for a weak solution of fully nonlinear elliptic systems in dimension three and four

On the BMO and C ^1,γ -regularity

for a weak solution of fully nonlinear elliptic systems in dimension three and four

Josef Danˇeˇcek

and Jana Stará

1Department of Applied Mathematics, FEECS, VŠB – Technical University of Ostrava 17. listopadu 15/2172, 70833 Ostrava-Poruba, Czech Republic

2Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University Sokolovská 83, 18600 Praha 8, Czech Republic

Received 4 October 2020, appeared 1 April 2021 Communicated by Patrizia Pucci

Abstract. We consider a nonlinear elliptic system of type

−DαA^α_i(x,Du) =Dαf_i^α

and give conditions guaranteeingC^1,γinterior regularity of weak solutions.

Keywords: nonlinear elliptic systems, regularity, Campanato–Morrey spaces.

2020 Mathematics Subject Classification: 35J60.

1 Introduction.

In this paper we give conditions guaranteeing that the first derivatives of weak solutions to the Dirichlet problem for a nonlinear elliptic system

(−DαA^α_i(x,Du) = Dαf_i^α, i=1, . . . ,N, α∈ _Rⁿ, |α|=1, x ∈_Ω,

u(x) = g(x), x∈ _∂Ω. (1.1)

Here Ω ⊂ _Rⁿ, n ≥ 3 is a bounded C^1,1 domain with points x = (x₁, . . . ,x_n), u : Ω → R^N, u(x) = (u¹(x), . . . ,u^N(x)), N ≥ 2 is a vector-valued function with gradient Du = (D₁u, . . . ,D_nu), D_α = ∂/∂x_α and coefficients A^α_i are continuously differentiable with respect to Du and Hölder continuous with respect to x and in the following we will specify our as- sumptions imposed on the function (f_i^α)and boundary datum g (throughout the whole text we use the summation convention over repeated indexes).

It is well known that elliptic systems in general do not conserve the regularizing property of Laplace equation and the attempts to find conditions guaranteeing the smoothness of weak

BCorresponding author. Email: danecek.j@seznam.cz

solutions as well as to construct counterexamples are rich and far reaching. The positive re- sults, i.e. proof that weak solutions of systems of order 2khave (under suitable assumptions) continuous partial derivatives of orderk, started already with pioneering work of Ch. B. Mor- rey in 1937 for domainsΩin R²(see [16]) and continued by deep results of E. De Giorgi (see [7]) who proved that weak solutions of one equation of second order with linear growth and bounded and measurable coefficients onΩ ⊂ _Rⁿ have continuous first derivatives. The case of nonlinear systems on plane domains was solved in paper by J. Stará (see [23]) in 1971 for systems of higher order.

In dimensionsn ≥ 3 analogous results do not hold as was shown by counterexamples of E. De Giorgi (see [8]) and E. Giusti and M. Miranda in 1968 (see [10]), J. Neˇcas in 1975 (see [19]) and L. Šverák and X. Yan in 2002 (see [24]).

The system (1.1) has been extensively studied in the papers [1,2,9,12,15,17,20] and for detailed and well-arranged informations, see [15]. If n ≥ 3, it is known that Du can be discontinuous. Campanato in [3] proved for the system (1.1) that Du ∈ L^2,θ_loc(_Ω,R^nN) with n−2< θ< n, and also u∈C^0,_loc^(θ−n+2)/2(_Ω,R^N)ifn=3, 4. More important for our work is a more general result from Kristensen–Melcher [13].

There are known many conditions on the coefficients which guarantee that solutions of nonlinear elliptic system of equations have required smoothness and, vice versa, counterex- amples illustrating that generally such assertions do not hold.

In the present paper, that is extending the articles [4], [5] and [6], we introduce another conditions on coefficients of a nonlinear elliptic system (1.1) and we show that if the first derivatives of weak solutionsuto Dirichlet problem for the system satisfy (1.11) with givenM andΨe from (1.10) then the gradient of weak solutions are locally BMO or Hölder continuous on domainsΩ in R³ andR⁴. The condition (1.11) shows that the our result is applicable to broader class of problems for smaller value ofM. Finally, the reality of our theoretical result is illustrated by means of numerical examples.

