On Morrey and BMO regularity for gradients of weak solutions to nonlinear elliptic systems with

On Morrey and BMO regularity for gradients of weak solutions to nonlinear elliptic systems with

non-differentiable coefficients

Josef Danˇeˇcek

and Eugen Viszus

1Institute of Mathematics and Biomathematics, Faculty of Science, University of South Bohemia Branišovská 31, ˇCeské Budˇejovice 3705, Czech Republic

2Department of Mathematical Analysis and Numerical Mathematics Faculty of Mathematics, Physics and Informatics, Comenius University

Mlynská dolina, Bratislava 84248, Slovak Republic Received 16 November 2016, appeared 24 February 2017

Communicated by Maria Alessandra Ragusa

Abstract. We consider weak solutions to nonlinear elliptic systems with non- differentiable coefficients whose principal parts are split into linear and nonlinear ones.

Assuming that the nonlinear part g(x,u,z)is equipped by sub-linear growth in zonly for big value of |z| (but the growth is arbitrarily close to the linear one), we prove the Morrey and BMO regularity for gradient of weak solutions.

Keywords: nonlinear equations, regularity, Morrey–Campanato spaces.

2010 Mathematics Subject Classification: 49N60, 35J60.

1 Introduction

In the paper, we consider the problem of interior everywhere regularity of gradients of weak solutions to the nonlinear elliptic system

−diva(x,u,Du) =b(x,u,Du), (1.1) where a : Ω×_R^N ×_R^nN → _R^nN, b: Ω×_R^N×_R^nN → _R^N are Caratheodorian mappings, Ω⊂ _Rⁿ is a bounded open set, N > 1, n ≥ 3. A function u ∈ W_loc^1,2(_Ω,R^N) is called a weak solution to (1.1) in Ωif

Z

Ωha(x,u,Du),Dϕ(x)i dx=

Z

Ωhb(x,u,Du),ϕ(x)i dx, ∀ϕ∈C^∞₀ (_Ω,R^N).

As it is shown by examples, in a case of general system (1.1), only partial regularity of weak solutions can be expected forn≥3 (see e.g. [2,7,12]). Under the assumptions specified below

BCorresponding author. Email: eugen.viszus@fmph.uniba.sk

we prove, in Campanato spaces, L^2,n–regularity (or, so calledBMO-regularity) of gradient of weak solutions for the system (1.1) whose coefficientsa can be written in the special form

a(x,u,Du) = A(x)Du+g(x,u,Du)_{, (1.2)} where A = (_A^αβ_ij )_{, i, j} = _{1, . . . ,N, α,} β = _{1, . . . ,}n, is a matrix of functions, the following condition of strong ellipticity

hA(x)ξ,ξi ≥ν|ξ|², a.e.x ∈_Ω, ∀ξ ∈_R^nN; ν >0 (1.3) holds, and g = g(x,u,z) are functions with sub-linear growth in z. In what follows, we formulate the conditions on the smoothness and the growth of the functions A, g and b precisely.

It is well known that in the case of linear elliptic systems with continuous (see [2]) or withVMO∩L^∞(see [8]) coefficients A, the gradient of weak solutions has theL^2,λ-regularity.

Supposing that the coefficients Aof the linear system belong to some Hölder class, the author of [2] proved that the gradient of weak solutions belongs to the BMO-class. The foregoing result has been refined in [1], where the coefficients Aare supposed to belong to the class of so-called “small multipliers ofBMO”. The both mentioned results from [2] and [1] have been generalized in [8], where the coefficients Abelong to some subclass ofVMO∩L^∞and in [13], where nonstandard growth conditions of p(_x)-type are considered.

Similar regularity results (L^2,λ-regularity for continuous coefficientsAandBMO-regularity for Hölder ones) were achieved in [2] for systems (1.1)–(1.2) in a case when g = g(x,u)(but does not depend on Du). The last mentioned results are generalized in [4], where the first author has proved the L^2,λ-regularity of the gradient of weak solutions to (1.1)–(1.2) when the coefficients A are continuous and the BMO-regularity of gradient in the case of Hölder continuous coefficients A under an assumption that the function g = g(x,u,z) grows sub- linearly inzand the growth is controlled by power function|z|^α, 0<α<1. TheL^2,λ-regularity result from [4] has been generalized to theVMO∩L^∞ coefficientsAin [5].

The present paper extends the results from [4] and [5] in two directions. The first one consists in the fact that, while the sub-linear inz growth of the function g(x,u,z) from (1.2) is controlled by the power function |z|^α, α ∈ (0, 1), the present paper offers the control by a function|z|/ ln^s/2(e+|z|²), s > 0, which is closer to the linear function then the power one.

