁢sin⁡θ1⁢cos⁡θ2,

⋮

xn-x0n=r⁢(sin⁡θ)1/2⁢sin⁡θ1⁢sin⁡θ2⁢sin⁡θ3⁢⋯⁢sin⁡θ2⁢n-4⁢sin⁡θ2⁢n-3⁢sin⁡θ2⁢n-2⁢cos⁡θ2⁢n-1,

yn-y0n=r⁢(sin⁡θ)1/2⁢sin⁡θ1⁢sin⁡θ2⁢sin⁡θ3⁢⋯⁢sin⁡θ2⁢n-4⁢sin⁡θ2⁢n-3⁢sin⁡θ2⁢n-2⁢sin⁡θ2⁢n-1,

t-t0=r2⁢cos⁡θ

for 0≤r≤R, θ,θi∈(0,π), i=1,2,…,2⁢n-2 and θ2⁢n-1∈(0,2⁢π). Then an easy computation shows that the Jacobian of the transformation is given by

J=r2⁢n+1⁢(sin⁡θ)n-1⁢(sin⁡θ1)2⁢n-2⁢(sin⁡θ2)2⁢n-3⁢⋯⁢sin⁡θ2⁢n-2.

Therefore,

⨍Bρ⁢(ξ0)(x1-x01)2⁢𝑑ξ=∫Bρ⁢(ξ0)(x1-x01)2⁢𝑑x|Bρ⁢(ξ0)|ℍn

=∫0ρr2⁢n+3⁢𝑑r⁢∫0π(sin⁡θ)n⁢𝑑θ⁢∫0π(sin⁡θ1)2⁢n-2⁢(cos⁡θ1)2⁢𝑑θ1⁢⋯⁢∫0πsin⁡θ2⁢n-2⁢d⁢θ2⁢n-2∫0ρr2⁢n+1⁢𝑑r⁢∫0π(sin⁡θ)n-1⁢𝑑θ⁢∫0π(sin⁡θ1)2⁢n-2⁢𝑑θ1⁢⋯⁢∫0πsin⁡θ2⁢n-2⁢d⁢θ2⁢n-2

=Q⁢ρ2Q+2⁢(1-∫0π(sin⁡θ1)2⁢n⁢𝑑θ1∫0π(sin⁡θ1)2⁢n-2⁢𝑑θ1)⁢∫0π(sin⁡θ)n⁢𝑑θ∫0π(sin⁡θ)n-1⁢𝑑θ

=Q⁢ρ2Q+2⁢1Q-2⁢∫0π(sin⁡θ)n⁢𝑑θ∫0π(sin⁡θ)n-1⁢𝑑θ,

which yields

b1=Q+2ρ2⁢Q-2c0⁢Q⁢∫Bρ⁢(ξ0)uα⁢(x1-x01)⁢𝑑ξ,where c0=∫0π(sin⁡θ)n⁢𝑑θ∫0π(sin⁡θ)n-1⁢𝑑θ.

So we conclude that

lξ0,ρ⁢(ξ¯0)=uξ0,ρ=⨍Bρ⁢(ξ0)u⁢𝑑ξ and X⁢lξ0,ρ=Q-2c0⁢Q⁢Q+2ρ2⁢⨍Bρ⁢(ξ0)u⊗(ξ¯-ξ¯0)⁢𝑑ξ.∎

Since

⨍Bρ⁢(ξ0)(u-lξ0,ρ⁢(ξ¯))⁢𝑑ξ=0,

we have another version of the Poincaré inequality (2.2), that is,

(3.3)(⨍Bρ⁢(ξ0)|u-lξ0,ρ(ξ¯)|pdξ)1/p≤Cpρ(⨍Bρ⁢(ξ0)|Xu|qdξ)1/q,

where 1<q<Q, 1≤p≤q⁢Q/(Q-q).

With the results of Lemma 3.1, we have the following estimates.

Lemma 3.2.

Let u∈L2⁢(Bρ⁢(ξ0),RN) and θ∈(0,1). We denote by lξ0,ρ and lξ0,θ⁢ρ, the horizontal affine functions defined as above for the radii ρ and θ⁢ρ, respectively. Then we have

(3.4)|Xlξ0,ρ-Xlξ0,θ⁢ρ|2≤(Q-2)2c0⁢QQ+2(θ⁢ρ)2⨍Bθ⁢ρ⁢(ξ0)|u-lξ0,ρ|2dξ

and, more generally,

(3.5)|Xlξ0,ρ-Xl|2≤(Q-2)2c0⁢QQ+2ρ2⨍Bθ⁢ρ⁢(ξ0)|u-l|2dξ

for every horizontal affine function l:R2⁢n→RN.

Proof.

Estimate (3.4) holds, since

|Xlξ0,ρ-Xlξ0,θ⁢ρ|2=(Q-2c0⁢QQ+2(θ⁢ρ)2)2|⨍Bθ⁢ρ⁢(ξ0)(u-lξ0,ρ(ξ¯0)-Xlξ0,ρ(ξ¯-ξ¯0))⊗(ξ¯-ξ¯0)dξ|2

≤(Q-2c0⁢Q⁢Q+2(θ⁢ρ)2)2⁢⨍Bθ⁢ρ⁢(ξ0)|u-lξ0,ρ⁢(ξ¯0)-X⁢lξ0,ρ⁢(ξ¯-ξ¯0)|2⁢𝑑ξ⁢⨍Bθ⁢ρ⁢(ξ0)|ξ¯-ξ¯0|2⁢𝑑ξ

=(Q-2)2c0⁢QQ+2(θ⁢ρ)2⨍Bθ⁢ρ⁢(ξ0)|u-lξ0,ρ|2dξ,

where we have used the fact that

(3.6)⨍Bθ⁢ρ⁢(ξ0)|ξ¯-ξ¯0|2dξ=c0⁢Q⁢(θ⁢ρ)2Q+2

with c0 defined in (3.2).

Using again (3.6), estimate (3.5) follows. In particular, we have

|Xlξ0,ρ-Xl|2=(Q-2c0⁢QQ+2ρ2)2|⨍Bρ⁢(ξ0)(u-l(ξ¯0)-Xl(ξ¯-ξ¯0))⊗(ξ¯-ξ¯0)dξ|2

≤(Q-2c0⁢QQ+2ρ2)2⨍Bρ⁢(ξ0)|u-l(ξ¯0)-Xl(ξ¯-ξ¯0)|2dξ⨍Bρ⁢(ξ0)|ξ¯-ξ¯0|2dξ

=(Q-2)2c0⁢QQ+2ρ2⨍Bθ⁢ρ⁢(ξ0)|u-l|2dξ.∎

Furthermore, estimate (3.5) implies that lξ0,ρ has the following quasi-minimizing property for the Lp-norm.