By a weak solution to the Dirichlet problem for (1.1) we understandu∈W^1,2(_Ω,R^N)such thatu−g∈W₀^1,2(_Ω,R^N), g∈W^1,2(_Ω,R^N), f ∈ L²(_Ω,R^nN)and

Z

ΩA^α_i (x,Du(x))D_αϕⁱ(x)dx=−

Z

Ω f_i^α(x)D_αϕⁱ(x)dx, ∀ϕ∈W₀^1,2(_Ω,R^N). (1.2) Further the symbolΩo ⊂⊂_Ωstands forΩo ⊂_Ω,d_Ω =diam(_Ω)and for the sake of simplicity we denote by| · | the norm inRⁿas well as inR^N andR^nN. If x ∈_Rⁿ andr is a positive real number, we setB_r(x) ={y∈_Rⁿ : |y−x|<r}, (i.e., the open ball inRⁿ),Ωr(x) =_Ω∩B_r(x). Denote byux,r = ur =R

Ωr(x)u(y)dy/mn(_Ωr(x)) =R

Ωr(x)

− u(y)dythe mean value of the func- tionu∈ L(_Ω,R^N)over the setΩr(x)_{. Here}m_n(_Ωr(x))_{is the}n-dimensional Lebesgue measure of Ωr(x) and we set Ur(x) = R

Ωr(x)|Du(y)−(Du)_x,r|²dy/rⁿ = R

Ωr(x)

− |Du(y)−(Du)_x,r|²dy, φ(_x,r) =R

Ωr(x)|Du(_y)−(_Du)_x,r|²dy.

The coefficients (A^α_i)_{i=1,...,N,α=1,...,n} have linear controlled growth and satisfy strong uni- form ellipticity condition. Without loss of generality we can suppose that A^α_i(x, 0) = 0. We suppose that A^α_i(x,p)∈C¹(_R^nN)for allx∈ _Ωand

(i) the strong ellipticity condition holds, i.e. there existν, M >0 such that for every x ∈_Ω andp, ξ ∈_R^nN

ν|ξ|²≤ ^∂A

αi

∂p^j_β

(x,p)ξⁱ_αξ_β^j ≤ M|ξ|², (1.3)

(ii)

|A^α_i(x,p)| ≤M(1+|p|),

∑

i,j,α,β

∂A^α_i

∂p^j_β(x,p)

≤ M, (1.4)

for all(x,p)∈ _Ω×_R^nN, (iii) for allx,y ∈_Ωand p∈_R^nN

|A^α_i(x,p)−A^α_i(y,p)| ≤C_H|x−y|^χ|p|, C_H >0 (1.5) whereχ=1 forn=3, 4,

(iv) there is a real function ω continuous on[0,∞), which is bounded, nondecreasing, con- cave, ω(0) =0 and such that for all x∈_Ωandp,q∈_R^nN

∂A^α_i

∂p^j_β(x,p)−^∂A

iα

∂p^j_β(x,q)

≤ω(|p−q|). (1.6)

We denoteω_∞=_limt→∞ω(t)and clearlyω(t)≤2M.

It is well known (see [9], p.169) that for uniformly continuous∂A^α_i/∂p^j_β there exists a real function ω satisfying (iv) and, viceversa, (1.6) implies uniform continuity of ∂A^α_i/∂p^j_β and absolute continuity of ω on [_0,∞). By pointwise derivative ω⁰ we will understand the right derivative ofωwhich is finite on(0,∞).

Here we will consider the functionωfrom (1.6) given by the formula

ω(t) =











ω_o(_t)_{, for 0}≤t ≤to, to >₀ ω₁(t) =

√ ε

t^γ_o t^γ, forto < t<t₁, ω_∞ fort ≥t₁

(1.7)

where ω_o is arbitrary continuous, concave, nondecreasing function such thatω_o(0) = 0 and the constants 0 <γ≤0.44,t_o >0 are selected in such a way thatωis continuous and concave on [0,∞).

For example we can choose

ω_o(t) = ²

√ ε

2+ln^t_t^o for 0<t ≤t_o,

and this function fail to satisfy Dini condition. It is obvious that in such case the coefficients

∂A^α_i/∂p^j_β are only continuous.

