local regularity for the solutions of the p-Laplacian on the Heisenberg group. The case 1 +

C

local regularity for the solutions of the p-Laplacian on the Heisenberg group. The case 1 +

5

< p ≤ 2

Silvana Marchi

Abstract. We prove the H¨older continuity of the homogeneous gradient of the weak solutionsu∈W_loc^1,pof the p-Laplacian on the Heisenberg groupHⁿ, for 1 +√¹5 < p≤2.

Keywords: degenerate elliptic equations, weak solutions, regularity, higher differentia- bility

Classification: 35D10, 35J60, 35J70

1. Introduction

In this paper we deal with the regularity of the weak solutionsu∈W_loc^1,p(Ω, X), 1 +√¹

5 < p≤2, of the equation

(1) div_H~a(Xu) = 0,

where div_H~a(Xu) = P_2n

k=1X_ka^k(Xu) and a^k(q) = |q|^p−2q_k, k = 1, . . . ,2n.

Here Ω is an open subset of the Heisenberg group Hⁿ, the vector fields X_k, k = 1, . . . ,2n, are the generators of the corresponding Lie algebra with their commutators up to the first order andXu= (X1u, . . . , X2nu).

Our main object is the local H¨older continuity of the homogeneous gradientXu.

To this aim we consider approximate equations and we prove the property uni- formly for their solutions. Then we gain the result for the solutions u of the equation (1) via a limit argument.

Let us recall the definitions of the functional spaces needed (see [7]). For any positive integer j, let us set s = (s₁, . . . , s_j), where s_i ∈ {1, . . . ,2n} for any i= 1, . . . , j, and set|s|=j.

Let us denote by Xs the operator Xs1Xs2. . . Xsj. For any q ≥ 1 and any positive integerh,W^h,q(Hⁿ, X) denotes the set of functionsf ∈L^q(Hⁿ) such that Xsf ∈L^q(Hⁿ) for|s| ≤h, with normkfkh,q=kfkL^q(Hⁿ)+P

|s|≤hkXsfkL^q(Hⁿ).

This work has been performed as a part of a National Research Project supported by MIUR.

W_loc^h,q(Ω, X) is the set of functions f such that ϕf ∈ W^h,q(Hⁿ, X) for any ϕ∈ C₀^∞(Ω).

We say thatu∈W_loc^1,p(Ω, X) is a local weak solution of (1) if (2)

Z

Ωa^k(Xu)X_k(ϕ)dx= 0

for all ϕ ∈ W^1,p(Ω, X) with suppϕ ⊂ Ω. Here and in the following repeated indices denote summation.

We can now state the main results of this paper.

From now on Ω^′ will denote an arbitrary open subset of Ω such that Ω^′ ⊂⊂Ω andB(ρ),ρ >0, will denote any homogeneous ball of radiusρ(see Section 3).

Theorem 1.1. Letu∈W_loc^1,p(Ω, X),1 +√¹

5 < p≤2, be a local weak solution of (1). Then for anyσ∈(0,1) there exists a positive constantγ(σ)depending only onσand the data such that, for any homogeneous ballB(R)⊂⊂Ω^′,

(3) kXuk_∞,B(R−σR)≤γ(σ) 1

|B(R)| Z

B(R)|Xu|^pdx 1/p

.

In particular |Xu| ∈ L^∞_loc(Ω^′) and for every compact K ⊂ Ω^′, there exists a constantC₀>0depending only on the data and ondist(K, ∂Ω^′)such that

kXuk_∞,K ≤C0. Theorem 1.2. Letu∈W_loc^1,p(Ω, X),1 +√¹

5 < p≤2, be a local weak solution of (1). Then for any homogeneous ballB(R)⊂⊂Ω^′ there exist positive constantsν andη∈(0,1)depending only on the data and ondist(B(R), ∂Ω^′)such that

(4) max

i=1,...,2nosc_B(ρ) X_iu≤νρ R

η

sup

B(R/2)|Xu|

for allρ < R/2. In particularXuis locally H¨older continuous inΩ^′, i.e. for every compactK ⊂Ω^′ there existC1 >0 andα∈(0,1) depending only on data and ondist(K, ∂Ω^′)such that

|Xu(x)−Xu(y)| ≤C₁d(x, y)^α, x, y∈K,

whereddenotes the homogeneous distance associated toHⁿ (see Section 3).

Our results extend to the Heisenberg group setting some properties which hold, in the Euclidean context, for the solutions of the p-Laplacian, but even of more general nonlinear elliptic equations.

Let us recall in particular on this subject the papers of K. Uhlenbeck [27], N.N. Ural’tzeva [28], L. Evans [6] forp≥2, and P. Tolksdorf [26], E. Di Benedetto [5] and J.L. Lewis [14] for 1< p <+∞.

In general these methods consist in differentiating the equation and proving that the derivatives of the solutions solve another partial differential equation.

But this procedure does not fit the Heisenberg context due to the lack of commutativity of the vector fields. In fact, even difference quotiens along any left- invariant vector field produce derivatives in the second commutator’s direction.

Ifp= 2 L. Capogna [1] solved this problem for sub-elliptic equations having the p-Laplacian as a prototype, establishing at first a control on the L² norm of the derivatives in the commutator’s direction. This is the key point in the matter. Thanks to this result he could prove the differentiability of the equation and gradient’s H¨older continuity.

