# Regularity for a Nonlinear Discontinuous Subelliptic System with Drift on the Heisenberg Group

Authors: Junli Zhang; Jialin Wang
Journal: Advances in Mathematical Physics
 (2022)

---

## Abstract

In this paper, we prove the partial Hölder regularity of weak solutions and the partial Morrey regularity to horizontal gradients of weak solutions to a nonlinear discontinuous subelliptic system with drift on the Heisenberg group by theA-harmonic approximation, where the coefficients in the nonlinear subelliptic system are discontinuous and satisfy the VMO condition for x, ellipticity and growth condition with the growth index 1<p<2 for the Heisenberg gradient variable, and the nonhomogeneous terms satisfy the controllable growth condition and the natural growth condition, respectively.


---

## Body

## 1. Introduction

Kohn in [1] proved L2 estimates for the operator
(1)Lu=∑j=1kXj2u+X0u+cuconstructed by Hörmander’s vector fields X1,X2,⋯,Xq,X0 (see [2]) based on the energy estimate and a subelliptic estimate. Moreover, some authors also inspected the regularity of solutions to linear degenerate elliptic equations with drift term by establishing singular integral estimates. For example, Folland and Stein in [3] established Lp estimates and Lipschitz estimates to the operator on the Heisenberg group
(2)L=−14∑j=1nXj2+Yj2+iαTfor suitable α, where T is the vertical vector field. To the nondivergence linear degenerate elliptic operator
(3)£=∑i,j=1qaijxXiXj+a0xX0constructed by Hörmander’s vector fields, Bramanti and Zhu in [4] established Lp estimates with aijx and a0x belonging to VMO spaces related to X1,X2,⋯,Xq,X0 and Schauder estimates with aijx and a0x being in Hölder spaces for strong solutions. It is important in [4] that the difference between equations without X0 and with X0 was pointed out. When X1,X2,⋯,Xq in (3) is basis vector fields and X0 is the drift vector field on homogeneous groups, many scholars have obtained regularities to the operator £ with coefficients aij and a0 satisfying appropriate conditions, such as [5–8]. In addition, Austin and Tyson in [9] achieved the C∞-smoothness for the operator on the Heisenberg group ℍn(4)L=−14∑i=1nXj2+Yj2±3Tby using the geometric analysis method.Note that the equations studied in the above-cited papers are linear. In this paper, we consider the regularity to the weak solution of discontinuous subelliptic systems with drift termTu on ℍn(5)−∑i=12nXiAikx,u,∇Hu−Tu=Bkx,u,∇Hu,x∈Ω,k=1,2,⋯,N,where Ω is the bounded domain in ℍn, Aik belongs to the vanishing mean oscillation space (which is abbreviated as VMO) and satisfies the ellipticity on ℝ2n×N and polynomial growth conditions with the growth index 1<p<2 for ∇Hu, and also Aik is continuous for u and differentiable for ∇Hu with continuous derivatives,
(6)∇Hu=X1u,X2u,⋯X2nu,Xii=1,2,⋯,2n is the horizontal vector field and T is the vertical vector field in ℍn. For more information about ℍn, see Section 2. The nonhomogeneous term Bk satisfies the controllable growth condition or natural growth condition. We will use the A-harmonic approximation method to conclude the partial Hölder regularity to the weak solutions and the partial Morrey regularity to the horizontal gradients of the weak solutions.More regularity for the elliptic system without drift term, one can refer to [10–12] (Euclidean space) and [13–15] (Heisenberg group).Now, for anyx∈Ω,u,u0∈ℝN,P,P0∈ℝ2n×N, and the growth index 1<p<2, we list the hypotheses that the system satisfies.(H1). LetAik satisfy the following ellipticity and polynomial growth conditions (growth index 1<p<2):
(7)DPAikx,u,PP0,P0≥λ1+Pp−2P02,(8)Aikx,u,P+1+PDPAikx,u,P≤Λ1+Pp−1,where DPAik denote the usual derivative of Aik with respect to the variable P, 0<λ≤1≤Λ<∞.(H2). Assume thatAikx,u,P/1+Pp−1 is continuous for u. More precisely, there exists a bounded, concave, and nondecreasing continuous modulus ω:0,∞⟶0,1 with lims⟶0ωs=0=ω0 such that
(9)Aikx,u,P−Aikx,u0,P≤Λωu−u0p1+Pp−1.(H3). LetAik be differentiable for the variable P with continuous derivatives, that is, there exists a bounded, concave, and nondecreasing continuous modulus ϑ:0,∞⟶0,1 with ϑs≤s,lims⟶0ϑs=0=ϑ0 such that
(10)DPAikx,u,P−DPAikx,u,P0≤ΛϑP−P01+P+P01+P+P0p−2.(H4). For allx∈Brx0, Aikx,u,P/1+Pp−1 satisfies the following VMO condition:
(11)Aikx,u,P−Aik·,u,Px0,r≤νx0x,r1+Pp−1,where νx0:ℝ2n+1×0,r0⟶0,2Λ is a bounded function and satisfies
(12)limr0⟶0νr0=limr0⟶0supx0∈Ωsup0≤r≤r0νx0x,rBrx0∩Ω=0.Here, we have used in (11) and (12) the notation
(13)fx0,r=1Brx0∫Brx0fdx.(HC) (controllable growth condition). The nonhomogeneous termBk satisfies the following controllable growth condition
(14)Bkx,u,P≤c1+up∗−1+Pp1−1/p∗,where c is a positive constant,
(15)p∗=p℘℘−pfor1<p<℘,and ℘ denotes the homogeneous dimension of the Heisenberg group.Obviously, we can see that system (5) includes the system
(16)−∑i=12nXiAikx1+∇Hu2p−2/2Xiuk−Tu=Bkx,u,∇Hu,x∈Ω,k=1,2,⋯,N.We state the main result.Theorem 1.
Assume thatAikx,u,∇Hu and Bkx,u,∇Hu satisfy the assumptions (H1)-(H4) and (HC). If 1<p<2 and u∈HW1,2Ω,ℝN is a weak solution to system (5), i.e., for all φ∈HW01,2Ω,
(17)∫ΩAikx,u,∇Hu∇Hφdx+∫Ωu·Tφdx=∫ΩBkx,u,∇Huφdx,
then, there exists a relatively closed singular setΩ0⊂Ω such that for any α∈0,1, we have
(18)u∈Cloc0,αΩ\Ω0,ℝN.
Moreover, for anyβ=℘−p1−α, we have
(19)∇Hu∈Llocp,βΩ\Ω0,ℝ2n×N,where Llocp,β is a local Morrey space. The singular set Ω0 satisfies
(20)Ω0=0,Ω0⊂∑1⋃∑2,where
(21)∑1=x0∈Ω:limsupr⟶0∇Hux0,r=∞,∑2=x0∈Ω:liminfr⟶01Brx0∫Brx0∇Hu−∇Hux0,r2dx>0.Corollary 2.
Assume thatAikx,u,∇Hu and Bkx,u,∇Hu satisfy the assumptions (H1)-(H4) and the following assumption:
(HN). The nonhomogeneous termBkx,u,∇Hu satisfies the following p natural growth condition
(22)Bkx,u,P≤aPp+b
for∣u∣≤M, where a and b are constants depending only on M.
Then, we have(23)u∈Cloc0,αΩ\Ω0,ℝNand∇Hu∈Llocp,βΩ\Ω0,ℝ2n×Nfor weak solution u∈HW1,2Ω,ℝN to system (5) under the assumption 2aM<λ, where 1<p<2 and Ω0 is same as in Theorem 1.Its proof is direct by combining the proof of Theorem1 in this paper with the proof of Theorem 1.2 in [15].Let us recall that theA-harmonic approximation method was first introduced by Duzaar and Steffen in [16] and then extended to other cases by some authors, see [17–19]. In this paper, we use the A-harmonic approximation method described in [15] to conclude Theorem 1. Different from [15], the system considered by us has a drift term, which brings new challenges to our research. Actually, the processing of drift term are different from that the processings of other terms in the system. Moreover, Lemmas 11–14 in Section 3 used in proving Theorem 1 are different from the corresponding lemmas in [15] and will be rebuilt.This paper is organized as follows: in Section2, we introduce the related knowledge of the Heisenberg group, some function spaces on the Heisenberg group, horizontal affine functions, and some necessary lemmas. In Section 3, we show a Caccioppoli-type inequality for weak solution to (5), the approximately A-harmonic lemma, the decay estimate, and iteration relations. In Section 4, the proof of Theorem 1 is given.