Lemma 3.3.

Let lξ0,ρ be the minimizer of (3.1). For any horizontal affine functions l:R2⁢n→RN and p≥2, we have

⨍Bρ⁢(ξ0)|u-lξ0,ρ|pdξ≤c(p,Q)⨍Bρ⁢(ξ0)|u-l|pdξ.

Proof.

We write l⁢(ξ¯):=l⁢(ξ0¯)+X⁢l⁢(ξ¯-ξ0¯), and we note that

⨍Bρ⁢(ξ0)|u-lξ0,ρ|p⁢𝑑ξ≤⨍Bρ⁢(ξ0)(|u-l|+|l-lξ0,ρ|)p⁢𝑑ξ

(3.7)≤3p-1(|uξ0,ρ-l(ξ0¯)|p+ρp|Xlξ0,ρ-Xl|p+⨍Bρ⁢(ξ0)|u-l|pdξ)

and

|uξ0,ρ-l(ξ0¯)|p=|⨍Bρ⁢(ξ0)(u-l(ξ0¯)-Xl(ξ¯-ξ0¯))dξ|p≤(⨍Bρ⁢(ξ0)|u-l|dξ)p≤⨍Bρ⁢(ξ0)|u-l|pdξ,

where we have used the fact that ⨍Bρ⁢(ξ0)X⁢l⁢(ξ¯-ξ0¯)⁢𝑑ξ=0 in the first equality, and Hölder’s inequality in the last inequality.

We note that the second term of (3.7) can be estimated by using (3.5). This completes the proof. ∎

4 Caccioppoli type inequality

In this section, we prove a Caccioppoli type inequality with super-quadratic natural growth conditions. Let us first present the following 𝒜-harmonic approximation lemma by Föglein [17].

Lemma 4.1.

Let λ and L be fixed positive numbers, and let n,N∈N with n≥2. Assume that for any given ε>0, there exists δ=δ⁢(n,N,p,λ,L,ε)∈(0,1] with the following properties:

for any 𝒜∈Bil⁢(ℝ2⁢n×N),
(4.1)𝒜⁢(ν,ν)≥λ⁢|ν|2 𝑎𝑛𝑑 𝒜⁢(ν,ν¯)≤L⁢|ν|⁢|ν¯|,ν,ν¯∈ℝ2⁢n×N,
for any w∈HW1,p⁢(Bρ⁢(ξ0),ℝN),
⨍Bρ⁢(ξ0)(|X⁢w|2+γp-2⁢|X⁢w|p)⁢𝑑ξ≤1
and
|⨍Bρ⁢(ξ0)𝒜(Xw,Xφ)dξ|≤δsupBρ⁢(ξ0)|Xφ| for all φ∈C0∞(Bρ(ξ0),ℝN).

Then there exists an A-harmonic function h∈C∞⁢(Bρ/2⁢(ξ0),RN) such that

⨍Bρ/2⁢(ξ0)(|X⁢h|2+γp-2⁢|X⁢h|p)⁢𝑑ξ≤2Q+1

and

⨍Bρ/2⁢(ξ0)(|w-hρ/2|2+γp-2⁢|w-hρ/2|p)⁢𝑑ξ≤ε.

Shores [27] established a priori estimates for weak solutions u to homogeneous sub-elliptic systems with constant coefficients in general Carnot groups (see also [4] for Carnot groups of step 2). For convenience, in order to use Shores’s estimate in the sequel, we state it in a slightly different form that follows easily from Poincaré’s inequality, that is,

(4.2)supBρ/2⁢(ξ0)⁡(|u|2+ρ2⁢|X⁢u|2+ρ4⁢|X2⁢u|2)≤C0⁢ρ2⁢⨍Bρ⁢(ξ0)|X⁢u|2⁢𝑑ξ.

Lemma 4.2.

Let u∈HW1,p⁢(Ω,RN)∩L∞⁢(Ω,RN) be a weak solution of the nonlinear sub-elliptic systems (1.2) under the assumptions (H1)–(H5) with ∥u∥∞≤M. Then, for any ξ0=(x1,…,xn,y1,…,yn,t)∈Ω and r≤1 with Bρ(ξ0)⊂⊂Ω, and any horizontal affine function l:R2⁢n→RN with |l⁢(ξ¯0)|≤M, we have the estimate

⨍Br/2⁢(ξ0)(|X⁢u-X⁢l|2(1+|X⁢l|)2+|X⁢u-X⁢l|p(1+|X⁢l|)p)dξ≤Cc[⨍Br⁢(ξ0)(|u-l|2r2⁢(1+|X⁢l|)2+|u-l|prp⁢(1+|X⁢l|)p)dξ

+ω(⨍Br⁢(ξ0)|u-l(ξ0¯)|2dξ)+𝑽(r)+(ap′|Xl|p′+bp′)rp′],

where p′ is the Hölder conjugate exponent of p, that is, 1/p+1/p′=1, p>2 and p′<2.

Proof.

We test the sub-elliptic system (1.2) with the testing function φ=ϕp⁢(u-l) with l=l⁢(ξ¯0)-X⁢l⁢(ξ¯-ξ¯0), where ϕ∈C0∞⁢(Br⁢(ξ0)) is a cut-off function satisfying 0≤ϕ≤1, |X⁢ϕ|≤4/r and ϕ≡1 on Br/2⁢(ξ0). This yields

⨍Br⁢(ξ0)Aiα⁢(ξ,u,X⁢u)⁢ϕp⁢(X⁢u-X⁢l)⁢𝑑ξ=-p⁢⨍Br⁢(ξ0)Aiα⁢(ξ,u,X⁢u)⁢ϕp-1⁢(u-l)⁢X⁢ϕ⁢𝑑ξ+⨍Br⁢(ξ0)fα⁢(ξ,u,X⁢u)⁢φα⁢𝑑ξ.