In [18] we proved the same result for the p-Laplacian when 2≤p <1+√ 5. Here we extend it to 1 +√¹

5 < p <2. Because of the worsening of the degeneracy, in both cases we are forced to smooth the problem introducing regularized equations

(5) div_H~aǫ(Xuǫ) = 0

for small ǫ > 0, where~aǫ(q) = [(ǫ+|q|²)^(p−2)/2q]. Following an adaptation of Di Benedetto’s method [5] we attempt to obtain “uniform” H¨older continuity forXuǫ. However this method requires differentiability of equations (5) too, that in turn requires a control inL^p of some derivatives ofu_ǫ.

Ifp >2 we could limit ourselves to establish anL^p estimate forT uǫ (T is the second commutator of the vector fields). But the casep <2 is much more tricky.

In fact, besides theL^p estimate for T u_ǫ we proved in [19], here we further need an L^p-control of the derivatives of T u_ǫ along the vector fields. We prove this result via an iterated application of fractional difference quotiens and repeated inclusions between functional spaces. As this is a crucial step we treat it apart in Section 7. The limitp >1 +√¹

5 comes from [19]. A different technique could improve the result.

Finally we would give a brief description of the content of each section. Sec- tions 2 and 3 are devoted to recall basic knowledge and preliminary results. In Section 4 we prove the differentiability of equations (5). We multiply them by a particular test function, defined by double difference quotiens and apply a Lemma of Cutr´ı-Garroni [4] which enables us to commute the vector fields with the dou- ble difference quotiens. Thanks to this tool and the L^p estimate of T uǫ from Section 7, we can then apply Giusti’s method [Giusti] and conclude about the W^2,p local regularity for uǫ (see Theorem 4.1). This is enough to differentiate equations (5).

Thanks to this tool in Sections 5 and 6 we prove boundedness and local H¨older continuity ofX_iu_ǫ by the methods of Di Benedetto [5]. As these estimates are uniform inǫ, this enables us to establish Theorems 1.1 and 1.2 aboutuby standard arguments [13], [14], possibly up to subsequences.

The general plan of this paper is the same as of [19]. The principal differences concern the crucial Sections 4 and 7 and part of Section 6. They are outlined in detail.

We limit ourselves to sketch the remainder, referring the reader to [5], [18], [19]

for a closer examination.

2. Basic knowledge

The Heisenberg groupHⁿis the Lie group whose underlying manifold isR²ⁿ⁺¹ with the following group law: for allx= (x^′, t) = (x1, . . . , x2n, t), y = (y^′, s) = (y1, . . . , y2n, s),

x◦y= (x^′+y^′, t+s+ 2[x^′, y^′]) where [x^′, y^′] :=P_n

i=1(y_ix_i+n−x_iy_i+n).

Hⁿ is a homogeneous group, that is a group with dilations. A norm for Hⁿ which is homogeneous of degree 1 with respect to the dilations is

|x|⁴=|(x^′, t)|⁴=|x^′|⁴+t² for any x= (x^′, t)∈ Hⁿ and the associated distance is

d(x, y) :=|y⁻¹◦x|, x, y∈ Hⁿ, where y⁻¹=−y.

B(x, r) will denote the homogeneous ball centered inx∈ Hⁿwith radiusr >0.

The Lie algebra L(X) of left-invariant vector fields corresponding to Hⁿ is generated by

X_i=∂xi+ 2x_i+n∂t

X_i+n=∂_xi+n−2x_i∂_t T =−4∂t

for i = 1, . . . , n, where [X_i, X_i+n] = −[X_i+n, X_i] = T, i = 1, . . . , n, and [X_j, X_k] = 0 in any other case.

The vector fieldsX_i do not commute with right translations. In particular we cannot interchange them with difference quotiens operators

D_hw(x) = w(x◦h)−w(x)

|h| , D_−hw(x) = w(x◦h⁻¹)−w(x)

−|h|

for anyx∈ Hⁿ,h= (h^′,0),h_i≥0 for anyi= 1, . . . ,2n.

For anyi = 1, . . . ,2n lethⁱ, (hⁱ)⁻¹ be the elements of the group whose j-th component ish_i, or resp. −h_i, ifj=iand 0 otherwise. We have

(6) X_iD_±hi =D_±hiX_i

for everyi= 1, . . . ,2n, butX_kD_±hi6=D_±hiX_k ifk6=i.

For any s > 0 let h^∗_s, (h^∗_s)⁻¹ be the elements of the group whose (2n+1)-th component is s or −s respectively and 0 otherwise. For any s > 0 and any α∈(0,1) let

D_h∗s,αw(x) = w(x◦h^∗_s)−w(x)

s^α , D_−h∗s,αw(x) =w(x◦(h^∗_s)⁻¹)−w(x)

−s^α .

For everyi= 1, . . . ,2nwe have

(7) X_iD_±h∗s,α =D_±h∗s,αX_i. 3. Difference quotiens and a priori bounds

For more details on this argument see also [4], [1]. Let us consider anyw ∈ C₀^∞(Ω) and anyh= (h^′,0) = (h₁, . . . , h_2n,0) withh_i≥0 fori= 1, . . . ,2n.