It is well known that on the above assumptions the Dirichlet problem (div(A(x,Du) + f) =0 inΩ,

u−g ∈W₀^1,2(_Ω,R^N) ^(1.8) has for any function f,g∈W^1,2(_Ω,R^N)the unique solution uin the same space.

For the problem (1.8) the following estimate holds Z

− |Ω Du−(Du)_Ω|² dy

≤12 M

ν _2Z

− |Ω Dg−(Dg)_Ω|² dx+ 10E

ν ₂

E ν

+ ^M ν

+3 M

ν ₂!

Z

−|ΩDg|²dx +₂₀

nN ν

₂ 1+

4E ν

₂! Z

− |Ω f−(f)_Ω|² dx (1.9)

whereE=nNCHd^χ_Ω (see AppendixAfor the proof of (1.9)).

In the following we will use the functionΨe(u) =ue^u2/(2µ−1), hereu≥0,µ≥17 (for detailed information forΨe, see (2.6)) and we can define the value

M = sup

to<t<_∞

Ψe

ω²(t) ε

−Ψe

ω²(to) ε

t−to

<_∞ (1.10)

whereωis from (1.7),ε=ω²_∞/C_µ^α,α>₁−2/nandC_µ =⁽ⁿ⁻_2e^2)µµ_. Now we can formulate the main theorem.

Theorem 1.1.LetΩo ⊂⊂_Ω⊂_Rⁿ, d_o =dist(_Ωo,∂Ω)/2, n=3, 4. Assume that g∈W^1,2(_Ω,R^N), Dg∈ L^2,ζ(_Ω,R^nN),ζ >2, f ∈W^1,2∩ L^2,ξ(_Ω,R^N), n<ξ ≤ n+2, n≤ϑ<λ=min{2χ+ζ,ξ} and moreover divf ∈ L^ζ(_Ω,R^nN). Let u ∈ W^1,2(_Ω,R^N) be a weak solution to the system (1.1) satisfying the conditions

Z

− |Ω Du−(Du)_Ω|² dy< ¹

M^{2 , (1.11)}

(1.13)and

CMC²_H+ [f]²_L2,ξ(_Ω,R^nN) ≤

|_Ω|1−(4eo)^λϑ−1e³_oν²

8dⁿ_omax{d^λ_o,d^λ_o⁻ⁿ} M^{2 (1.12)} whereeo =1/4(2ⁿ⁺⁵L)^n+ϑ2−ϑ, the constants L, C_H, C_M come from Lemma2.5,(1.5)and(3.8), respec- tively. Then Du∈C^0,(ϑ−n)/2(_Ωo,R^nN)in the caseϑ> n and Du∈ BMO(_Ωo,R^nN)forϑ=n.

Remark 1.2. In the foregoing formulas the constantsµ≥17,α>1−2/nhave to be such that

C_µ^n−n2α−1≥₂ⁿ⁻⁶² 20C_SMω_∞ ν²

|_Ω| (2do)ⁿ

_2n¹ !_n²ⁿ₋₂ e⁻

n−n2

o . (1.13)

HereC_Sis the Sobolev embedding constant.

The theorem we formulated above tells that, if coefficients of a nonlinear system satisfy (iv) with someω given (1.7) and (1.11)–(1.13) are fulfilled, then the gradient of u is Hölder continuous onΩo.

In most partial regularity results for the system (1.1) the regular pointsx∈_Ωof solutionu are characterized in such a way that for somer_x >0 the quantityU_rx(x)(for its definition see first section) has to be sufficiently small, but our condition regularity (1.11) allowsUr(x)not to be necessarily small. Moreover, the condition (1.11) is global condition (we do not know an analogous condition from the literature) and has fundamental meaning for domain Ω in

which it is possible ensure that the ratio |_Ω|/(2d_o)ⁿ is not extremely great (e.g. for the ball, see (1.13)).

For the functionω from (1.7) the right-hand side of (1.11) can be chosen in the following form

1 M² = ^t

2o

4



min





 1

3γ,C(−1+_2γ¹)α µ

e^C

2µ2α−1 µ











2

. (1.14)

(see AppendixBfor more information and forµandαsee Remark1.2).