Adding this to the equation

-⨍Br⁢(ξ0)Aiα⁢(ξ,u,X⁢l)⁢ϕp⁢(X⁢u-X⁢l)⁢𝑑ξ=p⁢⨍BR⁢(ξ0)Aiα⁢(ξ,u,X⁢l)⁢ϕp-1⁢(u-l)⁢X⁢ϕ⁢𝑑ξ-⨍Br⁢(ξ0)Aiα⁢(ξ,u,X⁢l)⁢X⁢φ⁢𝑑ξ

and using

0=⨍Br⁢(ξ0)(Aiα⁢(⋅,l⁢(ξ¯0),X⁢l))ξ0,r⁢X⁢φ⁢𝑑ξ,

we have

(4.3)⨍Br⁢(ξ0)[Aiα⁢(ξ,u,X⁢u)-Aiα⁢(ξ,u,X⁢l)]⁢ϕp⁢(X⁢u-X⁢l)⁢𝑑ξ=I1+I2+I3+I4,

where

I1=p⁢⨍Br⁢(ξ0)[Aiα⁢(ξ,u,X⁢l)-Aiα⁢(ξ,u,X⁢u)]⁢ϕp-1⁢(u-l)⁢X⁢ϕ⁢𝑑ξ,

I2=⨍Br⁢(ξ0)[Aiα⁢(ξ,l⁢(ξ¯0),X⁢l)-Aiα⁢(ξ,u,X⁢l)]⁢X⁢φ⁢𝑑ξ,

I3=⨍Br⁢(ξ0)[(Aiα⁢(⋅,l⁢(ξ¯0),X⁢l))ξ0,r-Aiα⁢(ξ,l⁢(ξ¯0),X⁢l)]⁢X⁢φ⁢𝑑ξ,

I4=⨍Br⁢(ξ0)fα⁢(ξ,u,X⁢u)⁢φα⁢𝑑ξ.

Due to the alternative form (1.7) of the uniformly elliptic condition (H2), the left-hand side of (4.3) can be estimated as follows:

⨍Br⁢(ξ0)[Aiα⁢(ξ,u,X⁢u)-Aiα⁢(ξ,u,X⁢l)]⁢ϕp⁢(X⁢u-X⁢l)⁢𝑑ξ

(4.4)≥λ0⁢⨍Br⁢(ξ0)ϕp⁢[(1+|X⁢l|)p-2⁢|X⁢u-X⁢l|2+|X⁢u-X⁢l|p]⁢𝑑ξ.

We now estimate the terms I1–I4 of the right-hand side. For ϵ>0 to be fixed later, using (H1) and Young’s inequality we have

I1≤⨍Br⁢(ξ0)pϕp-1|∫01DPAiα(ξ,u,Xl+t(Xu-Xl))dt(Xu-Xl)||Xϕ||u-l|dξ

≤p⁢L⁢⨍Br⁢(ξ0)ϕp-1⁢(1+|X⁢l|+|X⁢u-X⁢l|)p-2⁢|X⁢u-X⁢l|⁢|X⁢ϕ|⁢|u-l|⁢𝑑ξ

≤ϵ⁢⨍Br⁢(ξ0)ϕp⁢[(1+|X⁢l|)p-2⁢|X⁢u-X⁢l|2+|X⁢u-X⁢l|p]⁢𝑑ξ

(4.5)+C⁢(p,L,ϵ)⁢⨍Br⁢(ξ0)[(1+|X⁢l|)p-2⁢|u-lr|2+|u-lr|p]⁢𝑑ξ.

Noting that |X⁢φ|≤C⁢(p)⁢(ϕ⁢|X⁢u-X⁢l|+|u-l|/r), and using the continuity condition (1.3) and Young’s inequality, we get

I2≤C⁢(p)⁢L⁢⨍Br⁢(ξ0)ω⁢(|u-l⁢(ξ¯0)|2)⁢(1+|X⁢l|)p-1⁢(ϕ⁢|X⁢u-X⁢l|+|u-l|r)⁢𝑑ξ

≤ϵ⁢⨍Br⁢(ξ0)ϕp⁢|X⁢u-X⁢l|p⁢𝑑ξ+⨍Br⁢(ξ0)|u-lr|p⁢𝑑ξ+C⁢(p,L,ϵ)⁢⨍Br⁢(ξ0)ωp′⁢(|u-l⁢(ξ¯0)|2)⁢(1+|X⁢l|)p

(4.6)≤ϵ⨍Br⁢(ξ0)ϕp|Xu-Xl|pdξ+⨍Br⁢(ξ0)|u-lr|pdξ+C(p,L,ϵ)(1+|Xl|)pω(⨍Br⁢(ξ0)|u-l(ξ¯0)|2dξ),

where we have used ωp′≤ω with 1/p+1/p′=1, and Jensen’s inequality in the last inequality.

We next estimate I3. Using the VMO-condition (H4), |X⁢φ|≤C⁢(p)⁢(ϕ⁢|X⁢u-X⁢l|+|u-l|/r), and Young’s inequality, we have

I3≤C⁢(p)⁢(1+|X⁢l|)p-1⁢⨍Br⁢(ξ0)𝐯ξ0⁢(ξ,r)⁢(ϕ⁢|X⁢u-X⁢l|+|u-l|r)

(4.7)≤ϵ⁢⨍Br⁢(ξ0)ϕp⁢|X⁢u-X⁢l|p⁢𝑑ξ+⨍Br⁢(ξ0)|u-lr|p⁢𝑑ξ+C⁢(p,L,ϵ)⁢(1+|X⁢l|)p⁢𝐕⁢(r),

where we have used the assumption 𝐯ξ0≤2⁢L. Finally, I4 can be estimated using assumption (H5) and Young’s inequality for ϵ>0 to be fixed later. This yields

I4≤⨍Br⁢(ξ0)(a⁢|X⁢u|p+b)⁢ϕp⁢|u-l|⁢𝑑ξ

≤a⁢⨍Br⁢(ξ0)(|X⁢u-X⁢l|+|X⁢l|)p⁢ϕp⁢|u-l|⁢𝑑ξ+b⁢⨍Br⁢(ξ0)ϕ⁢|u-l|⁢𝑑ξ

≤a⁢⨍Br⁢(ξ0)ϕp⁢[(1+ϵ′)⁢|X⁢u-X⁢l|p+(1+ϵ′⁣-1)⁢|X⁢l|p]⁢|u-l|⁢𝑑ξ+b⁢⨍Br⁢(ξ0)ϕ⁢r⁢|u-lr|⁢𝑑ξ

≤a⁢(1+ϵ′)⁢(2⁢M+r⁢|X⁢l|)⁢⨍Br⁢(ξ0)ϕp⁢|X⁢u-X⁢l|p⁢𝑑ξ+C⁢(ϵ)⁢⨍Br⁢(ξ0)|u-lr|p⁢𝑑ξ

(4.8)+ϵ⁢(1+|X⁢l|)p⁢rp′⁢[ap′⁢(1+1/ϵ′)p′⁢|X⁢l|p′+bp′].