The second extension is that in [4] and [5] the sub-linear growth is required for all|z|>0 and, on the other hand, here we prescribe it only for big values of|z|as it is visible in (3.2), (3.3), (3.5) below. The last mentioned assumption could be seen as a kind of asymptotic growth condition. Recently a few papers have appeared, which study regularity of weak solutions to nonlinear systems diva(Du) =0, where the coefficientsa =a(z)are so called asymptotically regular (for precise definitions and statements see [15] and references therein). Our growth condition is a bit different from the condition on asymptotic regularity of coefficients in [15]

because of structure of the systems. Here it is useful to mention a paper [10], where the authors deal with (beside other problems) the partial C^1,α-regularity of W^1,∞-weak solutions to quasi-monotone systems diva(x,Du) = 0, a = a(x,z) is C¹ in variable z, where they provide upper bounds for the Hausdorff dimension of the singular set (see [10, Chapter 6]).

If a(x,z) = a(z) and the coefficients a satisfy an asymptotic condition, which requires the differentiability of awith respect to z, then weak solutions to the previous systems belong to W_loc^1,∞(_Ω,R^N). A typical model example for reaching such a result is a(z) = z+b(z), where the derivativeb_z(z)→_{0 when}|z| →_∞(see [10, Chapter 6] as well). In this paper we provide

L^2,n_loc-regularity of gradients of weak solutions because of special structure of the system (but here we have a = a(x,u,Du)) and a = a(x,u,z) does not have to be differentiable in the variablezand so we can not suppose any condition of the type g_z(x,u,z)→0 for|z| →_∞.

2 Notation and definitions

We consider the bounded open set Ω ⊂ _Rⁿ with points x = (x₁, . . . ,x_n), n ≥ 3, u : Ω → R^N, N > 1, u(x) = (u¹(x), . . . ,u^N(x)) is a vector-valued function, Du = (D₁u, . . . ,D_nu), Dα = ∂/∂xα. The meaning of Ω0 ⊂⊂ _Ω is that the closure of Ω0 is contained in Ω, i.e.

Ω0 ⊂ Ω. For the sake of simplicity we denote by | · | the norm in Rⁿ as well as in R^N and R^nN. Ifx ∈ _Rⁿ andr is a positive real number, we write B_r(x) = {y∈_Rⁿ:|y−x|< r}, i.e., the open ball in Rⁿ with radius r > 0, centered at x and Ωr(x) = _Ω∩Br(x). Denote by u_x,r = |_Ωr(x)|⁻_n¹R

Ωr(x)u(y)dy =R−

Ωr(x)u(y)dy the mean value of the functionu ∈ L¹(_Ω,R^N) over the set Ωr(x)_{, where} |_Ωr(x)|_n is the n-dimensional Lebesgue measure of Ωr(x)_{. Beside} the usually Sobolev spacesW^k,p(_Ω,R^N),W_loc^k,p(_Ω,R^N),W₀^k,p(_Ω,R^N)(see, e.g. [11]), we use the following Morrey and Campanato spaces.

Definition 2.1. Let λ ∈ (0,n), q∈ [1,∞). A function u ∈ L^q(_Ω,R^N) is said to belong to the Morrey space L^q,λ(_Ω,R^N)if

kuk^q

L^q,λ(_Ω,R^N)= sup

x∈Ω,r>0

1 r^λ

Z

Ωr(x)

|u(y)|^qdy<_∞.

Let λ ∈ [0,n+q], q ∈ [1,∞). The Campanato space L^q,λ(_Ω,R^N) is the subspace of such functionsu∈ L^q(_Ω,R^N)for which

[u]^q_{Lq,λ(Ω,RN)} = sup

r>_0,x∈_Ω

1 r^λ

Z

Ωr(x)

|u(y)−ux,r|^qdy< _∞.

Proposition 2.2. For a domainΩ⊂_Rⁿof the classC^0,1 we have the following

(a) With the norms kuk_Lq,λ and kuk_Lq,λ = kuk_Lq + [u]_Lq,λ, L^q,λ(_Ω,R^N) and L^q,λ(_Ω,R^N) are Banach spaces.

(b) L^q,λ(_Ω,R^N)is isomorphic to theL^q,λ(_Ω,R^N),1≤q<_∞,0<λ<n.

(c) L^q,n(_Ω,R^N)is isomorphic to the L^∞(_Ω,R^N)(L^q,n(_Ω,R^N),1≤q<_∞.

(d) L^2,n(_Ω,R^N) is isomorphic to theL^q,n(_Ω,R^N)andL^q,n(Q,R^N) = BMO(Q,R^N), Q being a cube,1≤ q<_∞.

(e) If u∈W_loc^1,2(_Ω,R^N)and Du∈L^2,λ_loc(_Ω,R^nN), n−2<λ<n, then u∈C^{0,(λ+2−n)/2}(_Ω,R^N). (f) L^q,λ(_Ω,R^N)is isomorphic to the C^0,(λ−n)/q(_Ω,R^N)for n<λ≤ n+q.