## 2. Preliminaries

### 2.1. The Heisenberg Groupℍn and Some Function Spaces on ℍn

The Euclidean spaceℝ2n+1,n≥1 with the group multiplication
(24)x∘y=x1+y1,x2+y2,⋯,x2n+y2n,t+s+12∑i=1nxiyn+i−xn+iyi,where x=x1,x2,⋯,x2n,t,y=y1,y2,⋯,y2n,s∈ℝ2n+1 leads to the Heisenberg group ℍn. The left invariant vector fields generated by commutation the Lie algebra on ℍn are
(25)Xi=∂xi−xn+i2∂t,Xn+i=∂xn+i+xi2∂t,1≤i≤n,and the only nontrivial commutator of such fields is
(26)T=∂t=Xi,Xn+i=XiXn+i−Xn+iXi,1≤i≤n.We call thatX1,X2,⋯,X2n are the horizontal vector fields on ℍn and T the vertical vector field. Denote the horizontal gradient of a smooth function u on ℍn by
(27)∇Hu=X1u,X2u,⋯,X2nu.The homogeneous dimension ofℍn is ℘=2n+2. The Haar measure in ℍn is equivalent to the Lebesgue measure in ℝ2n+1. We denote the Lebesgue measure of a measurable set E⊂ℍn by E.The Carnot-Carathèodary metric (C-C metric) between two points inℍn is the shortest length of the horizontal curve joining them, denoted by d. The ball induced by the C-C metric is
(28)Bρx=y∈ℍn:dy,x<ρ.Forx=x1,x2,⋯,x2n,t∈ℍn, its Korànyi metric is denoted by
(29)xℍn=∑i=12nxi22+t21/4.The C-C metricd is equivalent to the Korànyi metric
(30)dx,y=x−1∘yℍn.For1≤p<∞,Ω⊂ℍn, the horizontal Sobolev space HWk,pΩ is defined as
(31)HWk,pΩ=u∈LpΩ:∇Hu∈LpΩ,∇H2u∈LpΩ,⋯,∇Hku∈LpΩ,which is a Banach space under the norm
(32)uHWk,pΩ=uLpΩ+∑m=1k∇HmuLpΩ.The local horizontal Sobolev spaceHWlock,pΩ is
(33)HWlock,pΩ≔u:u∈HWk,pΩ′,∀Ω′⊂⊂Ω,and the space HW0k,pΩ is the closure of C0∞Ω in HWk,pΩ.Similar to the definition in [20], Morrey space and Campanato space on Heisenberg group are defined as follows.Definition 3 (Morrey space).
Let1≤p<∞andβ≥0. For the function g∈LpΩ, if
(34)gLp,βΩ≔supx∈Ω,0<r<diamΩr−β∫Ωx,rgypdy1/p<∞,then, we say that g belongs to the Morrey space denoted by Lp,βΩ, where Ωx,r=Ω∩Brx.Definition 4 (Campanato space).
Let1≤p<∞andβ≥0. For the function g∈LpΩ, if
(35)gp,β≔supx∈Ω,0<r<diamΩr−β∫Ωx,rgy−gΩx,rpdy1/p<∞,then, we say that g belongs to the Campanato space denoted by Lp,βΩ, and its norm is defined as
(36)gLp,βΩ=gp,β+gLp.Lemma 5 (see [21, 22]).
If for any1<p<∞,0<α<1, we have g∈Lp,℘+pαΩ, then g∈C0,αΩ.Lemma 6 (Sobolev inequality, [23]).
ForBr⊂ℍn, 1≤q<℘=2n+2 and for any u∈HW01,qBr, it holds
(37)1Br∫Bru℘q/℘−qdx℘−q/℘q≤cr1Br∫Br∇Huqdx1/q,where c=c℘,q>0.Then, the following four lemmas are true.For the proof of Lemma6, see [24] and [25].