Remark 3.1. It is easy to show that

(8) D_−hD_hw(x) =2w(x)−w(x◦h)−w(x◦h⁻¹)

−|h|² =D_hD_−hw(x).

Remark 3.2. For any functionw∈L^p(Ω) with compact supportω⊂Ω, for any f ∈L^p/(p_loc ⁻¹⁾(Ω) and for anyhsuch that|h|< d(ω, ∂Ω) we have

(9)

Z

f D_±hw dx=− Z

wD_∓hf dx.

Lemma 3.3(see [4, Lemma 2.7], [18, Lemma 3.3]). For anyw∈C₀^∞(Ω)and for anyi= 1, . . . , n,

(10) X_i(D_−hD_hw(x)) =D_−hD_h(X_iw(x))−h_i+n

2|h|²[(T w)(x◦h)−(T w)(x◦h⁻¹)], (11) X_i+n(D_−hD_hw(x))

=D_−hD_h(X_i+nw(x)) + h_i

2|h|²[(T w)(x◦h)−(T w)(x◦h⁻¹)].

Lemma 3.4(see [4, Lemma 2.9], [18, Lemma 3.4]). For anyw∈C₀^∞(Ω)and for anyi= 1, . . . ,2n,

(12) lim

hi→0D_±hiw=X_iw.

Lemma 3.5([1, Proposition 2.3]). Letp >1and letψ∈L^p_loc(Ω)andg∈C₀^∞(Ω) withω = suppg ⊂⊂Ω. Let i∈ {1, . . . ,2n}. If there are some constants ǫ₀ >0 andC >0such that

(13) sup

0<hi<ǫ0

Z

ω|D_±hiψ|^pdx≤C^p

thenX_iψ∈L^p(ω)andkX_iψkL^p(ω)≤C. Conversely, if X_iψ∈L^p_loc(Ω), then(13) holds for anyω = suppg ⊂⊂Ω, g ∈ C₀^∞(Ω) and C = 2kX_iψkL^p(ω). The same result holds if we substituteD_±hi andX_i byD_±h∗

s,1 and∂_t, respectively.

Lemma 3.6 (see [1, Theorem 2.6], [19, Lemma 3.6]). Let ψ ∈ C^∞(Ω) and let g ∈ C₀^∞(ω), with suppω ⊂⊂ Ω. Then there exists a positive constant C such that, for any smallǫ₀>0and anyp >1

(14)

sup

0<s<ǫ0

Z

Ω|D_±h∗

s,1/2(ψg)|^pdx

≤C

2n

X

i=1

( sup

0<hi<ǫ0

Z

Ω|D_hi(ψg)|^pdx+ sup

0<hi<ǫ0

Z

Ω|D_−hi(ψg)|^pdx )

.

From Lemmas 3.5 and 3.6 we easily deduce

Corollary 3.7. Let the assumptions of Lemma3.6hold true. Then there exists a constantC >0 such that, for any smallǫ₀>0 and anyp >1

(15) sup

0<s<ǫ0

Z

Ω|D_±h∗

s,1/2(ψg)|^pdx≤C Z

Ω|X(ψg)|^pdx.

4. W_loc^2,p regularity for the solutions of the approximate equation This section is devoted to prove the W^2,p local regularity of the solutions of equations (5) and, as a by-product, to differentiate equations (5). This will be a basic tool in order to apply the Di Benedetto’s machinery [5] to obtain uniform boundedness and H¨older continuity of∇u_ǫ (see Sections 5, 6).

Here we will exploit a localW^1,p estimate ofT uǫwhose proof can be found in Section 7.

Theorem 4.1. Let1 +√¹

5 < p≤2and, for anyǫ∈(0,1), letu_ǫ ∈W_loc^1,p(Ω, X) be a local weak solution of(5).

Thenu_ǫ∈W_loc^2,p(Ω, X)and, for anyΩ^′′ ⊂⊂Ω^′ Z

Ω^′′

V_ǫ^p−2|X²uǫ|²dx≤C(Ω^′′,Ω^′, ǫ, Hǫ, p), whereHǫ =R

Ω^′(V_ǫ^p+|uǫ|^p)dx andV_ǫ²=ǫ+|Xuǫ|².

Proof: For notational simplicity we will drop the subscript ǫ and denote the solution of (5) byu. We briefly recall some piece of notation used in the previous sections; for anyǫ >0 and for anyz∈R²ⁿwe will denote

V²(z) =ǫ+|z|²,

W_h²i(x) =ǫ+|Xu(x)|²+|Xu(x◦hⁱ)|², z^hi(θ) =Xu+θh_iD_hiXu,

z^h_kⁱ(θ) =X_ku+θh_iD_hiX_ku.

Let B(3R) be a homogeneous ball of radius 3R such that B(3R) ⊂ Ω^′. For an arbitrary i = 1, . . . , n, let ϕ = −(D_−hiD_hi +D_−hi+nD_hi+n+D_hiD_−hi + D_hi+nD_−hi+n)w, where w=g¹²uandg is a cut-off function between B(R) and B(2R). Let us observe that the existence of cut-off functions in the Heisenberg group follows from standard methods whenever one observes that the horizontal gradient of the gauge distance has length less or equal than 1 (this is a trivial computation from the definition in Section 2). Let us recall that h_i is always assumed to be nonnegative.