Remark 1.3. We would like to point that, in the case of (1.8), the left-hand side of (1.11) could be substituted with the right-hand side of (1.9). We can present some consequences of our theorem that follow from estimate (1.9).

g =const. ∧ f =const. =⇒

Z

− |Ω Du−(Du)_Ω|² dy=0 =⇒ u= P₁ g=P₁ ∧ f =const. ∧ C_H =0 =⇒

Z

− |Ω Du−(Du)_Ω|² dy=0 =⇒ u= P₁ g =P₁ ∧ f =const. ∧ d_Ω &0 =⇒

Z

− |Ω Du−(Du)_Ω|² dy&0 g= P₁ ∧ f ∈ L^2,ξ(_Ω,R^nN), ξ >n ∧ CH =0 ∧ d_Ω&0 =⇒

Z

− |Ω Du−(Du)_Ω|² dy&0 where P₁ is a polynom of at most first degree. We note that the last mentioned condition involves the data of the problem (1.8) only.

Remark 1.4. It is useful to point out that in the case when the ratioω_∞/νis small enough, the regularity of solution to the problem (1.8) is guaranteed by the Proposition 2.4 from [4].

2 Preliminaries and notations

Beside the usually used space C₀^∞(_Ω,R^N), Hölder space C^0,α(_Ω,R^N) and Sobolev spaces W^k,p(_Ω,R^N), W_loc^k,p(_Ω,R^N), W₀^k,p(_Ω,R^N)(see, e.g.[22]) we use the following Campanato and Morrey spaces.

Definition 2.1(Campanato and Morrey spaces). Letυ∈[0,n]. The Morrey spaceL^2,υ(_Ω,R^N) is the subspaces of such functionsu∈ L²(_Ω,R^N)for which

kuk²_L2,υ(_Ω,R^N) = sup

r>0,x∈_Ω

r^−υ Z

Ωr(x)

|u(y)|² dy<_∞.

Let υ ∈ [_0,n+₂]. The Campanato space L^2,υ(_Ω,R^N) is the subspace of such functions u ∈ L²(_Ω,R^N)for which

[u]²_L2,υ(_Ω,R^N)= sup

r>0,x∈_Ω

r^−υ Z

Ωr(x)

|u(y)−u_x,r|² dy<_∞.

Remark 2.2. It is worth recalling the trivial but basic property that R

Ω|u−u_Ω|²dx = min_c∈RN

R

Ω|u−c|²dxholds for each u∈ L²(_Ω,R^N).

For more details see [1], [9] and [22]. In particular, we will use:

Proposition 2.3. For a bounded domainΩ⊂_Rⁿwith a Lipschitz boundary we have the following (a) L^2,υ(_Ω,R^N)is isomorphic to the C^0,(υ−n)/2(_Ω,R^N)_{, for n}<υ≤n+_2,

(b) L^2,υ(_Ω,R^N)is isomorphic to theL^2,υ(_Ω,R^N),0≤υ< n,

(c) the imbeddingL^2,υ1(_Ω,R^N)⊂ L^2,υ2(_Ω,R^N)is continuous for all0≤ υ₂ <υ₁ ≤n+2, (d) L^2,n(_Ω,R^N)is isomorphic to the L^∞(_Ω,R^N) L^2,n(_Ω,R^N).

The following lemma is a modification of a lemma from [5].

Lemma 2.4. Let A > 1, d be positive numbers, C, B₁, B₂ ≥ 0, n ≤ δ < β, δ < α ≤ n+2 and 0 < s ≤ 1. Then there exist positive constants k₁, k₂ so that for any nonnegative nondecreasing functionϕdefined on[0,d]and satisfying the inequalities

ϕ(σ)≤ Aσ R

α

ϕ(R) + ¹

2

1+Aσ R

α h

(B₁+B₂U_2R^s )ϕ(2R) +CR^βi

, ∀ 0<σ< R≤ ^d 2 (2.1) and

B₁+B₂U_d^s ≤ ¹

4τ^δ, B₂

Cm 2^βτ^δ(₁−τ^β−δ)

s

≤ ¹

4τ^δ (2.2)

where U_R= φ(R)/Rⁿ, m=max{d^β,d^β−n}andτ=1/(2^α+1A)^α−1δ. Then it holds

Uσ ≤σ^δ−n(k₁ϕ(d) +k2), ∀σ∈ (0,d]. (2.3) Proof. I. We will prove by induction that

ϕ(τ^kd)≤τ^kδ ϕ(d) + ^Cm 2^βτ^δ

k−1 j

∑

τ^(β−δ)j

!

