L 2,Φ regularity for nonlinear elliptic systems of second order ∗

L ^2,Φ regularity for nonlinear elliptic systems of second order ^∗

Josef Danˇ eˇ cek & Eugen Viszus

Abstract

This paper is concerned with the regularity of the gradient of the weak solutions to nonlinear elliptic systems with linear main parts. It demonstrates the connection between the regularity of the (generally dis- continuous) coefficients of the linear parts of systems and the regularity of the gradient of the weak solutions of systems. More precisely: If above- mentioned coefficients belong to the classL^∞(Ω)∩ L^2,Ψ(Ω) (generalized Campanato spaces), then the gradient of the weak solutions belong to L^2,Φloc(Ω,R^nN), where the relation between the functions Ψ and Φ is for- mulated in Theorems 3.1 and 3.2 below.

1 Introduction

In this paper, we consider the problem of the regularity of the first derivatives of weak solutions to the nonlinear elliptic system

−Dαa^α_i(x, u, Du) =ai(x, u, Du), i= 1, . . . , N, (1.1) wherea^α_i,aiare Caratheodorian mappings from (x, u, z)∈Ω×R^N ×R^nN into R, N >1, Ω⊂Rⁿ, n≥3 is a bounded open set. A function u∈W_loc^1,2(Ω,R^N) is called a weak solution of (1.1) in Ω if

Z

Ω

a^α_i(x, u, Du)Dαϕⁱ(x)dx= Z

Ω

ai(x, u, Du)ϕⁱ(x)dx, ∀ϕ∈C₀^∞(Ω,R^N).

We use the summation convention over repeated indices.

As it is known, in case of a general system (1.1), only partial regularity can be expected forn >2 (see e.g. [2, 6, 9]). Under the assumptions below we will prove L^2,Φ-regularity of gradient of weak solutions for the system (1.1) whose coefficientsa^α_i have the form

a^α_i(x, u, Du) =A^αβ_ij (x)Dβu^j+g^α_i(x, u, Du), (1.2)

∗Mathematics Subject Classifications: 49N60, 35J60.

Key words: Nonlinear equations, regularity, Morrey-Campanato spaces.

c2002 Southwest Texas State University.

Submitted May 31, 2001. Published February 19, 2002.

J. Danˇeˇcek was partially supported by the research project MSM no. 261100006

1

where i, j = 1, . . . , N, α, β = 1, . . . , n, A^αβ_ij is a matrix of functions, and the following condition of strong ellipticity

A^αβ_ij (x)ξⁱ_αξ^j_β≥ν|ξ|², a.e.x∈Ω, ∀ξ∈R^nN;ν >0 (1.3) holds, and g^α_i are functions with sublinear growth in z. In what follows, we formulate the conditions on the smoothness and the growth of the functions A^αβ_ij ,g_i^αandai precisely.

It is well known (see [2]) that, in the case of linear elliptic systems with con- tinuous coefficientsA^αβ_ij , the gradient of weak solutions has theL^2,λ-regularity and, if the coefficients A^αβ_ij belong to some H¨older class, then the gradient of weak solutions belongs to the BMO-class (functions with bounded mean os- cillations, see Definition 2.1). These results were generalized in [3] where the first author has proved the L^2,λ-regularity of the gradient of weak solutions to (1.1)-(1.3) in the situation where the coefficients A^αβ_ij are continuous and the BMO-regularity of gradient in the case where coefficients A^αβ_ij are H¨older continuous.

In the case of linear elliptic systems when the coefficients A^αβ_ij are “small multipliers of BM O(Ω)”, a class neither containing nor contained in C(Ω), Acquistapace in [1] proved global (under Dirichlet boundary condition) and local BMO-regularity for the gradient of solutions. In [1] the local BMO-regularity does not follow in a standard way from the global one, because there are no regularity results in the Morrey spaces L^2,λ, 0< λ < n. The last mentioned fact was a motive for [4] and [5]. In [4, 5] the Morrey regularity for the gradient of weak solutions for nonlinear elliptic systems of type (1.1) is proved when the coefficientsA^αβ_ij are generally discontinuous (not necessarily “small multipliers ofBM O(Ω)”).