(g) For p∈[_1,∞),Ω⁰ ⊂⊂_Ω,0< a≤_dist(_Ω⁰_,∂Ω)and u ∈ L^p,n(_Ω,R^N)set N_p,a(u;Ω⁰) = sup

x∈_Ω⁰,r≤a

−

Z

Br(x)

|u(y)−u_x,r|^pdy 1/p

. Then we have for each u∈ L^p,n(_Ω,R^N)

N_1,a(u;Ω⁰)≤ N_p,a(u;Ω⁰)≤c(p,n) _sup

x∈_Ω,r>0

−

Z

Ωr(x)|u(y)−u_x,r|²dy 1/2

.

For more details see [2,7,11,16].

Definition 2.3(see [14]). Let f ∈BMO(_Rⁿ)and η(f,R) =sup

ρ≤R

−

Z

Bρ(x)

|f(y)− fx,r|dy,

whereB_ρ(x)ranges over the class of the balls ofRⁿof radiusρ. We say that f ∈VMO(_Rⁿ)if

Rlim→0η(f,R) =0.

We can observe that substituting Rⁿ for Ω we obtain the definition of VMO(_Ω). Some basic properties of the above-mentioned classes are formulated in [1,14,16].

3 Main results

Suppose that for almost allx ∈_Ωand all u∈_R^N,z∈ _R^nN the following conditions hold:

|b(x,u,z)| ≤ f(x) +M(|u|^δ0+|z|^γ0), (3.1)

|g(x,u,z)| ≤F(x) +M

|u|^δ+h(|z|), (3.2) where

h(|z|) =







|z|

ln^s/2(e+t²₀) if|z| ≤t₀,

|z|

ln^s/2(e+|z|²) if|z|>t₀.

(3.3)

Here f ∈ L^2q0,λq0(_Ω), q₀ = n/(n+2), 0 < λ ≤ n, M is a positive constant, 1 ≤ δ₀ <

(n+₂)_/(n−₂)_{, 1} ≤ γ₀ < 1/q₀, F ∈ L^2,λ(_Ω)_{, 1} ≤ δ < n/(n−₂)_, s > _0, t₀ > 0. We remark thatt₀ =t₀(s)is chosen in such a way that, puttingh²(|z|) = H(|z|²), the function H= H(t) is nondecreasing on[0,∞), absolutely continuous on every closed interval of finite length and H(₀) = 0. The relationship between t₀ > _{0 and} s can be expressed through an inequality s≤(e+t₀)ln(e+t₀)/t₀.

Now we can state a result for the continuous case.

Theorem 3.1. Let u ∈W^1,2(_Ω,R^N)be a weak solution to the system(1.1)with(1.2)and the condi- tions(1.3),(3.1),(3.2)be satisfied. Suppose further that A∈C(_Ω,R^nN). Then

Du∈

(L^2,λ_loc(_Ω,R^nN), ifλ<n, L^2,λ_loc⁰(_Ω,R^nN)with arbitraryλ⁰ < n, ifλ=n.

Therefore,

u∈

(C^{0,(λ−n+2)/2}(_Ω,R^N), if n−2<λ<n, C^0,ϑ(_Ω,R^N)with arbitraryϑ<1, ifλ=n.

If the coefficients of the linear part of the system are supposed to be discontinuous, we have to modify the previous assumptions in the following way:

|b(x,u,z)| ≤ f(x) +M|z|^γ0_{, (3.4)}

|g(x,u,z)| ≤F(x) +M h(|z|), (3.5) hg(x,u,z),zi ≥ν₁h²(|z|)−l²(x), (3.6)

where f ∈ L^qq0,λq0(_Ω), F∈ L^q,λ(_Ω),q> 2,ν₁ is a positive constant,l∈ L^q,λ(_Ω)and the other constants and functions are supposed to be the same as in (3.1), (3.2).

The next theorem slightly extends the main result from [5].

Theorem 3.2. Let u∈ W^1,2(_Ω,R^N)be a weak solution to the system(1.1)with(1.2)and the condi- tions(1.3),(3.4),(3.5)and(3.6)be satisfied. Suppose further that A ∈L^∞∩VMO(_Ω,R^nN). Then

Du ∈

(L^2,λ_loc(_Ω,R^nN) ifλ<n, L^2,λ_loc⁰(_Ω,R^nN)with arbitraryλ⁰ <n ifλ=n.

Therefore,

u∈

(C^{0,(λ−n+2)/2}(_Ω,R^N) if n−2<λ<n, C^0,ϑ(_Ω,R^N)with arbitraryϑ<1 ifλ= n.

To obtain L^2,n-regularity for the first derivatives of the weak solution we strengthen the conditions on the coefficients gandb. Namely suppose that

|g(x,u,z₁)−g(y,v,z₂)| ≤M

|F(x)−F(y)|+ (|u|+|v|)^δ+h(|z₁−z₂|) (3.7) for a.e. x ∈ _Ωand allu,v∈ _R^N, z₁,z₂∈ _R^nN. Here F ∈ L^2,n(_Ω), g(·, 0, 0)∈ L^2,n(_Ω,R^nN). It is not difficult to see that (3.7) implies (3.2) withλ=n.