, U_τkd≤ τ^k(δ−n) U_d+ ^Cm 2^βτ^δ

k−1 j

∑

τ^(β−δ)j

!

. (2.4) Let k= 1. Puttingσ = τd, R= d/2 in (2.1) we obtain thanks to (2.2) and the assumption onτ

ϕ(τd)≤2^αAτ^αϕ d

2

+¹₂(1+2^αAτ^α)

(B₁+B₂U_d^s)ϕ(d) +C

d 2

β

≤(2^αAτ^α+B₁+B₂U_d^s)ϕ(d) +C

d 2

β

=τ^δ

ϕ(d) + ^Cm

2^βτ^δ

. Also by means of (2.2) we get

U_τd≤ τ^δ−n

U_d+ ^Cm

2^βτ^δ

, B₁+B₂U_τd^s ≤ ¹ 2τ^δ. Next putσ=τ^k+1d, R= τ^kd/2 into (2.1) we get

ϕ(τ^k+1d)≤2^αAτ^αϕ 1

2τ^kd

+¹₂(1+2^αAτ^α)^h B₁+B2U_τ^skd

ϕ(τ^kd) + ^Cdβ

2^β τ^kβ i

≤ 2^αAτ^α+B₁+B₂U_τ^skd

ϕ(τ^kd) + ^Cdβ

2^βτ^δτ^kβ+δ ≤τ^δϕ(τ^kd) + ^Cm

2^βτ^δτ^(k+1)δ

because 2^αAτ^α+B₁+B₂U^s

τ^kd ≤τ^δ. Using (2.4) we get ϕ(τ^k+1d)≤τ^δϕ(τ^kd) + ^Cdβ

2^βτ^δτ^(k+1)δ ≤τ^(k+1)δ ϕ(d) + ^Cm 2^βτ^δ

k−1 j

∑

τ^(β−δ)j

! + ^Cm

2^βτ^δτ^(k+1)δ

=τ^(k+1)δ ϕ(d) + ^Cm 2^βτ^δ

∑

τ^(β−δ)j

! . It immediately implies the estimate ofU_τk+1d.

II. Let nowσ be an arbitrary positive number less than d. Then there is an integerk such that τ^k+1d ≤σ<τ^kd. Using monotonicity of ϕ, this inequality and (2.4) we get

ϕ(σ)≤ ϕ(τ^kd)≤τ^kδ ϕ(d) + ^Cm 2^βτ^δ

k−1 j

∑

τ^j(β−δ)

!

≤ ^σ

δ

(τd)^δ

ϕ(d) + ^Cm 2^βτ^δ(1−τ^β−δ)

and this estimate together with the choice of k₁ = _1/(τd)^δ_, k₂ = Cm/(₂^βd^δτ^2δ(₁−τ^β−δ)) completes the proof.

For the statement of following Lemma see e.g. [1,9,20].

Lemma 2.5. Consider system of the type (1.1) with A^α_i(x,p) = A^αβ_ij p^j_β, A^αβ_ij ∈ _R (i.e. linear system with constant coefficients) satisfying (i), (ii) and (iii). Then there exists a constant L = L(n,N,M/ν) ≥ ₁ such that for every weak solution v ∈ W^1,2(_Ω,R^N) and for every x ∈ _Ω and 0<σ ≤R≤dist(x,∂Ω)the following estimate

Z

Bσ(x)

|Dv(y)−(Dv)_x,σ|² dy≤ Lσ R

n+2Z

BR(x)

|Dv(y)−(Dv)_x,R|² dy holds.