### 2.2. Horizontal Affine Function and Some Lemmas

Letu∈L2Brx0,ℝN, x and x0∈ℝ2n+1. Denote the horizontal components of x,x0 by
(38)x′=x1,⋯,x2n,x′0=x10,⋯,x2n0.Letl:ℝ2n⟶ℝN be a horizontal affine function. Following [13], if the horizontal affine function
(39)lx0,rx′=lx0,rx′0+∇Hlx0,rx′−x′0is a minimizer of the functional
(40)l⟶1Brx0∫Brx0u−l2dx,then, we have
(41)lx0,rx′0=ux0,r=1Brx0∫Brx0udx,∇Hlx0,r=℘−2c0℘℘+2r21Brx0∫Brx0u⊗x′−x′0dx,where u⊗x′−x′0 stands for the matrix ukxi−xi0N×2n,k=1,⋯,N,i=1,⋯,2n, and c0 is a positive constant defined as
(42)c0=∫0πsinθndθ∫0πsinθn−1dθ=2m−2!!22m−1!!2m−3!!2π,n=2m−1,2m−1!!22m!!2m−2!!π2,n=2m.According to the meaning oflx0,r, one has the following Poincaré inequality ([13]):
(43)1Brx0∫Brx0u−lx0,rx′pdx1/p≤Cpr1Brx0∫Brx0∇Hu−∇Hlx0,rqdx1/q,where 1<q<℘,1≤p≤q℘/℘−q.Throughout the paper, we define(44)Vς=1+ς2p−2/4ς,1<p<2,ζ∈ℝN.Lemma 7 (see [26]).
For anyζ1,ζ2∈ℝN and s>0, it holds
(1)
2−1/2minζ1,ζ1p/2≤2p−2/4minζ1,ζ1p/2≤Vζ1≤minζ1,ζ1p/2(2)
Vsζ1≤maxs,sp/2Vζ1(3)
Vζ1+ζ2≤cpVζ1+Vζ2(4)
p/2ζ1−ζ2≤Vζ1−Vζ2/1+ζ12+ζ22p−2/4≤cp,kζ1−ζ2(5)
Vζ1−Vζ2≤cp,kVζ1−ζ2(6)
Vζ1−ζ2≤cp,MVζ1−Vζ2 for ζ2 with ζ2≤MLemma 8 (Sobolev-Poincaré-type inequality, [15]).
Let1<p<2 and u∈HW1,pBrx0,ℝN with Brx0⊂Ω. Then, it follows
(45)1Brx0∫Brx0Vu−ux0,rr2p∗/pdxp/2p∗≤Cp1Brx0∫Brx0V∇Hu2dx1/2,where p∗=p℘/℘−p and Cp depends only on ℘,N,p. In particular, we have
(46)1Brx0∫Brx0Vu−ux0,rr2dx1/2≤Cp1Brx0∫Brx0V∇Hu2dx1/2.
LetA∈BilΩ×ℝN×ℝ2n×N,ℝ2n×N be a bilinear form with constant tensorial coefficients. We recall that a map h∈C∞Brx0,ℝN is A-harmonic if and only if it holds
(47)1Brx0∫Brx0A∇Hh,∇Hϕdx=0for any testing function ϕ∈C0∞Brx0,ℝN.Lemma 9 (see [15]).
Leth∈HW1,1Ω,ℝN be a weak solution of the constant coefficient system
(48)−∑i=12nXiAik∇Hh=0,k=1,⋯,N.
Then,h is smooth and there exists c≥1 such that for any Brx0⊂Ω,
(49)supBr/2x0∇Hh2+∇H2h2≤cr−21Brx0∫Brx0∇Hh2dx.Lemma 10 (see [15]).
Given0<υ≤L,1<p<2, for any ε>0, there exist constants ρ∈0,1 and δ=δ℘,N,p,υ,L,ε∈0,1 and a bilinear form A on ℝ2n×N satisfying that for P,P¯∈ℝ2n×N,
(50)AP,P≥υP2,AP,P¯≤LPP¯.
Ifw∈HW1,pBrx0,ℝN is an approximate A-harmonic map, i.e., for any ϕ∈C0∞Brx0,ℝN, it holds
(51)1Brx0∫Brx0A∇Hw,∇Hϕdx≤δρsupBrx0∇Hϕ,(52)1Brx0∫Brx0V∇Hwdx≤ρ2,
then, there exists aA-harmonic map h∈C∞Brx0,ℝN satisfying
(53)1Brx0∫Brx0Vw−ρhr2dx≤ρ2ε,1Brx0∫Brx0V∇Hh2dx≤1.


## 2.1. The Heisenberg Groupℍn and Some Function Spaces on ℍn

The Euclidean spaceℝ2n+1,n≥1 with the group multiplication
(24)x∘y=x1+y1,x2+y2,⋯,x2n+y2n,t+s+12∑i=1nxiyn+i−xn+iyi,where x=x1,x2,⋯,x2n,t,y=y1,y2,⋯,y2n,s∈ℝ2n+1 leads to the Heisenberg group ℍn. The left invariant vector fields generated by commutation the Lie algebra on ℍn are
(25)Xi=∂xi−xn+i2∂t,Xn+i=∂xn+i+xi2∂t,1≤i≤n,and the only nontrivial commutator of such fields is
(26)T=∂t=Xi,Xn+i=XiXn+i−Xn+iXi,1≤i≤n.We call thatX1,X2,⋯,X2n are the horizontal vector fields on ℍn and T the vertical vector field. Denote the horizontal gradient of a smooth function u on ℍn by
(27)∇Hu=X1u,X2u,⋯,X2nu.The homogeneous dimension ofℍn is ℘=2n+2. The Haar measure in ℍn is equivalent to the Lebesgue measure in ℝ2n+1. We denote the Lebesgue measure of a measurable set E⊂ℍn by E.The Carnot-Carathèodary metric (C-C metric) between two points inℍn is the shortest length of the horizontal curve joining them, denoted by d. The ball induced by the C-C metric is
(28)Bρx=y∈ℍn:dy,x<ρ.Forx=x1,x2,⋯,x2n,t∈ℍn, its Korànyi metric is denoted by
(29)xℍn=∑i=12nxi22+t21/4.The C-C metricd is equivalent to the Korànyi metric
(30)dx,y=x−1∘yℍn.For1≤p<∞,Ω⊂ℍn, the horizontal Sobolev space HWk,pΩ is defined as
(31)HWk,pΩ=u∈LpΩ:∇Hu∈LpΩ,∇H2u∈LpΩ,⋯,∇Hku∈LpΩ,which is a Banach space under the norm
(32)uHWk,pΩ=uLpΩ+∑m=1k∇HmuLpΩ.The local horizontal Sobolev spaceHWlock,pΩ is
(33)HWlock,pΩ≔u:u∈HWk,pΩ′,∀Ω′⊂⊂Ω,and the space HW0k,pΩ is the closure of C0∞Ω in HWk,pΩ.Similar to the definition in [20], Morrey space and Campanato space on Heisenberg group are defined as follows.Definition 3 (Morrey space).
Let1≤p<∞andβ≥0. For the function g∈LpΩ, if
(34)gLp,βΩ≔supx∈Ω,0<r<diamΩr−β∫Ωx,rgypdy1/p<∞,then, we say that g belongs to the Morrey space denoted by Lp,βΩ, where Ωx,r=Ω∩Brx.Definition 4 (Campanato space).
Let1≤p<∞andβ≥0. For the function g∈LpΩ, if
(35)gp,β≔supx∈Ω,0<r<diamΩr−β∫Ωx,rgy−gΩx,rpdy1/p<∞,then, we say that g belongs to the Campanato space denoted by Lp,βΩ, and its norm is defined as
(36)gLp,βΩ=gp,β+gLp.Lemma 5 (see [21, 22]).
If for any1<p<∞,0<α<1, we have g∈Lp,℘+pαΩ, then g∈C0,αΩ.Lemma 6 (Sobolev inequality, [23]).
ForBr⊂ℍn, 1≤q<℘=2n+2 and for any u∈HW01,qBr, it holds
(37)1Br∫Bru℘q/℘−qdx℘−q/℘q≤cr1Br∫Br∇Huqdx1/q,where c=c℘,q>0.Then, the following four lemmas are true.For the proof of Lemma6, see [24] and [25].