Joining estimates (4.4)–(4.8) with (4.3), we obtain

Λ⁢⨍Br⁢(ξ0)ϕp⁢[(1+|X⁢l|)p-2⁢|X⁢u-X⁢l|2+|X⁢u-X⁢l|p]⁢𝑑ξ

≤C⁢(p,L,ϵ,ϵ′)⁢⨍Br⁢(ξ0)[(1+|X⁢l|)p-2⁢|u-lr|2+|u-lr|p]⁢𝑑ξ

+C⁢(p,L,ϵ)⁢(1+|X⁢l|)p⁢[ω⁢(⨍Br⁢(ξ0)|u-l⁢(ξ¯0)|2⁢𝑑ξ)+𝐕⁢(r)]

(4.9) +ϵ⁢(1+|X⁢l|)p⁢rp′⁢[ap′⁢(1+1/ϵ′)p′⁢|X⁢l|p′+bp′],

where Λ=λ0-a⁢(1+ϵ′)⁢(2⁢M+r⁢|X⁢l|)-3⁢ϵ. Dividing (4.9) by (1+|X⁢l|)p, we arrive at the desired Caccioppoli type inequality by choosing suitable ϵ and ϵ′ such that Λ>0. ∎

5 Proofs of Theorems 1.1 and 1.2

For sake of simplicity, we set

Φ(ξ0,r,l):=⨍Br/2⁢(ξ0)(|X⁢u-X⁢l|2(1+|X⁢l|)2+|X⁢u-X⁢l|p(1+|X⁢l|)p)dξ,

Ψ(ξ0,r,l):=⨍Br⁢(ξ0)(|u-l|2r2⁢(1+|X⁢l|)2+|u-l|prp⁢(1+|X⁢l|)p)dξ

and

Ψ*⁢(ξ0,r,l):=Ψ⁢(ξ0,r,l)+ω⁢(⨍Br⁢(ξ0)|u-l⁢(ξ0¯)|2⁢𝑑ξ)+𝐕⁢(r)+(a⁢|X⁢l|+b)p′⁢rp′.

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). To apply the 𝒜-harmonic approximation lemma, we need to establish the following lemma, which provides a linearization strategy for nonlinear sub-elliptic systems.

Lemma 5.1.

Under the assumptions of Theorem 1.1 and the assumption B2⁢ρ⁢(ξ0)⊆Ω with ρ≤ρ0, for an arbitrary horizontal function l⁢(ξ¯):R2⁢n→RN, we define

𝒜:=(DP⁢Aiα⁢(⋅,l⁢(ξ¯0),X⁢l))ξ0,ρ(1+|X⁢l|)p-1 𝑎𝑛𝑑 w~:=u-l(1+|X⁢l|).

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

⨍Bρ⁢(ξ0)𝒜(Xw~,Xφ)dξ≤C1[μ1/2(Ψ*⁢(2⁢ρ))Ψ*⁢(2⁢ρ)+Ψ*(2ρ)+ρ(a|Xl|p+b)]supBρ⁢(ξ0)|Xφ|,

where we have used the more compact notation Ψ*⁢(2⁢ρ):=Ψ*⁢(ξ0,2⁢ρ,l), and C1 is a positive constant.

Proof.

Without loss of generality, we may also assume that supBρ⁢(ξ0)⁡|X⁢φ|≤1. We denote

v:=u-l⁢(ξ¯0)-X⁢l⁢(ξ¯-ξ¯0).

Using the weak formulation of our sub-elliptic system (1.2) and noting that (Aiα⁢(⋅,l⁢(ξ¯0),X⁢l))ξ0,ρ is constant, we have

(5.1)(1+|X⁢l|)p-1⁢⨍Bρ⁢(ξ0)𝒜⁢(X⁢v,X⁢φ)⁢𝑑ξ≤I1′+I2′+I3′+I4′,

where

I1′=⨍Bρ⁢(ξ0)[∫01((DPAiα(⋅,l(ξ¯0),Xl))ξ0,ρ-(DPAiα(⋅,l(ξ¯0),Xl+tXv))ξ0,ρ)Xvdt]dξsupBρ⁢(ξ0)|Xφ|,

I2′=⨍Bρ⁢(ξ0)[(Aiα(⋅,l(ξ¯0),Xu))ξ0,ρ-Aiα(ξ,l(ξ¯0),Xu)]supBρ⁢(ξ0)|Xφ|,

I3′=⨍Bρ⁢(ξ0)[Aiα(ξ,l(ξ¯0),Xu)-Aiα(ξ,u,Xu)]dξsupBρ⁢(ξ0)|Xφ|,

I4′=⨍Bρ⁢(ξ0)fα⁢(ξ,u,X⁢u)⁢φα⁢𝑑ξ.

Using assumption (1.4), we first note that

∫01[(DP⁢Aiα⁢(⋅,l⁢(ξ¯0),X⁢l))ξ0,ρ-(DP⁢Aiα⁢(⋅,l⁢(ξ¯0),X⁢l+t⁢X⁢v))ξ0,ρ]⁢𝑑t

≤∫01⨍Bρ⁢(ξ0)[(DP⁢Aiα⁢(ζ,l⁢(ξ¯0),X⁢l))-(DP⁢Aiα⁢(ζ,l⁢(ξ¯0),X⁢l+t⁢X⁢v))]⁢𝑑t

≤C⁢(L)⁢μ⁢(X⁢u-X⁢l1+|X⁢l|)⁢(1+|X⁢l|+|X⁢u-X⁢l|)p-2.

This leads us to estimate the term I1′ as follows:

I1′≤C⁢(L,p)⁢(1+|X⁢l|)p-1⁢⨍Bρ⁢(ξ0)μ⁢(X⁢u-X⁢l1+|X⁢l|)⁢[|X⁢u-X⁢l|1+|X⁢l|+|X⁢u-X⁢l|p-1(1+|X⁢l|)p-1]⁢𝑑ξ

≤C⁢(L,p)⁢(1+|X⁢l|)p-1⁢[⨍Bρ⁢(ξ0)μ2⁢(X⁢u-X⁢l1+|X⁢l|)⁢𝑑ξ]1/2⁢[⨍Bρ⁢(ξ0)|X⁢u-X⁢l|2(1+|X⁢l|)2⁢𝑑ξ]1/2

+C⁢(L,p)⁢(1+|X⁢l|)p-1⁢[⨍Bρ⁢(ξ0)μp⁢(X⁢u-X⁢l1+|X⁢l|)⁢𝑑ξ]1/p⁢[⨍Bρ⁢(ξ0)|X⁢u-X⁢l|p(1+|X⁢l|)p⁢𝑑ξ]1/p′

≤C⁢(L,p)⁢(1+|X⁢l|)p-1⁢[μ1/2⁢(Φ⁢(ρ))⁢Φ1/2⁢(ρ)+μ1/p⁢(Φ⁢(ρ))⁢Φ1/p′⁢(ρ)]

(5.2)≤C⁢(L,p)⁢(1+|X⁢l|)p-1⁢[μ1/2⁢(Φ⁢(ρ))⁢Φ1/2⁢(ρ)+Φ⁢(ρ)],

where in the second last inequality we have used the fact that μ2,μp≤μ as μ≤1, Jensen’s inequality and

⨍Bρ⁢(ξ0)X⁢u-X⁢l1+|X⁢l|⁢𝑑ξ≤Φ1/2⁢(ρ),

which in turn holds by Hölder’s inequality. For the last inequality, we have used the fact that

a1/p⁢b1/p′=(a1/p⁢b1/p)⁢(bp-2/p)≤a1/2⁢b1/2+b,

by Young’s inequality.