Now we can formulate the main result of the paper.

Theorem 3.3. Let u ∈ W^1,2(_Ω,R^N) be a weak solution to the system(1.1) with (1.2) and suppose that the conditions (1.3), (3.1) with f ∈ L^2q0,nq0(_Ω) and (3.7) with 0 < s ≤ 1 hold. Let further A∈C^0,α(_Ω,R^nN)for someα∈ (0, 1]. Then Du∈ L^2,n_loc(_Ω,R^nN).

4 Some lemmas

In this section we present results needed for the proofs of the theorems. In B_R(x) ⊂ _Rⁿ we consider a linear elliptic system (here the summation convention over repeated indices is used)

−Dα(_A^αβ_{ij Dβu}^j) =_{0, i}=_{1, . . . ,N (4.1)} with constant coefficients (according to the introduced denotation, the previous system can be written in the form −div(A·Du) =0) for which (1.3) holds.

Lemma 4.1([2, pp. 54–55]). Let u∈W^1,2(B_R(x)_,R^N)be a weak solution to the system(4.1). Then, for each0<σ≤ R,

Z

Bσ

|Du(y)|² dy≤L₁ σ R

nZ

BR

|Du(y)|² dy, Z

Bσ

|Du(y)−(Du)_σ|² dy≤L2

σ R

n+2Z

BR

|Du(y)−(Du)_R|² dy hold with constants L₁, L2 independent of the homothety.

The following lemma is fundamental for proving the theorems.

Lemma 4.2([9, pp. 537–538]). Letφbe a nonnegative function on(0,d]and let E₁, E₂, D,α, βbe nonnegative constants. Suppose thatφ(d)<_∞and

φ(σ)≤E₁σ R

α

+E2

φ(R) +DR^β, ∀0<σ≤ R≤d hold. Further let the constant k∈(0, 1)exist such thate=E₁k^α−β+E₂k^−β <1. Then

φ(σ)≤Cσ^β, ∀σ∈(0,d], where

C=max

( D

(1−e)k^β, sup

σ∈[kd,d]

φ(σ) σ^β

) . We set

v₀ =min

n

1− ⁿ−2 n+2δ₀

,n(1−q₀γ₀)

. (4.2)

Lemma 4.3([2, pp. 106–107]). Let u∈W^1,2(_Ω,R^N), Du∈ L^2,η(_Ω,R^nN),0≤η<n and(3.1)or (3.4)be satisfied. Then b∈ L^2q0,λ0(_Ω,R^N)and for each ball BR(x)⊂_Ωwe have

Z

BR(x)

|b(y,u,Du)|^2q0 dy≤c R^λ0, (4.3) where c = c(n,M,δ₀,γ₀,q₀, diamΩ,kfk_L2q0,λq0(_Ω),kuk_L1(_Ω,R^N),kDuk_L2,η(_Ω,R^nN)), λ₀ = min{λq₀, v₀+ηq₀}in the case(3.1) or c = c(n,M,γ₀,q₀, diamΩ,kfk_Lqq0,λq0(_Ω),kDuk_L2,η(_Ω,R^nN)) andλ₀ = min{n(1−2/q) +2λq0/q,n−(n−η)q0γ0}in the case(3.4).

In the case of discontinuous coefficients of the linear part of the system (1.1) with (1.2) we will use a result about higher integrability of the gradient of a weak solution to the system.

Proposition 4.4([7, p. 138]). Let u∈W_loc^1,2(_Ω,R^N)be a weak solution to the system(1.1)with(1.2) and the conditions(1.3), (3.4)–(3.6) be satisfied. Then there exists an exponent 2 < r < q such that u ∈W_loc^1,r(_Ω,R^N). Moreover there exists a constant c= c(ν,ν₁,L,kAk_L∞)andRe > 0such that, for all balls B_R(x)⊂_{Ω, R}<R, the following inequality is satisfiede

−

Z

BR/2(x)

|Du|^rdy 1/r

≤c (

−

Z

BR(x)

|Du|²dy 1/2

+

−

Z

BR(x)

(|l|^r+|F|^r)dy 1/r

+R

−

Z

BR(x)

|f|^rq0dy

1/rq0) .

Lemma 4.5 ([17, p. 37]). Letφ : [0,∞) → [0,∞)be a nondecreasing function which is absolutely continuous on every closed interval of finite length,φ(0) = 0. If w ≥ 0 is measurable and E(t) = {y∈_Rⁿ :w(y)>t}then

Z

Rⁿφ◦w dy=

Z _∞

0 m(E(t))φ⁰(t)dt.