## 2.2. Horizontal Affine Function and Some Lemmas

Letu∈L2Brx0,ℝN, x and x0∈ℝ2n+1. Denote the horizontal components of x,x0 by
(38)x′=x1,⋯,x2n,x′0=x10,⋯,x2n0.Letl:ℝ2n⟶ℝN be a horizontal affine function. Following [13], if the horizontal affine function
(39)lx0,rx′=lx0,rx′0+∇Hlx0,rx′−x′0is a minimizer of the functional
(40)l⟶1Brx0∫Brx0u−l2dx,then, we have
(41)lx0,rx′0=ux0,r=1Brx0∫Brx0udx,∇Hlx0,r=℘−2c0℘℘+2r21Brx0∫Brx0u⊗x′−x′0dx,where u⊗x′−x′0 stands for the matrix ukxi−xi0N×2n,k=1,⋯,N,i=1,⋯,2n, and c0 is a positive constant defined as
(42)c0=∫0πsinθndθ∫0πsinθn−1dθ=2m−2!!22m−1!!2m−3!!2π,n=2m−1,2m−1!!22m!!2m−2!!π2,n=2m.According to the meaning oflx0,r, one has the following Poincaré inequality ([13]):
(43)1Brx0∫Brx0u−lx0,rx′pdx1/p≤Cpr1Brx0∫Brx0∇Hu−∇Hlx0,rqdx1/q,where 1<q<℘,1≤p≤q℘/℘−q.Throughout the paper, we define(44)Vς=1+ς2p−2/4ς,1<p<2,ζ∈ℝN.Lemma 7 (see [26]).
For anyζ1,ζ2∈ℝN and s>0, it holds
(1)
2−1/2minζ1,ζ1p/2≤2p−2/4minζ1,ζ1p/2≤Vζ1≤minζ1,ζ1p/2(2)
Vsζ1≤maxs,sp/2Vζ1(3)
Vζ1+ζ2≤cpVζ1+Vζ2(4)
p/2ζ1−ζ2≤Vζ1−Vζ2/1+ζ12+ζ22p−2/4≤cp,kζ1−ζ2(5)
Vζ1−Vζ2≤cp,kVζ1−ζ2(6)
Vζ1−ζ2≤cp,MVζ1−Vζ2 for ζ2 with ζ2≤MLemma 8 (Sobolev-Poincaré-type inequality, [15]).
Let1<p<2 and u∈HW1,pBrx0,ℝN with Brx0⊂Ω. Then, it follows
(45)1Brx0∫Brx0Vu−ux0,rr2p∗/pdxp/2p∗≤Cp1Brx0∫Brx0V∇Hu2dx1/2,where p∗=p℘/℘−p and Cp depends only on ℘,N,p. In particular, we have
(46)1Brx0∫Brx0Vu−ux0,rr2dx1/2≤Cp1Brx0∫Brx0V∇Hu2dx1/2.
LetA∈BilΩ×ℝN×ℝ2n×N,ℝ2n×N be a bilinear form with constant tensorial coefficients. We recall that a map h∈C∞Brx0,ℝN is A-harmonic if and only if it holds
(47)1Brx0∫Brx0A∇Hh,∇Hϕdx=0for any testing function ϕ∈C0∞Brx0,ℝN.Lemma 9 (see [15]).
Leth∈HW1,1Ω,ℝN be a weak solution of the constant coefficient system
(48)−∑i=12nXiAik∇Hh=0,k=1,⋯,N.
Then,h is smooth and there exists c≥1 such that for any Brx0⊂Ω,
(49)supBr/2x0∇Hh2+∇H2h2≤cr−21Brx0∫Brx0∇Hh2dx.Lemma 10 (see [15]).
Given0<υ≤L,1<p<2, for any ε>0, there exist constants ρ∈0,1 and δ=δ℘,N,p,υ,L,ε∈0,1 and a bilinear form A on ℝ2n×N satisfying that for P,P¯∈ℝ2n×N,
(50)AP,P≥υP2,AP,P¯≤LPP¯.
Ifw∈HW1,pBrx0,ℝN is an approximate A-harmonic map, i.e., for any ϕ∈C0∞Brx0,ℝN, it holds
(51)1Brx0∫Brx0A∇Hw,∇Hϕdx≤δρsupBrx0∇Hϕ,(52)1Brx0∫Brx0V∇Hwdx≤ρ2,
then, there exists aA-harmonic map h∈C∞Brx0,ℝN satisfying
(53)1Brx0∫Brx0Vw−ρhr2dx≤ρ2ε,1Brx0∫Brx0V∇Hh2dx≤1.