The term I2′ can be estimated by the VMO-condition (1.5), Young’s inequality and the bound 𝐯0≤2⁢L as follows:

I2′≤⨍Bρ⁢(ξ0)𝐯0⁢(ξ,ρ)⁢(1+|X⁢l|+|X⁢u-X⁢l|)p-1⁢𝑑ξ

≤C⁢(p)⁢(1+|X⁢l|)p-1⁢[⨍Bρ⁢(ξ0)𝐯0⁢(ξ,ρ)⁢𝑑ξ+⨍Bρ⁢(ξ0)𝐯0p⁢(ξ,ρ)⁢𝑑ξ+⨍Bρ⁢(ξ0)|X⁢u-X⁢l|p(1+|X⁢l|)p⁢𝑑ξ]

≤C⁢(L,p)⁢(1+|X⁢l|)p-1⁢[𝐕⁢(ρ)+⨍Bρ⁢(ξ0)|X⁢u-X⁢l|p(1+|X⁢l|)p⁢𝑑ξ]

(5.3)≤C⁢(L,p)⁢(1+|X⁢l|)p-1⁢[Ψ*⁢(ρ)+Φ⁢(ρ)].

Similarly, we estimate the term I3′ by the continuity condition (1.3), Young’s inequality, the bound ω≤1 and Jensen’s inequality. This leads us to

I3′≤L⁢⨍Bρ⁢(ξ0)ω⁢(|u-l⁢(ξ¯0)|2)⁢(1+|X⁢l|+|X⁢u-X⁢l|)p-1⁢𝑑ξ

≤C⁢(L,p)⁢(1+|X⁢l|)p-1⁢[⨍Bρ⁢(ξ0)ω⁢(|u-l⁢(ξ¯0)|2)⁢𝑑ξ+⨍Bρ⁢(ξ0)ωp⁢(|u-l⁢(ξ¯0)|2)⁢𝑑ξ+⨍Bρ⁢(ξ0)|X⁢u-X⁢l|p(1+|X⁢l|)p⁢𝑑ξ]

≤C⁢(L,p)⁢(1+|X⁢l|)p-1⁢[ω⁢(⨍Bρ⁢(ξ0)|u-l⁢(ξ¯0)|2⁢𝑑ξ)+⨍Bρ⁢(ξ0)|X⁢u-X⁢l|p(1+|X⁢l|)p⁢𝑑ξ]

(5.4)≤C⁢(L,p)⁢(1+|X⁢l|)p-1⁢[Ψ*⁢(ρ)+Φ⁢(ρ)].

Finally, we estimate the last term I4′ by the growth condition (H5) and the assumption supBρ⁢(ξ0)|φ|≤ρ≤1 as follows:

I4′≤⨍Bρ⁢(ξ0)ρ⁢[a⁢(|X⁢l|+|X⁢u-X⁢l|)p+b]⁢𝑑ξ

≤C⁢(p)⁢[a⁢(1+|X⁢l|)p⁢⨍Bρ⁢(ξ0)|X⁢u-X⁢l|p(1+|X⁢l|)p+ρ⁢|X⁢l|p⁢a+ρ⁢b]⁢d⁢ξ

(5.5)≤C⁢(p)⁢a⁢(1+|X⁢l|)p⁢[Φ⁢(ρ)+ρ⁢(a⁢|X⁢l|p+b)].

Combining (5.1) with estimates (5.2)–(5.5), dividing by (1+|X⁢l|)p, and recalling the definition of w~, we have

⨍Bρ⁢(ξ0)𝒜(Xw~,Xφ)dξ≤C(L,p)[μ1/2(Φ⁢(ρ)|)Φ1/2(ρ)+Ψ*(ρ)+Φ(ρ)+ρ(a|Xl|p+b)]

≤C1⁢[μ1/2⁢(Ψ*⁢(2⁢ρ))⁢Ψ*1/2⁢(2⁢ρ)+Ψ*⁢(2⁢ρ)+ρ⁢(a⁢|X⁢l|p+b)],

where we have used the Caccioppoli type inequality Φ⁢(ρ)≤Cc⁢Ψ*⁢(2⁢ρ) and the concavity of μ to have μ⁢(c⁢s)≤c⁢μ⁢(s) for c≥1. This yields the claim with C1=C⁢(L,p)⁢Cc. ∎

The strategy of the proof is to approximate the given solution by 𝒜-harmonic functions, for which suitable decay estimates are available from the classical theory. Now we are in the position to establish the excess improvement.

Lemma 5.2.

Suppose that the conditions of Theorem 1.1 are satisfied and consider a ball Br⁢(ξ0)⊆Ω with r≤ρ0. Let θ∈(0,1/8] be arbitrary and impose the following smallness conditions on the excess:

[μ1/2⁢(Ψ*⁢(ξ0,r,lξ0,r))+Ψ*⁢(ξ0,r,lξ0,r)]≤δ/2 with the constant δ=δ⁢(Q,N,p,L,λ,θQ+p+2)∈(0,1] from the 𝒜-harmonic approximation lemma (Lemma 4.1),
Ψ⁢(ξ0,r,lξ0,r)≤c0⁢Q⁢θQ+24⁢(Q-2)2⁢(Q+2),
γ:=[Ψ*p′/2⁢(ξ0,r,lξ0,r)+(δ/2)-p′⁢ρp′⁢(a⁢|X⁢lξ0,r|p+b)p′]1/p′≤1.

Then we have the excess improvement estimate

Ψ⁢(ξ0,θ⁢r,lξ0,r)≤C4⁢θ2⁢Ψ*⁢(ξ0,r,lξ0,r).

Proof.

We re-scale the map

w=u-lξ0,rC2⁢γ⁢(1+|X⁢lξ0,r|) with lξ0,r=uξ0,r+X⁢lξ0,r⁢(ξ¯-ξ¯0),

where C2=max⁡{Cc,C1,1}. We claim that w satisfies the assumptions of the 𝒜-harmonic approximation lemma.