In the proof of the theorems we will use a modification of Natanson’s lemma (for a proof see [6, pp. 8–9]). It can be read as follows.

Lemma 4.6. 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,∞) → _Rbe 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.

Remark 4.7. The foregoing estimate is optimal because if we put f(t) =1, t ∈[a,∞)then an equality will be achieved.

5 Proofs of the theorems

Proof of Theorem3.1. Let Ω0 ⊂⊂ _Ω, d₀ = dist(_Ω0,∂Ω), B_R = B_R(x₀) ⊂ _Ω, x₀ ∈ _Ω0 be an arbitrary ball and letw∈W₀^1,2(B_R/2(x0),R^N)be a solution to the system

Z

BR/2

h(A)_R/2Dw,Dϕi dx=

Z

BR/2

h((A)_R/2−A(x))Du,Dϕi dx

−

Z

B_R/2hg(x,u,Du)_,Dϕidx+

Z

B_R/2hb(x,u,Du)_,ϕidx for all ϕ ∈W₀^1,2(B_R/2,R^N). It is known (according to the linear theory and the Lax–Milgram theorem) that, under the assumption of this theorem, such solution exists and it is unique for all R < R⁰ (R⁰ ≤ 1 is sufficiently small). We can put ϕ = w in the previous equation and, using ellipticity, Hölder and Sobolev inequalities, we get

ν Z

BR/2

|Dw|²dx≤c Z

BR/2

|A_R/2−A(x)|²|Du|²dx+

Z

BR/2

|g(x,u,Du)|² dx

+ Z

BR/2

|b(x,u,Du)|^2q0 dx

1/q0!

=:c(I+II+III). (5.1) Now we obtain

I ≤ω²(R)

Z

B_R/2

|Du|²dx, (5.2)

whereω(R) =sup_{x,y∈Ω,|x−y|<R}|A(x)−A(y)|.

From the assumption (3.2) (taking into account (3.3) and the comments below it), putting mR(_t) =_{m y}∈BR(_x0)_:|Du|²> _t , we can estimateIIas follows.

II≤3 Z

BR/2

|F|²dx+3M² Z

BR/2

|u|^2δdx+

Z

BR/2

h²(|Du|)dx

≤c R^λ+ Z

BR/2

|u|^2n/(n−2)dx

δ(n−2)/n

R^{n(1−δ(n−2)/n)}+

Z _∞

0

d

dt(H(t))m_R/2(t)dt

!

≤c

R^λ+R^{n(1−δ(n−2)/n)}+J

, (5.3)

wherec= c(n,M,δ, diamΩ,kFk_Lq,λ,kuk_L2n/(n−2)). By means of Lemma 4.5and Lemma4.6we get

J =

Z _t0

0

d dt

t ln^s(_e+t²₀)

m_R/2(t)dt+

Z _∞

t₀

d dt

t ln^s(e+t)

m_R/2(t)dt

≤ ^κnt0 2ⁿln^s(e+t²₀)^R

n+ _sup

t0<t<_∞

1 t−t₀

Z _t

t0

d dw

w ln^s(e+w)

dw

Z _∞

t0

m_R/2(_w)_dw

≤ ^κnt0 2ⁿln^s(e+t²₀)^R

n+ sup

t0<ξ<_∞

1 ln^s(e+ξ)

1− ^sξ

(e+ξ)ln(e+ξ) _Z

B_R/2

|Du|²dy

≤ ^κnt0 2ⁿln^s(e+t²₀)^R

n+ ¹

ln^s(e+t₀)

Z

BR/2

|Du|²dx

≤ ¹

ln^s(e+t₀)

Z

BR

|Du|²dx+ ^t0 ln^s(e+t²₀)^R

n. (5.4)

From (5.3) and (5.4) we have II≤ c

1 ln^s(e+t0)

Z

B_R

|Du|²dx+ ^t0 ln^s(e+t²₀)^R

n+R^λ+R^{n(1−δ(n−2)/n)}

. (5.5)

We can estimateIIIby means of Lemma4.3(withη=0) and we have

III≤cR^λ0/q0. (5.6)

Together we have ν²

Z

BR/2

|Dw|²dx≤c

ω²(R) + ¹ ln^s(e+t₀)

Z

BR

|Du|²dx + ^t0

ln^s(e+t²₀)^R

n+R^λ+R^{n(1−δ(n−2)/n)}+R^λ0/q0

. (5.7)

The functionv=u−w∈W^1,2(B_R/2,R^N)is the solution to the system Z

BR/2

h(A)_R/2Dv,Dϕi dx=0, ∀ϕ∈W₀^1,2(B_R/2,R^N) and from Lemma4.1we have, for 0< σ≤R/2,

Z

Bσ

|Dv|²dx≤c σ R

nZ

B_R/2

|Dv|²dx.