Remark 2.6. The constant Lfrom the previous lemma can be stated as L=c(n,N)

M ν

2(2+[ⁿ₂])

and, because of a better presentment, choosing n = 3, N = 2 we can compute L < 1.4· 10⁸(M/ν)⁶.

In the paper [4, p. 108] a system forn = N =3 of type (1.1) was presented for which we can compute L≈10⁸.

Lemma 2.7. [25, p. 37] Let φ : [0,∞] → [0,∞] be non decreasing function which is absolutely continuous on every closed interval of finite length, φ(₀) = _{0. If w} ≥ 0is measurable and E(_t) = {y ∈_Rⁿ:w(y)>t}then

Z

Rⁿφ◦w dy=

Z _∞

0 mn(E(t))φ⁰(t)dt.

In the proof of Theorem1.1we will use an inequality which is a consequence of Natanson’s lemma (see e.g. [18, p. 262]) and Fatou’s lemma. It can be read as follows.

Lemma 2.8. Let f : [a,∞) → _R be a nonnegative function which is integrable on [a,b] for all a<b<_∞and

N = sup

0<h<_∞

1 h

Z _a+h

a f(t)dt<_∞

is satisfied. Let g: [a,∞)→ _R be an arbitrary nonnegative, non-increasing and integrable function.

Then Z _∞

a f(t)g(t)dt exists and

Z _∞

a f(t)g(t)dt≤ N

Z _∞

a g(t)dt holds.

In the proof of Theorem1.1 we use an inequality which can be read as follows.

Proposition 2.9 (see [4]). Let u ∈ W^1,2(_Ω,R^N) be a weak solution to(1.1) satisfying (i), (ii), (iii) and (iv). Then for every ball B_2R(x)⊂ _Ωand arbitrary constantsµ≥2, b> 0,1< q≤n/(n−2) and c∈_R^nN we have

Z

BR(x)

|Du−(Du)_x,R|²ln^µ₊ b|Du−(Du)_x,R|² dy

≤2^n(q−1)

5C_SM ν

2q µ (q−1)e

µ b (2R)ⁿ

Z

B2R(x)

|Du−c|²dy q−1Z

B2R(x)

|Du−c|²dy (2.5) where C_Sis the Sobolev embedding constant.

Hereafter we shall use conjugate Young functionsΦ, Ψ Φ(u) =uln^µ₊(au) foru≥0, Ψ(u)≤_Ψ(u) = ¹

aue^u

2µ2−1

foru≥0, (2.6) wherea>0 andµ≥2 are constants,

ln+(au) =

(0 for 0≤u< ¹_a, ln(au) foru≥ ¹_a. Then Young inequality forΦ,Ψreads as

xy≤_Φ(x) +_Ψ(y), ∀ x,y∈_R. (2.7)

3 Proof of Theorem 1.1

Let x_o be any point of Ωo∩ S (it means that R

B_R(xo) |Du−(Du)_xo,R|² dx > 0) and R ≤ d_o. Where no confusion can result, we will use the notationBR, UR, φ(R)and (Du)_R instead of B_R(x_o),U_R(x_o),φ(x_o,R)and(Du)_xo,R. Denoting A^αβ_ij,0= A^αβ_ij (x_o,(Du)_R),

A˜^αβ_ij =

Z ₁

0 A^αβ_ij (xo,(Du)_R+t(Du−(Du)_R))dt, we can rewrite the system (1.1) as

−Dα

A^αβ_ij,0D_βu^j

=−Dα

A^αβ_ij,0−A^˜αβ_ij D_βu^j−D_βu^j

R

−D_α(A^α_i(x_o,Du)−A^α_i(x,Du)) +D_α(f_i^α(x)−(f_i^α)_R).