## 3. Some Lemmas

For convenience, we introduce some notations:(54)p′=pp−1,p∗=p℘℘−p,p∗′=p∗p∗−1,fx0,r=rp′1Brx0∫Brx0∇Hup+up∗+1dxp′/p∗′+r21Brx0∫Brx0∇Hup+up∗+1dx2/p∗′,Φr≔Φx0,r,l≔1Brx0∫Brx0V∇Hu−∇Hl2dx,Ψr≔Ψx0,r,l≔1Brx0∫Brx0Vu−lr2dx+1Brx0∫Brx0u−lr2dx,Ψ∗r≔Ψ∗x0,r,l≔Ψx0,r,l+ω1Brx0∫Brx0u−lx′0pdx+νr+fx0,r, where
(55)νr=supx0∈Ωsup0≤r′≤rνx0x,r′Br′x0∩Ω.Lemma 11 (Caccioppoli-type inequality).
Letu∈HW1,2Ω,ℝN be a weak solution to (5) under the assumptions (H1)-(H4) and (HC). Then, for any x0=x′0,t0∈Ω,Brx0⊂⊂Ω and the horizontal affine function l:ℝ2n⟶ℝN with lx′0+∇Hl≤M0, we have
(56)Φx0,r2,l≤CcΨ∗x0,r,l,
whereCc is a positive constant depending on ℘,p,λ,Λ,M0.Proof.
We choose a standard cut-off functionη∈C0∞Brx0,0,1 with η≡1 on Br/2x0 and
(57)∇Hη≤4r,Tη≤cr2.
Taking a testing functionφ=η2u−l in (17), we have
(58)∫Brx0η2Aikx,u,∇Hu∇Hu−∇Hldx=−2∫Brx0ηAikx,u,∇Huu−l∇Hηdx+∫Brx0η2Bkx,u,∇Huu−ldx−∫Brx0u·Tη2u−ldx.
Dividing the equality above by the measure of the ball, it yields(59)1Brx0∫Brx0η2Aikx,u,∇Hu∇Hu−∇Hldx=−21Brx0∫Brx0ηAikx,u,∇Huu−l∇Hηdx+1Brx0∫Brx0η2Bkx,u,∇Huu−ldx−1Brx0∫Brx0u·Tη2u−ldx.
Note thatAik·,lx′0,∇Hlx0,r is a constant, so it infers by using the integration by parts that
(60)0=1Brx0∫Brx0Aik·,lx′0,∇Hlx0,r∇Hφdx.
Owing to(61)−1Brx0∫Brx0η2Aikx,u,∇Hl∇Hu−∇Hldx=21Brx0∫Brx0ηAikx,u,∇Hlu−l∇Hηdx−1Brx0∫Brx0Aikx,u,∇Hl∇Hφdx,we substitute the left and right hand sides of (60) and (61) into the left and right hand sides of (59), respectively, to obtain
(62)1Brx0∫Brx0η2Aikx,u,∇Hu−Aikx,u,∇Hl∇Hu−∇Hldx=21Brx0∫Brx0ηAikx,u,∇Hl−Aikx,u,∇Huu−l∇Hηdx−1Brx0∫Brx0Aikx,u,∇Hl∇Hφdx+1Brx0∫Brx0Aik·,lx′0,∇Hlx0,r∇Hφdx+1Brx0∫Brx0η2Bkx,u,∇Huu−ldx−1Brx0∫Brx0u·Tη2u−ldx.
Then, adding and subtracting the same term1/Brx0∫Brx0Aikx,lx′0,∇Hl∇Hφdx on the right hand side of the above equality, it gets
(63)I0≔1Brx0∫Brx0η2Aikx,u,∇Hu−Aikx,u,∇Hl∇Hu−∇Hldx=21Brx0∫Brx0ηAikx,u,∇Hl−Aikx,u,∇Huu−l∇Hηdx+1Brx0∫Brx0Aikx,lx′0,∇Hl−Aikx,u,∇Hl∇Hφdx+1Brx0∫Brx0Aik·,lx′0,∇Hlx0,r−Aikx,lx′0,∇Hl∇Hφdx+1Brx0∫Brx0η2Bkx,u,∇Huu−ldx−1Brx0∫Brx0u·Tη2u−ldx≕2I1+I2+I3+I4+I5.
The treatments to the termsI0,I1,⋯,I4 in (63) are similar to that of Lemma 4.1 in [15], and we simply write the processes of proofs. By (7), (44) and the known inequality 1+a+b−a2≤31+a2+b−a2, it gains
(64)I0≥λ31+M02p−2/21Brx0∫Brx0η2V∇Hu−∇Hl2dx.
By (8), ∇Hη≤4/r, Young’s inequality and Lemma 7, we have
(65)I1≤2ε1Brx0∫Brx0η2V∇Hu−∇Hl2dx+cp,Λ,M0ε1/1−p1Brx0∫Brx0Vu−lr2dx.
It implies from (9), Young’s inequality, Jensen’s inequality and Lemma 7 that
(66)I2≤2ε1Brx0∫Brx0η2V∇Hu−∇Hl2dx+2ε1Brx0∫Brx0Vu−lr2dx+cp,Λ,M0ε1/1−pω1Brx0∫Brx0u−lx′0pdx.
Using (11), Young’s inequality and Lemma 7, it follows
(67)I3≤2ε1Brx0∫Brx0η2V∇Hu−∇Hl2dx+2ε1Brx0∫Brx0Vu−lr2dx+cp,Λ,M0ε1/1−pνr.
We have by using (14), Hölder’s inequality, Lemma 6, Young’s inequality and Lemma 7 that
(68)I4≤4cpε1Brx0∫Brx0η2V∇Hu−∇Hl2dx+4cpε1Brx0∫Brx0Vu−lr2dx+cp,εrp′1Brx0∫Brx0∇Hup+up∗+1dxp′/p∗′+cp,εr21Brx0∫Brx0∇Hup+up∗+1dx2/p∗′.
The remaining task is to deal withI5. Noting l is independent of t and so
(69)Tl=0,we use Tη≤c/r2 to obtain
(70)I5=−1Brx0∫Brx0u·Tη2u−ldx=−1Brx0∫Brx0u−l·Tη2u−ldx=−1Brx0∫Brx0ηu−l2Tηdx≤cBrx0∫Brx0u−lr2dx.
Now, substituting (64)–(70) into (63), and taking ε small enough, it implies
(71)1Br/2x0∫Br/2x0V∇Hu−∇Hl2dx≤Cc1Brx0∫Brx0Vu−lr2dx+1Brx0∫Brx0u−lr2dx+Ccω1Brx0∫Brx0u−lx′0pdx+Ccνr+Ccrp′1Brx0∫Brx0∇Hup+up∗+1dxp′/p∗′+Ccr21Brx0∫Brx0∇Hup+up∗+1dx2/p∗′.
Then, (56) is proved.Lemma 12 (approximatelyA-harmonic lemma).
Assume the assumptions of Theorem1 are satisfied. For B2rx0⊂Ω with r≤r0 and a horizontal affine function l:ℝ2n⟶ℝN with lx′0+∇Hl≤M0, we define
(72)A=DPAik·,lx′0,∇Hlx0,r,w=u−lthen, for all ϕ∈C0∞Brx0,ℝN, it follows
(73)1Brx0∫Brx0A∇Hw,∇Hϕdx≤c1Ψ∗2r+Ψ∗1/22r+Ψ∗1/p2r+ϑΨ∗1/22r+ϑΨ∗1/p2rsupBrx0∇Hϕ,where c1=Cp,M0,Λ,Cc. Here, we say that w is an approximately A-harmonic map.Proof.