By means of (5.7) and the last estimate we obtain, for all 0<σ≤ R, the following estimate:

Z

Bσ

|Du|²dx≤c₁ σ

R n

+ω²(R) + ¹ ln^s(e+t₀)

Z

BR

|Du|²dx +c₂

t₀ ln^s(e+t²₀)^R

n+R^λ+R^{n(1−δ(n−2)/n)}+R^λ0/q0

≤c₁ σ

R n

+ω²(R) + ¹ ln^s(e+t₀)

_Z

BR

|Du|²dx+c₂R^λ0,

where the constants c₁ and c₂ only depend on the above-mentioned parameters and λ⁰ = min{n,λ,n(1−δ(n−2)/n),λ₀/q₀} (λ⁰ < n). For η = 0 (see Lemma 4.3) we have λ⁰ = min{λ,n−(n−2)δ,(n+2)−(n−2)δ0,(n+2)−nγ0}. Set

φ(σ) =

Z

Bσ

|Du|²dx, E₁ =c₁, E₂ =c₁

ω²(R) + ¹ ln^s(_e+t₀)

, D=c₂.

Further we can choose k < 1 such that E₁k^n−λ0 < 1/2. It is obvious (the coefficients A are continuous) that the constants R0 > 0 and t0 > 0 exist such that E2k^−λ0 < 1/2, then E₁k^n−λ0+E₂k^−λ0 < 1. For all 0 < σ ≤ R ≤ min{d₀,R₀} the assumptions of Lemma 4.2 are satisfied and therefore

Z

Bσ

|Du|²dx≤cσ^λ

0

, ∀σ ≤min{d0,R0}.

If min{d₀,R₀}<diamΩ0, it is easy to check that for min{d₀,R₀} ≤σ ≤diamΩ0we have Z

Ωσ(x0)

|Du|²dx≤c

σ min{d₀,R₀}

λ⁰Z

Ω |Du|²dx, and thus we get

kDuk_L2,λ0(_Ω0,R^nN) ≤ckDuk_L2(_Ω,R^nN).

Ifλ= λ⁰the Theorem is proved. Ifλ⁰ <λthe previous procedure can be repeated withη=λ⁰ in Lemma 4.3. It is clear that after a finite number of steps (since λ⁰ increases in each step as it follows from Lemma4.3) we obtainλ⁰ =λ.

Proof of Theorem3.2. Using the same procedure as in the foregoing proof we get the inequality (5.1). The terms I,IIandIIIwe can estimate as follows.

From Proposition4.4with 2 <r < q, Hölder inequality (r⁰ =r/(r−2)) and from the fact that, for a BMO-function, all L^r norms, 1 ≤ r < _∞are equivalent (see Proposition 2.2(g)) we obtain

I ≤ _Z

B_R/2|A(x)−A_R/2|^2r0 dx

1/r⁰_Z

B_R/2

|Du|^rdx 2/r

≤c _Z

B_R/2

|A(x)−A_R/2|^2r0 dx 1/r⁰

× (

R^−n/r0 Z

BR

|Du|²dx+ _Z

BR

(|l|^r+|F|^r)dx 2/r

+R^{2+2n(1−1/q0)/r} _Z

BR

|f|^rq0dx

2/rq0)

≤cN_2r²0,R(A;Ω0) _Z

B_R

|Du|²dx+R^{2n/r−2(n−λ)/q}+R^{2+2n(1/r−1/q)−4/q+2λ/q} R^n/r0

≤cN_2r²0,R(A;Ω0) Z

BR

|Du|²dx+R^{n−2(n−λ)/q}

, (5.8)

wherec=c(r,klk_Lq,λ,kFk_Lq,λ,kfk_Lqq0,λq0)_.

From assumption (3.5) (taking into account (3.3) and the comments below it) we can esti- mateIIas follows.

II≤2 Z

BR/2

|F|²dx+2M² Z

BR/2

h²(|Du|)dx≤ c

R^{n−2(n−λ)/q}+

Z _∞

0

d

dt(H(t))m_R/2(t)dt

=:c

R^{n−2(n−λ)/q}+J

. (5.9)

The term J in the previous inequality can be estimated in the same way as in (5.4) and so (5.9) and (5.4) give us

II≤c

1 ln^s(e+t0)

Z

B_R

|Du|²dx+ ^t0 ln^s(e+t²₀)^R

n+R^{n−2(n−λ)/q}

. (5.10)

We can estimateIIIby means of Lemma4.3(withη=0) and we have

III≤cR^λ0/q0. (5.11)

Now (5.1) implies ν²

Z

B_R/2

|Dw|²dx≤c

N_2r²0,R(A;Ω0) + ¹ ln^s(_e+t₀)

_Z

BR

|Du|²dx + N_2r²0,R(A;Ω0) +1

R^{n−2(n−λ)/q}+ ^t0 ln^s(e+t²₀)^R

n+R^λ0/q0

. (5.12) The functionv=u−w∈W^1,2(B_R/2,R^N)is the solution to the system

Z

B_R/2

h(A)_R/2Dv,Dϕi dx=0, ∀ϕ∈W₀^1,2(B_R/2,R^N) and Lemma4.1gives us, for 0<σ ≤R/2,

Z

Bσ

|Dv|²dx≤c σ R

nZ

BR/2

|Dv|²dx.