The purpose of this paper is a generalization of the results from [4] and [5].

Result of this paper may open a way to proving the BMO-regularity for the gradient of solutions of (1.1).

If we want to sketch our method of proof, we must say that its crucial point is the assumption onA^αβ_ij : A^αβ_ij ∈L^∞(Ω)∩ L^2,Ψ(Ω) (for the definition see below).

Taking into account higher integrability of gradient Du (for some r > 2), we obtainL^2,Φ-regularity of the gradient.

2 Notation and definitions

We consider the bounded open set Ω⊂Rⁿ with pointsx= (x1, . . . xn),n≥3, uΩ → R^N, N ≥ 1, u(x) = (u¹(x), . . . , u^N(x)) is a vector-valued function, Du = (D1u, . . . , Dnu), 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. If x∈Rⁿ and r is a positive real number, we writeB_r(x) ={y∈Rⁿ:|y−x|< r}, i.e., the open ball inRⁿ, Ω(x, r) = Ω∩B_r(x). Denote byu_x,r =|Ω(x, r)|⁻n¹

R

Ω(x,r)u(y)dy= R

Ω(x,r)

− u(y)dythe mean value of the functionu∈L¹(Ω,R^N) over the set Ω(x, r),

where|Ω(x, r)|nis the n-dimensional Lebesgue measure of Ω(x, r). The bounded domain Ω⊂ Rⁿ is said to be of type A if there exists a constantA >0 such that, for everyx∈Ω and all 0< r <diam Ω, it holds|Ω(x, r)|n≥ Arⁿ. Beside the usually used spaceC₀^∞(Ω,R^N), the H¨older spacesC^0,α(Ω,R^N),C^0,α(Ω,R^N) and the Sobolev spaces W^k,p(Ω,R^N),W_loc^k,p(Ω,R^N),W₀^k,p(Ω,R^N) (see, e.g.[8]), 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 Morrey spaceL^q,λ(Ω,R^N) if

||u||^q_Lq,λ(Ω,R^N)= sup

x∈Ω,r>0

1 r^λ

Z

Ω(x,r)

|u(y)|^qdy <∞.

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

[u]^q

L^q,λ(Ω,R^N)= sup

r>0,x∈Ω

1 r^λ

Z

Ω(x,r)

|u(y)−ux,r|^qdy <∞.

LetQ0⊂Rⁿ is a cube whose edges are parallel with the coordinate axes. The BM O(Q0,R^N) space (bounded mean oscillation space) is the subspace of such functions u∈L¹(Q0,R^N) for which

huiQ₀= sup

Q⊂Q₀

1

|Q| Z

Q

|u(y)−uQ|dy <∞, where uQ=R

−Qu(y)dy andQ⊂Q0 is the cube homotetic withQ0.

Remark u∈L^q,λ_loc(Ω,R^N) if and only ifu∈L^q,λ(Ω₀,R^N) for each Ω₀⊂⊂Ω.

Proposition 2.1 For a domainΩ⊂Rⁿ of the class C^0,1 we have the following (a) With the normskukL^q,λ andkuk_L^q,λ =kukL^q+[u]_Lq,λ,kukBM O=kukL¹+ huiQ,L^q,λ(Ω, R^N),L^q,λ(Ω, R^N)andBM O(Q0,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 theL^∞(Ω,R^N)(L^q,n(Ω,R^N),1≤q <∞. (d) L^2,n(Ω,R^N)is isomorphic to the L^q,n(Ω,R^N)and

L^q,n(Q,R^N) =BM O(Q,R^N),Qbeing 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)forn < λ≤n+q.

For more details see [2, 6, 8, 9].

The generalization of Campanato spacesL^q,λ (see [2]) are the classes L^2,Ψ introduced by Spanne [10] and [11].