Splituasv+wwherevis the solution of the Dirichlet problem

−D_α

A^αβ_ij,0D_βv^j

=0 in B(R) v−u ∈W₀^1,2

B_R,R^N . andw∈W₀^1,2(B_R,R^N)is the weak solution of the system

−Dα

A^αβ_ij,0D_βw^j

=−Dα

A^αβ_ij,0−A^˜αβ_ij D_βu^j−D_βu^j

R

−Dα(A^α_i(x_o,Du)−A^α_i(x,Du)) +Dα(f_i^α(x)−(f_i^α)_R). For every 0<σ≤Rfrom Lemma2.5it follows

Z

Bσ

|Dv−(Dv)_σ|² dx≤ Lσ R

n+2Z

BR

|Dv−(Dv)_R|² dx hence

Z

Bσ

|Du−(Du)_σ|² dx≤2Lσ R

n+2Z

BR

|Dv−(Dv)_R|² dx+4 Z

BR

|Dw|²dx

≤4Lσ R

n+2Z

BR

|Du−(Du)_R|² dx+4

1+2Lσ R

n+2Z

BR

|Dw|²dx. (3.1) Noww∈W₀^1,2(BR,R^N)satisfies

Z

BR

A^αβ_ij,0D_βw^jD_αϕⁱdx ≤

Z

BR

A^αβ_ij,0−A^˜αβ_ij

D_βu^j−(D_βu^j)_R||D_αϕⁱ dx +

Z

BR

|A^α_i(xo,Du)−A_i^α(x,Du)|Dαϕⁱ dx

≤ _Z

B_Rω²(|Du−(Du)_R|)|Du−(Du)_R|² dx

_1/2Z

B_R

|Dϕ|²dx _1/2

+ _Z

BR

|A^α_i(xo,Du)−A^α_i(x,Du)|² dx

1/2_Z

BR

|Dϕ|²dx 1/2

+ _Z

BR

|f − f_R|² dx

1/2_Z

BR

|Dϕ|²dx 1/2

for any ϕ∈W₀^1,2(B_R,R^N). Hence, choosing ϕ=w, we get ν²

Z

BR

|Dw|²dx≤2 Z

BR

ω²(|Du−(Du)_R|)|Du−(Du)_R|² dx +4

Z

BR

|A^α_i(x_o,Du)−A^α_i(x,Du)|² dx+4 Z

BR

|f − f_R|² dx. (3.2) From (3.1) and (3.2) we have

φ(σ) =

Z

Bσ

|Du−(Du)_σ|² dx≤4Lσ R

n+2Z

BR

|Du−(Du)_R|² dx

+⁸

1+_2L ^σ_Rⁿ⁺² ν²

Z

BR

ω²(|Du−(_Du)_R|)|Du−(_Du)_R|² dx +2

Z

BR

|A^α_i(xo,Du)−A^α_i(x,Du)|² dx+2 Z

BR

|f − f_R|² dx

=4Lσ R

n+2

φ(R) + 8

1+2L _R^σn+2

ν² (I₁+2I₂+2I₃) (3.3)

We use the Young inequality (2.7) (here complementary functions are defined through (2.6)) and for any 0<ε<ω²_∞ we obtain

I₁=

Z

BR

ω²(|Du−(Du)_R|)|Du−(Du)_R|² dx

≤ε Z

BR

|Du−(Du)_R|²ln+

aε|Du−(Du)_R|² dx+

Z

BR

Ψ ω²_R

ε

dx=εJ₁+J₂ (3.4) whereω_R²(x) =ω²(|Du(x)−(Du)_R|).

The term J₁ can be estimated by means of Proposition2.9(hereq=n/(n−2)) and we get J₁ ≤CC_µ(aεU_2R)^q−1φ(2R) _(3.5) where

C=2^(q−1)n

5C_SM ν

2q

, C_µ=

n−2 2e µ

µ

.

Taking in Lemma2.7w(y) =|v(y)−v_x,R|onB_R(x)andw=0 otherwise, we haveE_R(t) = {y∈ B_R(x) : |v(y)−v_x,R|> t}and for the the second integral J₂ we get

J₂= ¹ a

Z _∞

0

d dtΨe

ω²(t) ε

m_n(E_R(t))dt (3.6)

whereΨe = _aΨ.