In [19] we provedT w∈L^p_loc(Ω^′) (see Theorem 7.1 in Section 7 of the present paper). Thanks to this fact and to Lemma 3.3 we obtainϕ ∈W₀^1,p(Ω, X); this makesϕa right test function for equation (5).

Let us multiply equation (5) for the test functionϕ. On account of Remark 3.2 and Lemma 3.3 we obtain

(16)

0 =

2n

X

k=1

Z

Ω

D_±hia^kD_±hiX_kw dx+

2n

X

k=1

Z

Ω

D_±hi+na^kD_±hi+nX_kw dx

+ Z

Ω

hD_±hiaⁱ⁺ⁿ−D_±hi+naⁱi T w dx

=I₁+I₂+I₃,

where±in I₁, I₂ andI₃ means the sum of the terms corresponding to both the signs. Let us observe thatD_±hia^kD_±hiX_kwdenotes the product of the functions D_±hia^k and D_±hiX_kw. Here and in the following we omit the parentheses for sake of simplicity.

Estimates of I₁ and I₂. Let us observe that, for anyi, k= 1, . . . ,2n

(17)

D_hia^k= 1 h_i

Z 1 0

d

dθa^k(Xu+θh_iD_hiXu)dθ

= Z 1

0 a^k_j(Xu+θh_iD_hiXu)D_hiX_ju dθ

=α^kj_hiD_hiX_ju

whereα^kj_hi :=

Z ₁

0

a^k_j(Xu+θh_iD_hiXu)dθand the sum overj is understood even if not explicitly written. Herea^k_j denotes the derivative of a^kwith respect to its j-th variable.

Using the previous notation we have

(18) a^k_j(z^hi) = (p−2)V^p−4(z^hi)z_k^hiz_j^hi+V^p−2(z^hi)δ_kj whereδ_kj= 1 ifk= 1 andδ_kj= 0 ifk6=j. An easy calculation gives

(19)

2n

X

k,j=1

a^k_j(z^hi)D_hiX_ku D_hiX_ju≥(p−1)V^p−2(z^hi)|D_hiXu|².

In virtue of (17) and (19) we easily obtain

(20)

2n

X

k=1

D_hia^kD_hiX_ku=

2n

X

k,j=1

α^kj_hiD_hiX_ku D_hiX_ju

≥c Z ₁

0

V^p−2(z^hi)dθ|D_hiXu|². By [9, Lemma 8.3] we have

(21)

Z ₁

0

V^p−2(z^hi)dθ≥c W_h^p_i⁻². Hence, from (20) and (21) we have

(22)

2n

X

k,j=1

D_hia^kD_hiX_ku≥cW_h^p_i⁻²|D_hiXu|².

Let us observe that

(23) D_hiX_kw=g¹²D_hiX_ku+ 12g¹¹X_kuD_hig

+ 12g¹¹D_hiuX_kg+ 12g¹¹uD_hiX_kg+ 132ug¹⁰D_higX_kg.

Then, from (22) and (23) we obtain

(24)

2n

X

k=1

Z

Ω

D_hia^kD_hiX_kw dx≥c Z

Ω^′

g¹²W_h^p_i⁻²|D_hiXu|²dx + 12

Z

Ω^′

g¹¹D_hia^kX_ku D_hig dx+ 12 Z

Ω^′

g¹¹D_hia^kD_hiu X_kg dx + 12

Z

Ω^′

g¹¹u D_hia^kD_hiX_kg dx + 132 Z

Ω^′

g¹⁰u D_hia^kD_hiX_kg dx

=J₁+J₂+J₃+J₄+J₅.

Estimate of J2, . . . , J5. Let us observe that, for anyk, j= 1, . . . ,2n, (25) |α^kj_hi| ≤c W_h^pi⁻².

On account of (17), (25), H¨older inequality and the decompositionp−1 = ^p+(p₂⁻²⁾, we have

(26)

|J2|= 12| Z

Ω^′

g¹¹α^kj_hiD_hiX_ju X_ku D_hig dx|

≤c Z

Ω^′g¹¹W_h^pi⁻¹|D_hiX_ju||D_hig|dx

≤δ Z

Ω^′g¹²W_h^pi⁻²|D_hiXu|²dx+cδ⁻¹ Z

Ω^′g¹⁰W_h^pi|D_hig|²dx.

As forh_i < R (27)

Z

B2R

W_h^p_idx≤ Z

B3R

V^pdx,

it follows from (26) and (27) that (28) |J2| ≤δ

Z

Ω^′

g¹²W_h^pi⁻²|D_hiXu|²dx+cδ⁻¹R⁻² Z

Ω^′

V^pdx.

We choose another suitable approach to estimate|J₃|,|J₄| and|J₅|.

To this end, let us observe that for anyi, k= 1, . . . ,2n (29) D_hia^k= 1

h_i Z ₁

0 ∇Ha^k(Xu(x◦δ_θhⁱ))·hⁱdθ

= Z 1

0

X_ia^k(Xu(x◦δ_θhⁱ))dθ=X_iα^k_hi

where the functionsα^k_hi :=R₁

0 a^k(Xu(x◦δ_θhⁱ))dθcan be estimated as (30) |α^k_hi| ≤Y_hi:=

Z ₁

0

(ǫ+|Xu(x◦δ_θhⁱ)|²)^p−12 dθ.