First, for our choice of the bilinear form, note that

𝒜:=(DP⁢Aiα⁢(⋅,lξ0,r⁢(ξ0),X⁢lξ0,r))ξ0,ρ(1+|X⁢lξ0,r|)p-1.

Condition (4.1) is valid due to assumptions (H1)–(H2).

Next, we check, by Lemma 5.1 with ρ=r/2 and l≡lξ0,r, that the map w is approximately 𝒜-harmonic in the sense that

⨍Br/2⁢(ξ0)𝒜(Xw,Xφ)dξ≤C1C2[μ1/2(Ψ*⁢(ξ0,r,lξ0,r))+Ψ*⁢(ξ0,r,lξ0,r)+δ2]supBr/2⁢(ξ0)|Xφ|

(5.6)≤δsupBr/2⁢(ξ0)|Xφ|

for all φ∈C0∞⁢(Br/2⁢(ξ0),ℝN), with the constant δ determined by Lemma 4.1 for the choice ε=θQ+p+2. Moreover, w satisfies the following energy bound:

⨍Br/2⁢(ξ0)(|Xw|2+γp-2|Xw|p)dξ=⨍Br/2⁢(ξ0)|X⁢u-X⁢lξ0,r|2c22⁢(1+|X⁢lξ0,r|)2⁢γ2+γp-2|X⁢u-X⁢lξ0,r|pc2p⁢(1+|X⁢lξ0,r|)p⁢γpdξ

≤Φ⁢(r/2,lξ0,r)C22⁢γ2

≤Cc⁢Ψ*⁢(r,lξ0,r)C22⁢Ψ*⁢(r,lξ0,r)

≤1.

So we are in the situation of Lemma 4.1 with ε=θQ+p+2 and ρ=r/2. The lemma ensures the existence of an 𝒜-harmonic map h∈C∞⁢(Br/4,ℝN) such that

(5.7)⨍Br/4⁢(ξ0)(|X⁢h|2+γp-2⁢|X⁢h|p)⁢𝑑ξ≤2Q+1

and

⨍Br/4⁢(ξ0)(|w-hr/4|2+γp-2|w-hr/4|p)dξ≤θQ+p+2.

Since h is 𝒜-harmonic, from (4.2) we have the estimate

r2supBr/8⁢(ξ0)|X2h|2≤C0⨍Br/4⁢(ξ0)|Xh|2dξ.

For p>2, by Hölder’s inequality, this yields

γp-2rpsupBr/8⁢(ξ0)|X2h|p≤γp-2(r2supBr/8⁢(ξ0)X2h|2)p/2

≤γp-2C0p/2(⨍Br/4⁢(ξ0)|Xh|2dξ)p/2

≤C0p/2γp-2⨍Br/4⁢(ξ0)|Xh|pdξ.

From these estimates we infer the following decay estimate using in turn the Poincaré type inequality (3.3) and (5.7) for s=2 and s=p:

γs-2⨍Bθ⁢r⁢(ξ0)|h-hξ0,θ⁢r-(X⁢h)ξ0,θ⁢r⁢(ξ¯-ξ¯0)θ⁢r|sdξ≤γs-2Cps⨍Bθ⁢r⁢(ξ0)|Xh-(Xh)ξ0,θ⁢r|sdξ

≤γs-2Cp2⁢s(θr)s⨍Bθ⁢r⁢(ξ0)|X2h|sdξ

≤Cp2⁢sγs-2θsrssupBr/8⁢(ξ0)|X2h|s

≤C3θsγs-2⨍Br/4⁢(ξ0)|Xh|sdξ

≤C3⁢2Q+1⁢θs

with C3=max⁡{C0⁢Cp4,C0p/2⁢Cp2⁢p}, where θ∈(0,1/8] can be chosen arbitrary. Abbreviating

l(h)⁢(ξ):=hξ0,θ⁢r+(X⁢h)ξ0,θ⁢r⁢(ξ¯-ξ¯0),

we conclude

≤C⁢(4⁢θ)-Q-s⁢θQ+p+2+C3⁢2Q+1⁢θs

≤C⁢θ2.

Scaling back to u, we infer

≤C⁢C2s⁢θ2⁢(1+|X⁢lξ0,r|)s⁢Ψ*⁢(r),

where we have used the definition of γ from (iii) with |X⁢l| being constant.

≤C⁢C2s⁢θ2⁢(1+|X⁢lξ0,r|)s⁢Ψ*⁢(r).

Here, we want to replace the term |X⁢lξ0,r| on the right-hand side by |X⁢lξ0,θ⁢r|. For this we estimate

|Xlξ0,r-Xlξ0,θ⁢r|2≤(Q-2)2c0⁢Q(Q+2)(θ⁢r)2⨍Bθ⁢r⁢(ξ0)|u-lξ0,r|2dξ

≤(Q-2)2c0⁢Q⁢(Q+2)θQ+2⁢⨍Br⁢(ξ0)|u-lξ0,rr2|2⁢𝑑ξ

≤(Q-2)2c0⁢Q⁢(Q+2)θQ+2⁢(1+|X⁢lξ0,r|)2⁢Ψ⁢(r)

≤14⁢(1+|X⁢lξ0,r|)2,

by assumption (ii). This yields

1+|X⁢lξ0,r|≤1+|X⁢lξ0,θ⁢r|+|X⁢lξ0,r-X⁢lξ0,θ⁢r|≤1+|X⁢lξ0,θ⁢r|+12⁢(1+|X⁢lξ0,r|),

and, after reabsorbing the last term from the right-hand side on the left, we have

1+|X⁢lξ0,r|≤2⁢(1+|X⁢lξ0,θ⁢r|).

So we deduce

⨍Bθ⁢r⁢(ξ0)|u-lξ0,θ⁢rθ⁢r|s⁢𝑑ξ≤2⁢C⁢C2s⁢θ2⁢(1+|X⁢lξ0,θ⁢r|)s⁢Ψ*⁢(r) for ⁢s∈{2,p}.

Dividing by (1+|X⁢lξ0,θ⁢r|)s and adding the corresponding terms for s=2 and s=p, we deduce the claim, that is,

Ψ⁢(ξ0,θ⁢r,lξ0,r)≤C4⁢θ2⁢Ψ*⁢(ξ0,r,lξ0,r),

with C4=2⁢C⁢C2s. ∎

We fix an arbitrary Hölder exponent γ∈(0,1) and define the Campanato type excess

Cγ(ξ0,ρ):=ρ-2⁢γ∫Bρ⁢(ξ0)|u-uξ0,ρ|2dξ.

In the following lemma, we iterate the excess improvement estimate from Lemma 5.2.

Lemma 5.3.