A direct calculation gives(74)1Brx0∫Brx0A∇Hw,∇Hϕdx=1Brx0∫Brx0∫01DPAik·,lx′0,∇Hlx0,r−DPAik·,lx′0,∇Hl+s∇Hwx0,r∇Hw·∇Hϕdsdx+1Brx0∫Brx0∫01DPAik·,lx′0,∇Hl+s∇Hwx0,r∇Hw·∇Hϕdsdx≕J1+J2.
The treatment ofJ1 is similar to that of Lemma 4.2 in [15]. In fact, we use (10), the monotonicity of ϑ, Lemma 7, Young’s inequality, Jensen’s inequality and Hölder’s inequality to gain
(75)J1≤Cp,M0,ΛΦr+ϑΦ1/2r+ϑΦ1/prsupBrx0∇Hϕ.
Now, let us estimateJ2. Since
(76)1Brx0∫Brx0Aik·,lx′0,∇Hlx0,r∇Hφdx=0,we see
(77)J2=1Brx0∫Brx0Aik·,lx′0,∇Hux0,r−Aik·,lx′0,∇Hlx0,r∇Hϕdx=1Brx0∫Brx0Aik·,lx′0,∇Hux0,r∇Hϕdx.
Noting from (17) that
(78)1Brx0∫Brx0Aikx,u,∇Hu∇Hϕdx+1Brx0∫Brx0u·Tϕdx−1Brx0∫Brx0Bkx,u,∇Huϕdx=0,so we have
(79)J2=1Brx0∫Brx0Aik·,lx′0,∇Hux0,r∇Hϕdx−1Brx0∫Brx0Aikx,u,∇Hu∇Hϕdx+1Brx0∫Brx0Bkx,u,∇Huϕdx−1Brx0∫Brx0u·Tϕdx=1Brx0∫Brx0Aik·,lx′0,∇Hux0,r−Aikx,lx′0,∇Hu∇Hϕdx+1Brx0∫Brx0Aikx,lx′0,∇Hu−Aikx,u,∇Hu∇Hϕdx+1Brx0∫Brx0Bkx,u,∇Huϕdx−1Brx0∫Brx0u·Tϕdx≕J21+J22+J23+J24.
The treatments ofJ21 and J22 are similar to that of Lemma 4.2 in [15]. To be specific, by (11), Lemma 7, Young’s inequality and lx′0+∇Hl≤M0, one has
(80)J21≤Cp,M0,Λνr+ΦrsupBrx0∇Hϕ.
We use (9), Young’s inequality, Jensen’s inequality and Lemma 7 to get
(81)J22≤Cp,M0,Λω1Brx0∫Brx0u−lx′0pdx+ΦrsupBrx0∇Hϕ.
It is worth noting that the treatments ofJ23 and J24 are different from that in [15]. Using the assumption (14), Hölder’s inequality and Lemma 6, we obtain
(82)J23≤c1Brx0∫Brx0∇Hup+up∗+11/p∗′·ϕdx≤cr1Brx0∫Brx0∇Hup+up∗+1dx1/p∗′1Brx0∫Brx0ϕrp∗dx1/p∗≤cr1Brx0∫Brx0∇Hup+up∗+1dx1/p∗′1Brx0∫Brx0∇Hϕpdx1/p≤cΨ∗1/2rsupBrx0∇Hϕ.
Noting(83)T=XiXn+i−Xn+iXi,it implies
(84)J24=−1Brx0∫Brx0u−l·Tϕdx=1Brx0∫Brx0u−l·Xn+iXiϕdx−1Brx0∫Brx0u−l·XiXn+iϕdx=−1Brx0∫Brx0Xn+iu−l·Xiϕdx+1Brx0∫Brx0Xiu−l·Xn+iϕdx≤supBrx0∇Hϕ1Brx0∫Brx0∇Hu−ldx.
In order to deal with1/Brx0∫Brx0∇Hu−ldx, we denote
(85)Ω1=∇Hu−∇Hl≤1∩Brx0,Ω2=∇Hu−∇Hl>1∩Brx0,so
(86)1Brx0∫Brx0∇Hu−ldx=1Brx0∫Ω1∇Hu−ldx+1Brx0∫Ω2∇Hu−ldx≤Ω1Brx01/21Brx0∫Ω1∇Hu−l2dx1/2+Ω2Brx01/p′1Brx0∫Ω2∇Hu−lpdx1/p≤1Brx0∫Ω1∇Hu−l2dx1/2+1Brx0∫Ω2∇Hu−lpdx1/p≤cΦ1/2r+Φ1/pr.
Then,(87)J24≤cΦ1/2r+Φ1/prsupBrx0∇Hϕ.
Now, we replace (80)–(87) in (79) to see
(88)J2≤Cp,M0,ΛΦr+Φ1/2r+Φ1/pr+ω1Brx0∫Brx0u−lx′0pdxsupBrx0∇Hϕ+Cp,M0,Λνr+Ψ∗1/2rsupBrx0∇Hϕ≤Cp,M0,ΛΦr+Φ1/2r+Φ1/pr+Ψ∗r+Ψ∗1/2rsupBrx0∇Hϕ.
Finally, we substitute (75) and (88) into (74) and then use Lemma 11 to get
(89)1Brx0∫Brx0A∇Hw,∇Hϕdx≤Cp,M0,ΛΦr+Φ1/2r+Φ1/pr+Ψ∗r+Ψ∗1/2r+ϑΦ1/2r+ϑΦ1/prsupBrx0∇Hϕ≤Cp,M0,Λ,CcΨ∗2r+Ψ∗1/22r+Ψ∗1/p2r+ϑΨ∗1/22r+ϑΨ∗1/p2rsupBrx0∇Hϕ,
i.e., (73) holds.Lemma 13 (decay estimate).
Assume the assumptions of Theorem1 are satisfied and Brx0⊂Ω with r≤r0. For constants θ∈0,1/4,δ=δ℘,N,p,υ,L,θ∈0,1 and ρ∈0,1 from the A-harmonic approximation Lemma 10, we impose the following smallness conditions:
(i)
Ψ∗1/2r<δ/2;(ii)
ρ≔Ψ∗r+δ/2−2Ψ∗1/2r+Ψ∗1/pr+ϑΨ∗1/2r+ϑΨ∗1/pr2≤1.
Then, it holds(90)Ψx0,θr,lx0,θr≤c3θ2Ψ∗x0,r,lx0,r,where lx0,θr and lx0,r denote the minimizing horizontal affine functions with lx0,θrx′0+∇Hlx0,θr≤M0 and lx0,rx′0+∇Hlx0,r≤M0, respectively, and the constant c3 depends only on n,N,p,λ,Λ,δ.Proof.
We divide several steps to prove (90).Step 1.
Let us take(91)w~=u−lx0,rc2,where lx0,r=ux0,r+∇Hlx0,rx′−x′0,c2=maxc1,Cc. We first claim that w~ satisfies the assumptions (51) and (52) Lemma 10.
In fact, forl=lx0,r and any ϕ∈C0∞Brx0,ℝN, we have by using Lemma 12, Lemma 7 (2), and assumptions (ii) and (i) that
(92)1Br/2x0∫Br/2x0A∇Hw~,∇Hϕdx≤ρc1c2Ψ∗r+Ψ∗1/2r+Ψ∗1/pr+ϑΨ∗1/2r+ϑΨ∗1/prρsupBr/2x0∇Hϕ≤ρΨ∗1/2r+δ2supBr/2x0∇Hϕ≤ρδsupBr/2x0∇Hϕ.
Now, it deduces from Lemma7 (2) and Lemma 11 that
(93)1Br/2x0∫Br/2x0V∇Hw~2dx≤1c221Br/2x0∫Br/2x0V∇Hu−∇Hlx0,r2dx=Φr/2,lx0,rc22≤CcΨ∗rc22≤ρ2.
Then, the assumptions (51) and (52) of Lemma 10 are satisfied by (92) and (93).
Using Lemma10, it follows that there exists an A-harmonic function h∈C∞Br/2x0,ℝNsatisfies
(94)1Br/2x0∫Br/2x0Vw~−ρhr2dx≤ρ2εand
(95)1Br/2x0∫Br/2x0V∇Hh2dx≤1.Step 2.
We estimate1/Bθrx0∫Bθrx0Vu−lx0,θr/θr2dx.