On account of (6), (29) and Remark 3.2 we have

(31)

J3= −12 Z

Ω^′α^k_hiX_i[g¹¹D_hiuX_kg]dx

= −12 Z

Ω^′α^k_hiD_hiX_iu g¹¹X_kg dx

−12 Z

Ω^′

α^k_hiD_hiu[11g¹⁰X_igX_kg+g¹¹X_iX_kg]dx and then, by (30)

(32) |J₃| ≤cR⁻¹ Z

Ω^′

g¹¹Y_hi|D_hiXu|dx+cR⁻² Z

B2R

g¹⁰Y_hi|D_hiu|dx.

The first integral on the right-hand side of (32) can be estimated taking into account the following decomposition

(33)

R⁻¹g¹¹Y_hi|D_hiXu|=R⁻¹g¹¹Y_hiW

2−p 2

hⁱ W

p−2 2

hⁱ |D_hiXu|

≤δg¹²W_h^pi⁻²|D_hiXu|²+cδ⁻¹R⁻²g¹⁰Y_h²iW_h²i^−p

≤δg¹²W_h^p_i⁻²|D_hiXu|²+cδ⁻¹R⁻²g¹⁰(W_h^p_i+Y

p p−1

hⁱ ).

To estimate the second integral on the right-hand side of (32) let us observe at first that

(34) Y_hi|D_hiu| ≤c(|D_hiu|^p+Y

p p−1

hⁱ ).

Moreover (35)

Z

B(2R)

g¹⁰Y

p p−1

hⁱ dx

≤ Z ₁

0 { Z

B(2R)

g¹⁰(1 +|Xu(x◦δ_θhⁱ)|²)^p2 dx}dt≤ Z

B(3R)

V^pdx.

From (27), Lemma 3.5 and (32),. . ., (35) we finally obtain (36) |J₃| ≤δ

Z

Ω^′

g¹²W_h^p_i⁻²|D_hiXu|²dx+cR⁻² Z

Ω^′

V^pdx.

Analogously we have

(37)

J₄= −12 Z

Ω^′

α^k_hiX_i[g¹¹u D_hiX_kg]dx

= −12 Z

Ω^′

α^k_hiX_iu g¹¹D_hiX_kg dx

−12 Z

Ω^′

α^k_hiu[11g¹⁰X_ig D_hiX_kg+g¹¹X_iD_hiX_kg]dx and then, by (30)

(38) |J₄| ≤12R⁻² Z

Ω^′

g¹¹Y_hi|Xu|dx+ 144R⁻³ Z

Ω^′

g¹⁰Y_hiu dx.

Estimating as in (34), (35) we finally obtain

(39) |J₄| ≤cR⁻³

Z

Ω^′

(V^p+|u|^p)dx.

The same holds for|J5|:

(40) |J₅| ≤cR⁻³

Z

Ω^′

(V^p+|u|^p)dx.

From (22), (28), (36), (39), (40) and choosing δ small, we obtain that there exist some positive constantsc andc^′ such that

(41)

2n

X

k=1

Z

Ω^′

D_hia^kD_hiX_kw dx

≥c Z

Ω^′g¹²W_h^pi⁻²|D_hiXu|²dx−c^′R⁻³ Z

Ω^′(V^p+|u|^p)dx.

An analogous result can be obtained switching betweenhⁱand−hⁱ. In conclusion there are some positive constantsc andc^′, possibly different from those in (41), such that

(42) I1=

2n

X

k=1

Z

Ω^′D_±hia^kD_±hiX_kw dx

≥c Z

Ω^′

g¹²W^p−2

±hⁱ|D_±hiXu|²dx−c^′R⁻³ Z

Ω^′

(V^p+|u|^p)dx.

The estimate ofI₂ proceeds exactly in the same way.

Estimate of I₃. By Theorem 7.2, we haveT w∈W_loc^1,p(Ω^′, X) and Z

Ω^′|XT w|^pdx≤C(ǫ, R, H, p)

for a certain positive constantC depending on ǫ, R, H, p. This inequality and the methods applied toJ3, . . . , J5 give now

(43) Z

Ω^′D_hiaⁱ⁺ⁿT w dx ≤c

Z

Ω^′|αⁱ⁺ⁿ_hi | |XT w|dx

≤c Z

Ω^′

Y_hi|XT w|dx≤C(ǫ, R, H, p) for some other constantC(ǫ, R, H, p)>0. The other three terms of I₃, that is R

Ω^′D_−hiaⁱ⁺ⁿT w dx, R

Ω^′D_+hi+naⁱT w dx, R

Ω^′D_−hi+naⁱT w dx can be estimated in the same way. So we obtain

(44) I₃≥ −C(ǫ, R, H, p).

From (42), the analogous estimate of I₂ and (44) we finally obtain, for any i= 1, . . . ,2n,

(45)

Z

Ω^′

g⁶W_h²_i^−p|D_hiXu|²dx≤C(ǫ, R, H, p).

If 2α=p(p−2) then, for anyi= 1, . . . ,2n

(46) |D_hiXu|^p=W_h^αiW_h⁻_i^α|D_hiXu|^p≤W_h^p_i+W_h^p_i⁻²|D_hiXu|².