Suppose that the assumptions of Theorem 1.1 are satisfied. For every γ∈(0,1), there exists constants ε*,κ*,ρ*>0 and θ∈(0,1/8], all depending at most on Q,N,λ,L,γ,ρ0,μ⁢(⋅),ω⁢(⋅),V⁢(⋅),a,b and M, such that

(A0)Ψ⁢(ξ0,r,lξ0,r)<ε* 𝑎𝑛𝑑 Cγ⁢(ξ0,r)<κ*

for r∈(0,ρ*) with Br(ξ0)⊂⊂Ω. Then

(Ak)Ψ⁢(ξ0,θk⁢r,lξ0,θk⁢r)<ε* 𝑎𝑛𝑑 Cγ⁢(ξ0,θk⁢r)<κ* for every k∈ℕ.

Proof.

We begin by choosing the constants. First, we let

(5.8)θ:=min⁡{[c0⁢Q16⁢(Q-2)2⁢(Q+2)]1/(2-2⁢α),12⁢C4}≤18

with the constant C4 determined in Lemma 5.2. In particular, note that the choice of θ>0 fixes the constant δ=δ⁢(Q,N,p,λ,L,θQ+p+2) from Lemma 4.1. Next, we fix an ε*>0 sufficiently small to ensure

(5.9)ε*≤c0⁢Q⁢θQ+216⁢(Q-2)2⁢(Q+2) and μ1/2⁢(4⁢ε*)+3⁢ε*≤δ2.

Then we choose κ*>0 so small that ω⁢(κ*)<ε*. Finally, we fix ρ*∈(0,1) small enough to guarantee

ρ*≤{ρ0,κ*1/2⁢(1-γ),1},V⁢(ρ*)<ε* and (δ2)-1⁢[a⁢(Q-2)1/2⁢κ*⁢(Q+2)c0⁢Q+b]⁢(θk⁢r)γ≤ε*<1.

Now we prove assertion (Ak) by induction. We assume that we have already established (Ak) up to some k∈ℕ, we begin by proving the first part of assertion (Ak+1), that is, the one concerning Ψ⁢(θk+1⁢r,lξ0,θk+1⁢r). First, using with l=uξ0,θk⁢r in (3.5), we have

|Xlξ0,θk⁢r|2≤(Q-2)2⁢(Q+2)c0⁢Q⁢(θk⁢r)2∫Bθk⁢r⁢(ξ0)|u-uξ0,θk⁢r|2dξ

=(Q-2)2⁢(Q+2)c0⁢Q⁢(θk⁢r)2⁢(γ-1)⁢Cγ⁢(ξ0,θk⁢r)

≤(Q-2)2⁢(Q+2)c0⁢Q⁢(θk⁢r)2⁢(γ-1)⁢κ*.

By assumption (Ak), the choice of κ* and ε*, and the above estimate, we infer

Ψ*⁢(ξ0,θk⁢r,lξ0,θk⁢r)≤Ψ⁢(ξ0,θk⁢r,lξ0,θk⁢r)+ω⁢(Cγ⁢(ξ0,θk⁢r))+V⁢(θk⁢r)+(a⁢|X⁢lξ0,θk⁢r|+b)p′⁢(θk⁢r)p′

≤ε*+ω⁢(κ*)+V⁢(ρ*)+[a⁢(Q-2)1/2⁢κ*⁢(Q+2)c0⁢Q+b]p′⁢(θk⁢r)p′⁢γ

≤4⁢ε*.

Here, we have used the fact that (θk⁢r)γ-1≥1. Now it is easy to check that our choice of ε* implies that the smallness condition assumptions (i)–(iii) in Lemma 5.2 are satisfied on the level θk⁢r, that is, we have

Ψ1/2⁢(ξ0,θk⁢r,lξ0,θk⁢r)+μ1/2⁢(Ψ1/2⁢(ξ0,θk⁢r,lξ0,θk⁢r))≤2⁢ε*+μ1/2⁢(4⁢ε*)<δ2

and

Ψ1/2⁢(ξ0,θk⁢r,lξ0,θk⁢r)<ε*≤θQ+2⁢c0⁢Q4⁢(Q-2)2⁢(Q+2)<1.

Furthermore, we also have the smallness condition (iii), that is,

γ⁢(θk⁢ρ):=[Ψ∗p′/2⁢(θk⁢ρ)+(δ2)-p′⁢(θk⁢ρ)p′⁢(a⁢|X⁢lξ0,θk⁢ρ|+b)p′]1/p′≤2⁢ε∗+ε∗≤3⁢ε∗≤δ2≤1.

Thus, we can apply Lemma 5.2 with the radius θk⁢r instead of r, which yields

Ψ⁢(ξ0,θk+1⁢r,lξ0,θk+1⁢r)≤C4⁢θ2⁢Ψ*⁢(ξ0,θk⁢r,lξ0,θk⁢r)≤4⁢C4⁢θ2⁢ε*≤ε*.

By the choice of θ in (5.8). So we establish the first part of assertion (Ak+1) and it remains to prove the second one, that is, the one concerning Cγ⁢(ξ0,θk+1⁢r). For this, we first estimate

where in the last inequality we used assumption (Ak). Noting that

lξ0,θk⁢r=uξ0,θk⁢r+lξ0,θk⁢r⁢(ξ1-ξ01),

we have

Cγ⁢(ξ0,θk+1⁢r)=(θk+1⁢r)-2⁢γ⁢⨍Bθk+1⁢r⁢(ξ0)|u-uξ0,θk+1⁢r|2⁢𝑑ξ

≤(θk+1r)-2⁢γ⨍Bθk+1⁢r⁢(ξ0)|u-uξ0,θk⁢r|2dξ

≤2(θk+1r)-2⁢γ[⨍Bθk+1⁢r⁢(ξ0)|u-lξ0,θk⁢r|2dξ+|Xlξ0,θk⁢r|2(θk+1r)2]

≤2(θk+1r)-2⁢α[θ-Q⨍Bθk⁢r⁢(ξ0)|u-lξ0,θk⁢r|2dξ+|Xlξ0,θk⁢r|2(θk+1r)2]

≤4⁢(θk⁢r)2⁢(1-α)⁢[θ-Q-2⁢α⁢ε*+|X⁢lξ0,θk⁢r|2⁢(θ-Q-2⁢α⁢ε*+θ2⁢(1-α))].

Using (5.9) and recalling the choice of ρ*, ε* and θ, we deduce

Cγ⁢(ξ0,θk+1⁢r)≤4⁢ρ*2⁢(1-γ)⁢θ-Q-2⁢γ⁢ε*+4⁢(Q-2)2⁢(Q+2)c0⁢Q⁢(θ-Q-2⁢γ⁢ε*+θ2⁢(1-γ))⁢κ*≤14⁢κ*+14⁢κ*+14⁢κ*<κ*.