Inequality (5.12) and the last estimate give us, for all 0 <σ≤ R, the following estimate:

Z

Bσ

|Du|²dx ≤c₁ σ

R n

+N_2r²0,R(A;Ω0) + ¹ ln^s(e+t₀)

Z

BR

|Du|²dx +c₂

N_2r²0,R(A;Ω0) +1

R^{n−2(n−λ)/q}+ ^t0 ln^s(e+t²₀)^R

n+R^λ0/q0

≤c₁ σ

R n

+N_2r²0,R(A;Ω0) + ¹ ln^s(e+t₀)

_Z

BR

|Du|²dx+c₂R^λ0,

where the constants c₁ and c₂ only depend on the above-mentioned parameters and λ⁰ = min{n−2(n−λ)/q,λ₀/q₀} (λ⁰ < n). For η = 0 (see Lemma 4.3) we have λ⁰ = min{n−2(n−λ)/q, 2+n(1−γ0)}. Set

φ(σ) =

Z

Bσ

|Du|²dx, E1= _{c1, E2}= _c1

N_2r²0,R(_A;Ω0) + ¹ ln^s(e+t₀)

, D=_c2. Further, we can choose k < 1 such that E₁k^n−λ0 < 1/2. It is obvious (the coefficients A are VMO) that the constants R0 > 0 and t0 > 0 exist such that E2k^−λ0 < 1/2, then E₁k^n−λ0+ E₂k^−λ0 <1. For all 0< σ≤ R≤min{d₀,R₀}the assumptions of Lemma 4.2are satisfied and therefore

Z

Bσ

|Du|²dx≤ cσ^λ0, ∀σ ≤min{d₀,R₀}.

The remaining part of the proof is analogous to the corresponding part of the proof of Theorem3.1.

Proof of Theorem3.3. Theorem3.1 gives that Du ∈ L^2,λ_loc(_Ω,R^nN)for arbitrary λ < n and, con- sequently, u ∈ C^0,α(_Ω,R^N)for each α ∈ (0, 1). Let B_R/2(x₀) ⊂ B_R(x₀) ⊂ _Ω be an arbitrary ball and letw ∈W₀^1,2(B_R/2(x₀),R^N)be a solution to the system (we denote B_R = B_R(x₀)and u_R=u_x0,R)

Z

BR/2

h(A)_R/2Dw,Dϕi dx=

Z

BR/2

h((A)_R/2−A(x))Du,Dϕidx

−

Z

B_R/2hg(x,u,Du)−(g(x,u,Du))_R/2,Dϕidx +

Z

B_R/2

hb(x,u,Du),ϕi dx (5.13) for every ϕ ∈ W₀^1,2(B_R/2,R^N). It is known that, under the assumption of the theorem, such solution exists and, it is unique for all R < R⁰ (R⁰ is sufficiently small, R⁰ ≤ 1). We can put ϕ=win (5.13) and using the ellipticity, Hölder’s and Sobolev’s inequalities, we get

ν² Z

BR/2

|Dw|²dx≤ c Z

BR/2

|A_R/2−A(x)|²|Du|²dx+

Z

BR/2

|g(x,u,Du)−(g(x,u,Du))_R/2|²dx +

Z

BR/2

|b(x,u,Du)|^2q0dx

1/q0!

=:c(I+II+III). (5.14) The estimate of I is analogous to that in the proof of Theorem 3.1, but here we have to use the Hölder continuity of coefficients, which is the crucial assumption for obtaining some reasonable estimate (using the information at the beginning of the proof).

I ≤cR^2α Z

B_R/2

|Du|²dx≤cRⁿ, whereα∈(0, 1]is a given constant.

Further, we estimate the second integral on the right hand side of (5.14). From the assump- tion (3.7) and by means of the Young inequality, we obtain

II≤ −

Z

B_R/2

_Z

B_R/2

|g(x,u(x)_,Du(x))−g(y,u(y)_,Du(y))|²dy

dx

≤3M−

Z

B_R/2

Z

B_R/2

|F(x)−F(y)|²dy

dx+3M−

Z

B_R/2

Z

B_R/2

(|u(x)|+|u(y)|)^2δ dy

dx +3M−

Z

BR/2

Z

BR/2

h²(|Du(x)−Du(y)|)dy

dx

≤12M Z

B_R/2

|F(x)−F_R/2|²dx+3·2^δ−nMκ_nkuk^2δ_C(B

R/2,R^N)Rⁿ +3M−

Z

BR/2

_Z

BR/2

|Du(x)−Du(y)|²

ln^s(e+|Du(x)−Du(y)|²)^dy

dx

=12M Z

BR/2

|F(x)−F_R/2|²dx+3·2^δ−nMκnkuk^2δ_C(B

R/2,R^N)Rⁿ+3M−

Z

BR/2

_Z

BR/2

K(x,y)dy

dx.