Let us denote(96)lhx=hx0,θr+∇Hhx0,θrx′−x′0,and compute by (3) and (2) of Lemma 7, Lemma 8 and (94) that
(97)1Bθrx0∫Bθrx0Vw~−ρlhθr2dx=1Bθrx0∫Bθrx0Vw~−ρh+ρh−hx0,θr−∇Hhx0,θrx′−x′0θr2dx≤cp1Bθrx0∫Bθrx0Vw~−ρhθr2dx+cp1Bθrx0∫Bθrx0ρVh−hx0,θr−∇Hhx0,θrx′−x′0θr2dx≤cpθ−22θ−℘1Br/2x0∫Br/2x0Vw~−ρhr2dx+cpCpρ21Bθrx0∫Bθrx0V∇Hh−∇Hhx0,θr2dx≤cpθ−22θ−℘ρ2ε+Cp2ρ2θr21Bθrx0∫Bθrx0V∇H2h2dx.
In order to estimate (97), we need to deal with 1/Bθrx0∫Bθrx0V∇H2h2dx. Noting Lemma 9, it derives h∈C∞Ω,ℝN and
(98)supBr/2x0∇Hh2+∇H2h2≤cr−21Brx0∫Brx0∇Hh2dx,so we have
(99)supBr/2x0∇H2h2≤cr−21Brx0∫Brx0∇Hh2dx,(100)∇Hh≤cr−21Brx0∫Brx0∇Hh2dx1/2≔MinBr/2x0.
Hence, forθ∈0,1/4, we use Lemma 7 (1), (99), (100), Hölder’s inequality and (95) to obtain
(101)1Bθrx0∫Bθrx0V∇H2h2dx≤1Bθrx0∫Bθrx0∇H2h2dx≤supBr/4x0∇H2h2≤cr−21Br/2x0∫Br/2x0∇Hh2dx≤cr−2M1Br/2x0∫Br/2x0∇Hhdx=cr−2M1Br/2x0∫Br/2x0∩∇Hh≤1∇Hhdx+1Br/2x0∫Br/2x0∩∇Hh>1∇Hhdx≤cr−2M1Br/2x0∫Br/2x0∩∇Hh≤1∇Hh2dx1/2+1Br/2x0∫Br/2x0∩∇Hh>1∇Hhpdx1/p≤2cr−2M1Br/2x0∫Br/2x0V∇Hh2dx1/2+1Br/2x0∫Br/2x0V∇Hh2dx1/p≤Cr−2.
By substituting (101) into (97), we get
(102)1Bθrx0∫Bθrx0Vw~−ρlhθr2dx≤cpθ−22θ−℘ρ2ε+CCp2ρ2θr2r−2≤cp,Cp,Cρ2θ−2−℘ε+θ2.
Sinceε>0 in (102) is arbitrary, we take especially ε=θ℘+4. Noting
(103)Ψ∗r<1and2p>1,we use (ii) and the monotonicity of ϑ to obtain
(104)ρ2=Ψ∗r+δ2−2Ψ∗1/2r+Ψ∗1/pr+ϑΨ∗1/2r+ϑΨ∗1/pr2≤Ψ∗r+16δ−2Ψ∗1/2r+Ψ∗1/pr2≤Ψ∗r+64δ−2Ψ∗r.
Therefore, it shows from (102) that
(105)1Bθrx0∫Bθrx0Vw~−ρlhθr2dx≤cp,Cp,Cθ21+64δ−2Ψ∗r.
Now, we substitutew~=u−lx0,r/c2 into (105) and use Lemma 7 (2) to gain
(106)1Bθrx0∫Bθrx0Vu−lx0,r−c2ρlhθr2dx≤c22cp,Cp,Cθ21+64δ−2Ψ∗r.
Sincelx0,θr is a minimizer of 1/Bθrx0∫Bθrx0u−l2dx, we have
(107)1Bθrx0∫Bθrx0Vu−lx0,θrθr2dx≤1Bθrx0∫Bθrx0Vu−lx0,r−c2ρlhθr2dx≤c22cp,Cp,Cθ21+64δ−2Ψ∗r≤c3θ2Ψ∗r.Step 3.
We estimate1/Bθrx0∫Bθrx0u−lx0,θr/θr2dx.
To do so, let us first deal with1/Bθrx0∫Bθrx0w~−ρlh/θr2dx. Since Vw~−ρh/r is bounded almost everywhere by (94), we denote its upper bound by M1. It implies by Lemma 7 (1) that
(1)
whenw~−ρh/r≤1, it follows(108)w~−ρhr≤2Vw~−ρhr≤2M1(2)
whenw~−ρh/r>1, we have(109)w~−ρhr≤2Vw~−ρhr2/p≤2M12/p
Hence,(110)w~−ρhr≤max2M1,2M12p≔M2.
Now, by using(111)lhx=hx0,θr+∇Hhx0,θrx′−x′0,
Lemma8, (101), Lemma 7 (1) and (2), (94) and the similar proof to (102), we get
(112)1Bθrx0∫Bθrx0w~−ρlhθr2dx=1Bθrx0∫Bθrx0w~−ρlh+ρh−hx0,θr−∇Hhx0,θrx′−x′0θr2dx≤21Bθrx0∫Bθrx0w~−ρhθr2dx+ρ21Bθrx0∫Bθrx0h−hx0,θr−∇Hhx0,θrx′−x′0θr2dx≤2θ−22θ−℘1Br/2x0∫Br/2x0w~−ρhr2dx+Cpρ21Bθrx0∫Bθrx0∇Hh−∇Hhx0,θr2dx≤22−℘M22θ−2−℘1Br/2x0∫Br/2x0w~−ρh2M2r2dx+Cp2ρ2θr21Bθrx0∫Bθrx0∇H2h2dx≤221−℘M22θ−2−℘1Br/2x0∫Br/2x0Vw~−ρh2M2r2dx+CCp2ρ2θr2r−2≤22−℘θ−2−℘1Br/2x0∫Br/2x0Vw~−ρhr2dx+CCp2ρ2θr2r−2≤2θ−2−℘2−℘ρ2ε+CCp2ρ2θr2r−2≤cp,Cp,Cρ2θ−2−℘ε+θ2≤cp,Cp,Cθ21+64δ−2Ψ∗r.
We substitutew~=u−lx0,r/c2 into the inequality above to obtain
(113)1Bθrx0∫Bθrx0u−lx0,r−c2ρlhθr2dx≤c22cp,Cp,Cθ21+64δ−2Ψ∗r,and know from that lx0,θr is a minimizer of 1/Bθrx0∫Bθrx0u−l2dx that
(114)1Bθrx0∫Bθrx0u−lx0,θrθr2dx≤c22cp,Cp,Cθ21+64δ−2Ψ∗r≤c3θ2Ψ∗r.Step 4.
Combining (107) with (114), we derive (90). Lemma 13 is proved.
Before stating a new lemma, we introduce Campanato-type functions. For the fixed Hölder exponentα∈0,1, define a Campanato-type function by
(115)Yαx0,r=r−pα1Brx0∫Brx0u−ux0,rpdx,1<p<2.
We can prove the following lemma from Lemma13.Lemma 14 (iteration relations).
Assume the assumptions of Theorem1 are satisfied. For any α∈0,1, there exist constants ε∗,κ∗,r∗ and θ∈0,1/8, such that if for 0<r<r∗ with Brx0⊂Ω, one has
D0Ψx0,r,lx0,r<ε∗andYαx0,r<κ∗,
then, for anyk∈ℕ, it holds
DkΨx0,θkr,lx0,θkr<ϵ∗andYαx0,θkr<κ∗.Proof.
Its proof is similar to Lemma 4.4 in [15]. Actually, Ψ∗θkr≤ε∗ has been proved in [15], so we only need to take ε∗≤minθ℘+pρ/82/p,δ2/4δ2+256 and change the estimate of ρθkr in [15] to
(116)ρθkr≔Ψ∗θkr+δ2−2Ψ∗1/2θkr+Ψ∗1/pθkr+ϑΨ∗1/2θkr+ϑΨ∗1/pθkr2≤4ε∗+16δ−24ε∗+4ε∗p2≤4ε∗+16δ−224ε∗2=4δ2+256δ2ε∗≤1.