Inequalities (46), together with (27) and (45) enable us to affirm that, for any i= 1, . . . ,2n

D_hiXuis bounded inL^p(B(R)).

By Lemma 3.4, possibly up to a subsequence,D_hiXuconverges inL^p_loc(B(R)) to XiXuforhⁱ→0 and thenu∈W_loc^2,p(B(R), X).

Moreover we can extract from it a subsequence converging a.e.x∈B(R). By Lemma 3.4

W_hi → (ǫ + 2|Xu|²)^1/2 a.e.x∈B(R) ashⁱ→ 0.

The proof of Theorem 4.1 is then finished passing to the limit hⁱ → 0 in (45) on account that Ω^′′ can be covered by a finite number of ballsB(R) forR small

enough.

Remark 4.2. We would like to point out that, thanks to Theorem 4.1, we can now differentiate formally equationsR

a^k_ǫ(Xu_ǫ)X_kϕ= 0 alongX_i,i= 1, . . . ,2n, obtaining

(47)

Z

B(R)

a^k_ǫ,j(Xuǫ)X_iX_juǫX_kϕ dx= 0 for anyϕ∈W₀^1,p(B(R), X),B(R)⊂Ω^′.

5. Local boundedness of the gradient

Here we are concerned with the uniform, local boundedness ofuǫand∇uǫ(see Propositions 5.1, 5.2, for the proofs we refer to [3] and [18] respectively).

We point out that the proof of Theorem 5.2 insists on the differentiability of equations (5), proved in Theorem 4.1, so its validity is limited to the range 1 +√¹

5 < p≤2. However Proposition 5.1 holds for any p >1.

Let us observe that Theorem 1.1 is an easy consequence of Theorem 5.2, stand- ing its uniform validity, via a standard limit argument ([13], [14]).

Proposition 5.1 ([3, Theorem 3.4]). Let p > 1. For any compact K ⊂ Ω^′ there exists a constantC >0depending only on the structural constants and on dist(K, ∂Ω^′)such that

ku_ǫk_∞,K ≤C for allǫ >0.

Letx₀ ∈Ω^′ be arbitrary fixed and, for anyρ >0, letB(ρ) be the ball centered atx₀ of radiusρ. LetB(R)⊂⊂Ω^′.

Theorem 5.2([18, Theorem 5.2]). Let1 +√¹

5 < p≤2. For anyσ∈(0,1)there exists a constantγ(σ)depending only on the structural constants andσsuch that

k[ǫ+|Xuǫ|²]k_∞,B(R−σR)≤γ(σ) 1

|B(R)| Z

B(R)

[ǫ+|Xuǫ|²]^p/2dx for allǫ >0.

Although the proof of Theorem 5.2 is referred to [18, Theorem 5.2], we want to underline its dependence on the differentiability of equation (5). In fact it is accomplished substituting in (47) the test functionϕ=X_iuǫV_ǫ^αg²,α >0, where g is a cut-off function between B(R−σR) and B(R), σ ∈ (0,1), and applying standard methods.

6. Local H¨older continuity of the gradient

Our purpose is to establish the H¨older continuity of Xu_ǫ at x₀, uniformly in ǫ >0. The technique is due to [6], [5], with few adaptations due to [17].

We will not deal with all the proofs in depth. We will mostly refer to [5], even if we will discuss all needed modifications in details. We outline that Proposi- tion 6.1 holds true for anyp >1, while the validity of Propositions 6.2, 6.4 and Theorem 6.5, which depend on the results of the former section, is limited to the range 1 +√¹

5 < p≤2.

Let us observe that Theorem 1.2 easily follows from Theorem 6.5 via a standard limit technique (see [13], [14]).

The following result can be found in [3, Theorem 3.35].

Proposition 6.1 (Local H¨older continuity of u_ǫ). For any compact K ⊂ Ω^′ there exist constantsC, β∈(0,1)depending only on the structural constants and dist(K, ∂Ω^′)such that

|uǫ(x)−uǫ(y)| ≤C|x−y|^β, x, y∈K, for allǫ >0.

As before letx₀be an arbitrary point of Ω^′and, for anyρ >0,B(ρ) be the ball centered atx₀ of radiusρ. We will chooseR >0 in such a way thatB(2R)⊂Ω^′. Let us now setϕ=±(X_iuǫ−k)^±ξ² in (47), fork∈Randi= 1, . . . ,2n, where ξis a cut-off function with support inB(R). We easily obtain

(48) Z

B(R)

V_ǫ^p−2|X(X_iu_ǫ−k)^±|²ξ²dx≤γ Z

B(R)

V_ǫ^p−2|(X_iu_ǫ−k)^±|²|Xξ|²dx for allǫ >0, where V_ǫ² =ǫ+|Xuǫ|² andγ is a structural constant independent onǫ,R.

Let us observe that, due to Theorem 5.1 and the results of [1], the solutionsuǫ

are now smooth. Therefore, for anyρ≤R,ǫ >0, we can set µ_ǫ(ρ) = max

i sup

B(ρ)|X_iu_ǫ| ω_ǫ(ρ) = max

i osc_B(ρ) X_iu_ǫ. Proposition 6.2. Let2ρ < R. Set

λ= µ_ǫ(2ρ) 2 .