This proves the second part of the assertion (Ak+1) and completes the proof of the lemma. ∎

Proof of Theorem 1.1.

By Lebesgue’s differentiation theorem, we have |Σ1∪Σ2|=0. So it suffices to show that every point ξ0∈Ω∖(Σ1∪Σ2) is regular. To this end, we first note that for every 0<ρ<dist⁡(ξ0,∂⁡Ω), the bound (3.5) with l=uξ0,ρ+(X⁢u)ξ0,ρ⁢(ξ¯-ξ¯0), and the Poincaré inequality (3.3) imply

|X⁢lξ0,ρ-(X⁢u)ξ0,ρ|≤(Q-2)2⁢(Q+2)c0⁢Q⁢ρ2⁢⨍Bρ⁢(ξ0)|u-uξ0,ρ-(X⁢u)ξ0,ρ⁢(ξ¯-ξ¯0)|2⁢𝑑ξ

≤Cp⁢(Q-2)2⁢(Q+2)c0⁢Q⁢⨍Bρ⁢(ξ0)|X⁢u-(X⁢u)ξ0,ρ|2⁢𝑑ξ,

and then, using again the Poincaré inequality (3.3), we obtain

Ψ(ξ0,ρ,lξ0,ρ)≤ρ-2⨍Bρ⁢(ξ0)|u-lξ0,ρ|2dξ+ρ-p⨍Bρ⁢(ξ0)|u-lξ0,ρ|pdξ

(5.10)≤Cp2⨍Bρ⁢(ξ0)|Xu-Xlξ0,ρ|2dξ+Cpp⨍Bρ⁢(ξ0)|Xu-Xlξ0,ρ|pdξ.

Moreover, for any γ∈(0,1) and ρ≤1,

Cγ(ξ0,ρ)=ρ-2⁢γ⨍Bρ⁢(ξ0)|u-(u)ξ0,ρ|2dξ

≤ρ2-2⁢γCp⨍Bρ⁢(ξ0)|Xu|2dξ

(5.11)≤2ρ2-2⁢γCp⨍Bρ⁢(ξ0)|Xu-(Xu)ξ0,ρ|2dξ+2ρ2-2⁢γCp|(Xu)ξ0,ρ|2.

Keeping in mind the definitions of Σ1 and Σ2, estimates (5.10) and (5.11) imply the existence of a radius 0<ρ<min⁡{ρ*,dist⁡(ξ0,∂⁡Ω)} such that

Ψ⁢(ξ0,ρ,lξ0,ρ)<ε* and Cγ⁢(ξ0,ρ)<κ*

for the constants κ*,ε*,ρ*>0 determined in Lemma 5.3. Using the absolute continuity of the integral, there exists a neighborhood U⊆Ω of ξ0 with

Ψ⁢(ξ,ρ,lξ0,ρ)<ε* and Cγ⁢(ξ,ρ)<κ*

for all ξ∈U. Using Lemma 5.3 in any point ξ∈U yields

Ψ⁢(ξ,θk⁢ρ,lξ0,θk⁢ρ)<ε* and Cγ⁢(ξ,θk⁢ρ)<κ*

for all ξ∈U and all k∈ℕ. This implies

supξ∈U,σ∈(0,ρ)σ-2⁢γ⨍Bσ⁢(ξ)|u-(u)ξ,ρ|2dη=supξ∈U,σ∈(0,ρ)Cγ(ξ,σ)<θ-Q-2⁢γκ*<∞,

and hence u∈C0,γ⁢(U,ℝN), by Campanato’s characterization of Hölder continuous functions. ∎

Proof of Theorem 1.2.

Analogously to (4.3), we have

⨍Br⁢(ξ0)[Aiα⁢(ξ,u,X⁢u)-Aiα⁢(ξ,u,X⁢l)]⁢ϕp⁢(X⁢u-X⁢l)⁢𝑑ξ=I1+I2+I3+I^4,

where I1, I2, I3 are the same terms as those in (4.3). We will just estimate the different term

I^4=⨍Br⁢(ξ0)fα⁢(ξ,u,X⁢u)⁢φα⁢𝑑ξ.

Employing Hölder’s inequality and Young’s inequality, we deduce

I^4≤⨍Br⁢(ξ0)(a⁢|X⁢u|p-ε+b)⁢ϕp⁢|u-l|⁢𝑑ξ

≤a[⨍Br⁢(ξ0)|Xu-Xl|pϕpdξ](p-ε)/p[⨍Br⁢(ξ0)|u-l|p/εdξ]ε/p+⨍Br⁢(ξ0)(b+a|Xl|p-ε)|u-l|ϕdξ

≤a(2M+r|Xl|)1-ε[⨍Br⁢(ξ0)|Xu-Xl|pϕpdξ](p-ε)/p[⨍Br⁢(ξ0)|u-l|pdξ]ε/p

+C⁢(ε1)⁢⨍Br⁢(ξ0)|u-lr|p⁢𝑑ξ+ε1⁢(b⁢r)p′+ε1⁢(a⁢|X⁢l|p-ε⁢r)p′

≤aε2(2M+r|Xl|)1-ε⨍Br⁢(ξ0)|Xu-Xl|pϕpdξ+C(ε1,ε2)⨍Br⁢(ξ0)|u-lr|pdξ+ε1(1+|Xl|)prp′(ap′|Xl|p′+bp′),

where we have used the inequality |u-l|≤(2⁢M+r⁢|X⁢l|). Then we also obtain the Caccioppoli type inequality in Lemma 4.2, and deduce Theorem 1.2 just by following the procedure of the proof of Theorem 1.1. ∎

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11201081

Funding source: Natural Science Foundation of Jiangxi Province

Award Identifier / Grant number: 20142BAB201001

Award Identifier / Grant number: S2016ZRMSB0350

Funding statement: The research of the first author is supported by the National Natural Science Foundation of China (no. 11201081) and the Natural Science Foundation of Jiangxi Province (no. 20142BAB201001 and no. S2016ZRMSB0350)

Acknowledgements

The first author would like to express his gratitude and appreciation to the China Scholarship Council for their support, and to Professor J. Manfredi and the Department of Mathematics for their hospitality and assistance at the University of Pittsburgh.

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Received: 2015-12-31

Revised: 2016-4-6

Accepted: 2016-5-4

Published Online: 2016-7-2

Published in Print: 2018-2-1

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Artikel in diesem Heft

https://doi.org/10.1515/anona-2015-0182

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Partial Hölder continuity; Heisenberg group; nonlinear sub-elliptic system; VMO-coefficient; super-quadratic natural growth

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