We have (we use Lemma2.8) for∀ ε>0 Z _∞

0

d dtΨe

ω²(t) ε

mn(ER(t)) dt

≤

Z _to

0

d dtΨe

ω²(t) ε

m_n(E_R(t))dt+

Z _∞

to

d dtΨe

ω²(t) ε

m_n(E_R(t))dt

≤κ_nRⁿ Z _to

0

d dtΨe

ω²(t) ε

dt+ sup

to<t<_∞

1 t−t_o

Z _t

to

d dsΨe

ω²(s) ε

ds

_{Z ∞}

to

m_n(E_R(s))ds

≤κ_nΨe

ω²(t_o) ε

Rⁿ+ _sup

to<t<_∞



 Ψe

ω²(t) ε

−Ψe

ω²(to) ε

t−to



 Z

BR

|Du−(Du)_R| dx

≤κ_nΨe

ω²(t_o) ε

Rⁿ+Mκ^1/2_n R^n/2φ^1/2(R)

≤



 κ_n 2ⁿ

Ψe

ω²(to) ε

U2R

+^κn 2ⁿ

1/2 M

√U_2R



φ(2R)≤



 Ψe

ω²(to) ε

U2R

+ M

√U_2R



φ(2R). (3.7) If for someR > 0 the averageU_R =0 then it is clear that x_o is the regular point. So next we can supposeU_Ris positive for allR>_0.

From [2] and [13] we have thatDu ∈ L^2,ζ(_Ω,R^nN)_,ζ ∈(_{2, 3})_{and also}

Z

BR

|Du|²dx≤ ^c

2(ζ,M/ν,C_H,χ,Ω) ν²

kfk²_L2,ζ(Ω,RnN)+kDgk²_L2,ζ(Ω,RnN)

R^ζ

=C_MR^ζ, ∀0< R≤ d_o. (3.8)

From the assumptions (iii) follows I₂≤C_MC²_HR^2χ

Z

BR

|Du|²dx ≤C_MC²_HR^2χ+ζ (3.9) and

I₃ ≤[f]²_L2,ξ(Ω

o,R^nN)R^ξ. (3.10)

We get from (3.3) and (3.4) by means of (3.5), (3.7), (3.9) and (3.10) φ(σ)≤4Lσ

R n+₂

φ(2R) +8

1+2Lσ R

n+2 CC_µε

ν² (aεU_2R)^q−1+ ¹ aν²

Ψe

ω²(t_o) ε

1

U_2R+ M

√U_2R

φ(2R) +2C_M

C_H ν

2

R^2+ζ+ ²

ν²[f]²_L2,ξ(_Ωo,R^nN)R^ξ )

≤4Lσ R

n+2

φ(2R) +8

1+2Lσ R

n+2

×

×

CC_µε

ν² (aεU2R)^q−1+ ¹ aν²

Ψe

ω²(to) ε

1

U_2R + M

√U_2R

φ(2R) + ²

ν²

C_MC²_H+ [f]²_L2,ξ(Ω

o,R^nN)

R^λ

(3.11) whereλ=_min{_2χ+_ζ,ξ}_.

In (3.11) we can choose ε= ^ω

∞2

C_µ^α, a= ¹²⁸|_Ω|^1/2

(2d_o)^n/2ν²e_oU_2R forU(2R)>0 (3.12) wheree_o = ¹

4(2ⁿ⁺⁵L)^{ϑ/(n+2−ϑ)} andµ≥_17, α>₁−2/nare suitable constants.

We setP=ω_∞/ν. Then we obtain forU_2R >0 φ(σ)≤4Lσ

R n+2

φ(R) + ¹ 2

1+2Lσ R

n+2

×









 16CP²

C_µ^α−1

128|_Ω|^1/2P² (2do)^n/2C_µ^αeo

!q−1

+(2d_o)^n/2ν² 8|_Ω|^1/2

Ψe

ω²(t_o) ε

+M^pU_2R

e_o



φ(2R)

+ ³² ν²

C_MC²_H+ [f]²_L2,ξ(Ω

o,R^nN)

R^λ







≤4Lσ R

n+2

φ(R) + ¹ 2

1+2Lσ R

n+2

×











2^7q−3CP^2q C_µ^qα−1e_o^q−1

|_Ω| (2d_o)ⁿ

^q−₂¹ + ¹

8

3+ (2do)^n/2

|_Ω|^1/2 M^pU2R

e0



φ(2R)

+ ³² ν²

C_MC²_H+ [f]²_L2,ξ(Ω

o,R^nN)

R^λ