Then there exists a positive constantC₀depending only on the data but inde- pendent ofǫ,R,λ, such that, if for some1≤i≤2n

|{x∈B(2ρ)|X_iu_ǫ< λ}| ≤C₀|B(2ρ)| then

X_iu_ǫ≥ λ

4, ∀x∈B(ρ).

Analogously if

|{x∈B(2ρ)|Xiuǫ>−λ}| ≤C0|B(2ρ)| then

X_iuǫ≤ −λ

4, ∀x∈B(ρ).

Proof: We will drop the subscriptǫ. As in [5, Proposition 4.1] we distinguish be- tweenǫ≥λ²andǫ < λ². In the first simpler case the proof is easily accomplished using (48) as in [5, Proposition 4.1]. Let nowǫ < λ².

Lemma 6.3. Letv=|X_iu|^p/2signX_iu. Then (49)

Z

B(r−σr)|X(v−h)⁻|²dx≤γh²₀(σr)⁻²|A⁻_h,r|

for anyσ ∈(0,1), r≤2ρ, h≤h₀ =λ^p/2 and for a suitable positive structural constantγ, independent onǫ, r,σ,h, whereA⁻_h,r =:{x∈B(r)|v(x)< h}. Proof of Lemma 6.3: We refer the reader to [5, Section 4] for the details of the proof. Here we recall only the main steps of it for convenience of the reader.

If we set in (47) the test functionϕη =−[(|X_iu|+η)^p−2X_iu−k]⁻ξ²,k∈R⁺, whereη is a small positive number which will be let tend to 0 andξ is a cut-off function with support inB(ρ), then we obtain

(50)

Z

B(ρ)

V^p−2|X(v−k

p

2(p−1))⁻|²ξ²dx

≤γ Z

B(ρ)|(|X_iu|^p−2X_iu−k)⁻|²|Xξ|²dx wherev:=|X_iu|^p/2signX_iu, for a structural constantγindependent onǫ. From (50), recalling the definition ofλwe deduce for anyr <2ρ,

(51) λ^p−2 Z

B(r)|X(v−k

p

2(p−1))⁻|²ξ²dx

≤γ Z

B(r)|(|X_iu|^p−2X_iu−k)⁻|²|Xξ|²dx for a new constantγindependent on ǫ,r, σ. If we choosek≤λ^p−1 in (51) and denote byhany number such thath≤h₀ =λ^p/2, then we obtain (49).

Let us now continue theproof of Proposition 6.2.

LetH = sup_B(2r)(v−h₀)⁻. Let us observe that if H < ^h₂⁰, then X_iu > ^λ₄, for anyx∈B(2ρ). Therefore we may assumeH ≥ ^h₂⁰. For any integerj≥0 let

(52) r_j =ρ+ ρ

2^j , h_j=h₀−H 4(1− 1

2^j), B_j =B(r_j), A_j =A⁻_h

j,r_j, µ_j =|A⁻_h

j,r_j|.

If we set in (49)h=h_j, r=r_j, r−σr=r_j+1, for an arbitrary j ≥0, then we obtain

(53)

Z

Bj+1

|X(v−h_j)⁻|²dx≤C2^2jh²₀ ρ² |A_j|.

Lets∈(p,_Q^pQ_−p). Applying Poincar´e’s inequality [15] to the function (v−h_j)₋ξ, whereξis a cut-off function betweenB_j+2 andB_j+1 we have, on account of the doubling property,

(54) Z

Aj+1

|(v−h_j)⁻ξ|^sdx 1/s

≤c ρ Z

Aj+1

|X(v−h_j)⁻|^pdx+ρ^−p2^pj Z

Aj+1

|(v−h_j)⁻|^pdx 1/p

|B(ρ)|^1/s−1/p. By H¨older inequality, (53) and (54) we obtain

(55) H

2^j+1|A_j+2| ≤ Z

Aj+2

|(v−h_j)⁻|dx

≤( Z

Aj+1

|(v−h_j)⁻ξ|^sdx)^1/s |A_j+1|^1−1/s

≤cρ{( Z

Aj+1

|X(v−hj)⁻|²dx)^p/2|Aj+1|^2−2p +ρ^−p2^pj

Z

Aj+1

|(v−h_j)⁻|^pdx}^1/p|B(ρ)|^1/s−1/p|A_j+1|^1−1/s

≤c{ρ(

Z

Aj+1

|X(v−h_j)⁻|²dx)^1/2|A_j+1|^22p−p + 2^j(

Z

Aj+1

|(v−h_j)⁻|^pdx)^1/p}|B(ρ)|^1/s−1/p|A_j+1|^1−1/s

≤c2^jH|A_j|^1/p|B(ρ)|^1/s−1/p|A_j+1|^1−1/s from which we obtain for anyj ≥0

(56) |A_j+2|

|B(ρ)| ≤c2^4j |A_j|

|B(ρ)| 1+χ

whereχ=¹_p −¹_s >0. In particular (56) gives for any l≥1

(57) |A_2l|

|B(ρ)| ≤c(2⁸)^(l−1) |A_2(l−1)|

|B(ρ)|

!1+χ

.

It follows from (57) and [12, Lemma 4.7, p. 66] that there exists a positive con- stant C₀ depending only on c and b = 2⁸ such that, if |A₀| ≤ C₀|B₀|, then lim_l→+∞A_2l= 0, which implies|{x∈B(ρ)|X_iu < _22/p^λ }|= 0, and thenX_iu≥ ^λ₄

for anyx∈B(ρ), so Proposition 6.2 is proved.