Using the fact that the functionl(t) =t²/ ln^s(e+t²)is nondecreasing and convex on[0,∞) if 0<s ≤1 and so the functionk(z) =l(|z|)is convex onR^nN, we can get the estimate

K(x,y) = |Du(x)−(Du)_R/2+ (Du)_R/2−Du(y)|² ln^s(e+|Du(x)−(Du)_R/2+ (Du)_R/2−Du(y)|²)

≤ ¹ 2

|2(Du(x)−(Du)_R/2)|²

ln^s(e+|2(Du(x)−(Du)_R/2)|²)+ ¹ 2

|2(Du(y)−(Du)_R/2)|² ln^s(e+|2(Du(y)−(Du)_R/2)|²)

which gives

−

Z

B_R/2

_Z

B_R/2K(x,y)dy

dx ≤

Z

B_R/2

4|Du(x)−(Du)_R/2|²

ln^s(e+4|Du(x)−(Du)_R/2|²)^dx=_: J_A and so

II≤12M Z

B_R/2

|F(x)−F_R/2|²dx+3.2^δ−nMκ_nkuk^2δ_C(B

R/2,R^N)Rⁿ+3M J_A. (5.15) The quantity J_A can be estimated in a way, analogous to that in (5.4). Putting m_R(t) = m({y ∈ B_R(x₀) : 4|Du−(Du)_R|² > t})and H(t) = t/ ln^s(e+t), t ∈ [0,∞), we get (through Lemmas4.5and4.6)

JA=

Z _∞

0

d

dt(H(_t))_mR/2(_t)_dt

=

Z _t0

0

d dt

t ln^s(e+t)

m_R/2(t)dt+

Z _∞

t₀

d dt

t ln^s(e+t)

m_R/2(t)dt

≤ ⁴

ln^s(e+t0)

Z

B_R

|Du−(Du)_R|²dx+ ^t0 ln^s(e+t0)^R

n. (5.16)

From (5.15) and (5.16) we have II≤c

4 ln^s(e+t₀)

Z

BR

|Du−(Du)_R|²dx+M

t₀

Mln^s(e+t₀)+kFk_L2,n(_Ω)+kuk^2δ_C(B

R/2,R^N)

Rⁿ

. The termIIIwe can estimate in the following manner. In Lemma 4.3(remember that λ= n), thanks to Theorem3.1, the parameterη< ncan be chosen arbitrarily close ton. Consequently, λ₀can be bigger than value nq₀and so

III≤cRⁿ. Now, using the estimates I,II,III, from (5.14) we have

ν² Z

B_R/2

|Dw|²dx≤c 1 ln^s(e+t0)

Z

B_R

|Du(x)−(Du)_R|²dx+c Rⁿ. (5.17) The functionv=u−w∈W^1,2(B_R/2,R^N)is the solution to the system

Z

B_R/2

h(A)_R/2Dv,Dϕi dx=0, ∀ϕ∈W₀^1,2(B_R/2,R^N) and Lemma4.1gives us, for 0<σ ≤R/2,

Z

Bσ

|Dv(x)−(Dv)_σ|²dx≤ cσ R

n+2Z

B_R/2

|Dv−(Dv)_R/2|²dx.

Inequality (5.17) and the last estimate give us, for all 0 <σ≤ R, the following estimate:

Z

Bσ

|Du(x)−(Du)_σ|²dx ≤c₁ σ

R n+2

+ ¹

ln^s(e+t₀) Z

BR

|Du(x)−(Du)_R|²dx+c2Rⁿ, where the constantsc₁andc₂only depend on the above-mentioned parameters.

If we putφ(R) = R

BR|Du(x)−(Du)_R|²dx,α=n+2, β=n, E₁= c₁, E₂= c₁/ ln^s(e+t₀), D = c₂ and use Lemma 4.2, the result follows in a standard way, analogous to those in the previous proofs. So we can conclude thatDu ∈ L^2,n_loc(_Ω,R^nN)_.

Remark 5.1. It is known that for weak solutions u ∈ W^1,∞(_Ω,R^N) to the system (1.1) the Hölder continuity of their gradients is, broadly speaking, equivalent to the fact that the con- dition of Liouville type is satisfied (see [12, Chapter 6] for precise information). Later the first author of the paper proved in [3] that the same holds under the assumption that gradients of weak solutions belong to the class L^2,n(_Ω,R^nN). So the paper [4] and the statement of Theorem3.3could be seen as contributions to the above mentioned theory.

Acknowledgements

This work was supported by the research projects Slovak Grant Agency No. 1/0071/14 and No. 1/0078/17.

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