## 4. Proof of Theorem1

Proof of Theorem1 is finished with two steps.Step 1.
We proveu∈Cloc0,αΩ\Ω0,ℝN. In fact, by Lebesgue’s differentiation theorem ([27]), we get ∑1∪∑2=0, so our aim is to show that u is Hölder continuous for every x0∈Ω\∑1∪∑2. For any 0<r0<distx0,∂Ω, we use Lemma 8, (43) and Lemma 7 (1) to gain
(117)Ψx0,r0,lx0,r0=1Br0x0∫Br0x0Vu−lx0,r0r02dx+1Br0x0∫Br0x0u−lx0,r0r02dx≤cp21Br0x0∫Br0x0V∇Hu−∇Hlx0,r02dx+1Br0x0∫Br0x0∇Hu−∇Hlx0,r02dx≤2cp21Br0x0∫Br0x0∇Hu−∇Hlx0,r02dx.
For anyα∈0,1 and r0≤1, using
(118)lx0,r0=ux0,r0+∇Hlx0,r0x′−x′0
(from it, one seesux0,r0=lx0,r0−∇Hlx0,r0x′−x′0), lx0,r0+∇Hlx0,r0≤M0 and Hölder’s inequality, it infers
(119)Yαx0,r0=r0−pα1Br0x0∫Br0x0u−ux0,r0pdx=r0p−pα1Br0x0∫Br0x0u−ux0,r0r0pdx≤r0p−pα1Br0x0∫Br0x0u−lx0,r0r0+∇Hlx0,r0pdx≤Cpr0p−pα1Br0x0∫Br0x0u−lx0,r0r0pdx+∇Hlx0,r0p≤Cp,M0r0p−pα1Br0x0∫Br0x0u−lx0,r0r02dxp/2+1=Cp,M0r0p−pαΨx0,r0,lx0,r0p/2+1,where x′−x′0≤dx,x0≤r0 is used in the third inequality.
By the definition of∑1 and ∑2, (117) and (119), we know that for any ε∗ and κ∗, there exists a radius r′:0<r′<minr∗,distx0,∂Ω such that
(120)Ψx0,r′,lx0,r′<ε∗,Yαx0,r′<κ∗.
Using the continuity of integrals, it follows that there exists a neighborhoodU⊆Ω of x0 so that for any x∈U,
(121)Ψx,r′,lx0,r′<ε∗,Yαx,r′<κ∗.
Then, Lemma14 shows
(122)Ψx,θkr′,lx0,θkr′<ε∗,Yαx,θkr′<κ∗,∀x∈U,k∈ℕ,so
(123)supx∈U,r∈0,r′r−pα1Brx∫Brxu−ux,rpdy=supx∈U,r∈0,r′Yαx,r<κ∗<∞,
i.e.,u∈Lp,℘+pαU,ℝ2n×N. Therefore, we have u∈Cloc0,αU,ℝN from Lemma 5.Step 2.
We prove∇Hu∈Lp,βΩ\Ω0,ℝ2n×N. For α∈0,1, it implies by Lemma 7 (1) that
(1)
if∇Hu−∇Hlx,r>1, then ∇Hu−∇Hlx,rp≤2V∇Hu−∇Hlx,r2(2)
if∇Hu−∇Hlx,r≤1, then ∇Hu−∇Hlx,rp≤1so
(124)supx∈U,r∈0,r′rp1−α1Brx∫Brx∇Hu−∇Hlx,rpdy≤supx∈U,r∈0,r′rp1−α2Brx∫Brx∩∇Hu−∇Hlx,r>1V∇Hu−∇Hlx,r2dy+1≤supx∈U,r∈0,r′rp1−α2Brx∫BrxV∇Hu−∇Hlx,r2dy+1≤supx∈U,r∈0,r′rp1−α2CcΨx,r,lx,r+ωYαx,r+νr+fx,r+1<∞.
Thus,(125)supx∈U,r∈0,r′r−℘−p1−α∫Brx∇Hupdy≤Cpsupx∈U,r∈0,r′rp1−α1Brx∫Brx∇Hu−∇Hlx,rpdy+∇Hlx,rp≤Cpsupx∈U,r∈0,r′rp1−α1Brx∫Brx∇Hu−∇Hlx,rpdy+M0p<∞,
i.e.,∇Hu∈Lp,βU,ℝ2n×N, where β=℘−p1−α.
Therefore, Theorem1 is proved.



---

*Source: 7853139-2022-01-24.xml*