Proposition 6.4 ([5, Proposition 4.2], [18, Proposition 6.4]). Let 2ρ < R. If the assumptions of Proposition 6.4 fail, then there exists a positive structural constantσ₀∈(0,1)independent on ǫ,ρ, such that

µ_ǫ(ρ/2)≤σ₀µ_ǫ(2ρ).

Theorem 6.5. There exist positive constants γ and η ∈(0,1) depending only on the data anddist(B(R), ∂Ω^′)such that

osc_B(ρ)X_iuǫ ≤ γ(ρ

R)^η sup

B(R/2) |Xuǫ|, i= 1, . . . ,2n for every2ρ < Rand everyǫ >0.

Proof: The proof is the same as that of [5, Proposition 4.3] using a result of [17]. Here we limit ourselves to describe the general idea of the proof and we refer the reader to [18] for any details.

We prove the existence of positive structural constants α∈(0,1), δ0 andσ0

independent ofǫ such that, for all smallρ >0, if the subset ofB(ρ) whereXu_ǫ degenerates is “small”, then the equation behaves in B(ρ) as a nondegenerate elliptic equation (see Proposition 6.2). In this case, by [17, Theorem 2.1], we obtainω_ǫ(ρ/2)≤δ₀ρ^α.

On the other hand if Xu_ǫ degenerates in a “thick” portion of B(ρ), then we haveµ_ǫ(ρ/2)≤µ_ǫ(2ρ) (see Proposition 6.4).

The H¨older continuity follows from both cases by a standard iteration tech- nique [12].

7. Estimate of T u_ǫ

In this section we prove that, for any 1 +√¹

5 < p≤2, the local weak solutions uǫ of equation (5) satisfyT uǫ, XT uǫ∈L^p_loc(Ω^′). Just as before, Ω^′ will denote an arbitrary open bounded subset of Ω such that Ω^′ ⊂⊂Ω.

Theorem 7.1([19, Theorem 1.1]). Let1 +√¹

5 < p≤2 and, for anyǫ∈(0,1), letuǫ∈W_loc^1,p(Ω, X) be a local weak solution of(5). LetB(3R)be an arbitrary homogeneous ball of radius 3R such that B(3R) ⊂ Ω^′ and let g be a cut-off function betweenB(R)andB(2R). ThenT(g⁴u_ǫ)∈L^p(Ω^′)and

(58)

Z

Ω^′|T(g⁴u_ǫ)|^p dx≤CR^−4p Z

Ω^′

(V_ǫ^p +|u_ǫ|^p)dx whereV_ǫ²=ǫ+|Xuǫ|².

Theorem 7.2. Let the assumptions of Theorem 7.1 hold. Then T(g¹²u_ǫ) ∈ W_loc^1,p(Ω^′, X)and

(59)

Z

Ω^′|XT(g¹²uǫ)|^p dx≤C(R, ǫ, Hǫ, p) whereHǫ =R

Ω^′(V_ǫ^p+|uǫ|^p)dx andV_ǫ²=ǫ+|Xuǫ|².

Proof: From Lemma 3.5 and Theorem 7.1 we easily deduce for any smalls >0 (60)

Z

Ω^′|D_h∗

s,1/2(g⁴u_ǫ)|^pdx≤cs^p/2C(R, H_ǫ, p).

Let us multiply the equation (5) by the test functionϕ=D_−h∗

s,1/2(g¹⁰D_h∗ s,1/2uǫ).

Let us observe thatϕ∈W₀^1,p(Ω, X). In the following we will drop the subscript ǫfor the sake of simplicity. On account of (7) and Remark 3.2 we obtain

(61) Z

Ω^′

D_h∗

s,1/2a^kg¹⁰X_kD_h∗

s,1/2u dx+ 10 Z

Ω^′

D_h∗

s,1/2a^kD_h∗

s,1/2u g⁹X_kg dx = 0.

For anyp >1 the first integral on the left-hand side of (61) can be estimated by the same argument we applied toJ₂ in Section 4: as

(62) D_h∗

s,1/2a^k=α^kj_h∗

s,1/2D_h∗ s,1/2X_ju whereα^kj_h∗

s,1/2 :=R1

0 a^k_j(Xu+θs^1/2D_h∗

s,1/2Xu)dθ, then we have (63)

Z

Ω^′

D_h∗

s,1/2a^kg¹⁰X_kD_h∗

s,1/2u dx≥c Z

Ω^′

g¹⁰W_h^p_∗⁻²

s |D_h∗

s,1/2Xu|²dx whereW_h²_∗

s(x) =ǫ+|Xu(x)|²+|Xu(x◦h^∗_s)|².

Let us now estimate the second integral on the left-hand side of (61). As

|s^1/2D_h∗

s,1/2Xu| ≤2W_h∗s, we obtain, forγ= ²⁻_p^p,

| Z

Ω^′

D_h∗

s,1/2a^kD_h∗

s,1/2u g⁹X_kg dx| (64)

≤ Z

Ω^′

W_h^p_∗⁻²

s |D_h∗

s,1/2Xu| |D_h∗

s,1/2u|g⁹|X_kg|dx