Definition 2.2 LetΨ be a positive function on (0,diam Ω]. A function u∈ L²(Ω,R^N) is said to belong toL^2,Ψ(Ω,R^N) if

[u]_2,Ψ,Ω= sup

x∈Ω,r∈(0,diam Ω]

1 Ψ(r)

Z

Ω(x,r)

|u(y)−u_x,r|²dy^1/2

<∞

and byl^2,Ψ(Ω,R^N) we denote the subspace of allu∈ L^2,Ψ(Ω,R^N) such that [u]2,Ψ,Ω,r0 = sup

x∈Ω,r∈(0,r0]

1 Ψ(r)

Z

Ω(x,r)

|u(y)−ux,r|²dy^1/2

=o(1) asr0&0.

Some basic properties of the above-mentioned spaces are formulated in the following proposition (for the proofs see [1, 10, 11]).

Proposition 2.2 For a domain Ω⊂Rⁿ of the classC^0,1we have the following (a) L^2,Ψ(Ω,R^N)is a Banach space with normkuk_L^2,Ψ(Ω,R^N)=kukL²(Ω,R^N)+

[u]_L2,Ψ(Ω,R^N).

(b) LetΨ(r) =r^n/2/(1 +|lnr|). Then C⁰(Ω,R^N)\ L^2,Ψ(Ω,R^N)and (L^∞(Ω,R^N)∩l^2,Ψ(Ω,R^N))\C⁰(Ω,R^N) are not empty.

In the sequel we assume thatΨ : (0, d]→(0,∞) has the form

Ψ(r) =r^ζ/2ψ(r), 0≤ζ≤n+ 2, (2.1) where ψ is a continuous, non-decreasing function such that lim_r→0+ψ(r) = 0 and r→ψ(r)/r^ξ for some ξ >0 is almost decreasing, i.e. there existsk_ψ ≥1 andd0≤dsuch that

k_ψψ(r)

r^ξ ≥ψ(R)

R^ξ , ∀0< r < R≤d₀. (2.2) Remark The functionψ(r) = 1/(1 +|lnr|) satisfies (2.2) with an arbitrary ξ >0.

3 Main results

Suppose that for all (x, u, z)∈Ω×R^N ×R^nN the following conditions hold:

|ai(x, u, z)| ≤fi(x) +L|z|^γ0, (3.1)

|g_i^α(x, u, z)| ≤f_i^α(x) +L|z|^γ, (3.2) g^α_i(x, u, z)zⁱ_α≥ν1|z|^1+γ−f²(x) (3.3) for almost all x ∈ Ω and all u ∈ R^N, z ∈ R^nN. Here L, ν₁ are positive constants, 1≤γ₀<(n+ 2)/n, 0≤γ <1,f, f_i^α ∈L^σ,λ(Ω),σ >2, 0 < λ≤n, fi ∈ L^σq0,λq0(Ω), q0 = n/(n+ 2). We set A = (A^αβ_ij ), g = (g_i^α), a = (ai), fe= (fi),e

fe= (f_i^α).

The next theorem is slightly generalizing the main result from [4].

Theorem 3.1 Letu∈W_loc^1,2(Ω,R^N)be a weak solution to the system (1.1) and the conditions (1.2), (1.3), (3.1), (3.2) and (3.3) be satisfied. Suppose further that A^αβ_ij ∈ L^∞(Ω)∩ L^2,Ψ(Ω), i, j = 1, . . . , N, α, β = 1, . . . , n and Ψ is a function satisfying the condition (2.1) with ζ=n. 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) ifn−2< λ < n C^0,ϑ(Ω, R^N)with arbitraryϑ <1 ifλ=n.

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

|a_i(x, u, z)| ≤f_i(x) +L|z|^γ0 (3.4)

|g^α_i(x, u, z1)−g^α_i(y, v, z2)| ≤L(|f_i^α(x)−f_i^α(y)|+|z1−z2|^γ) (3.5) g_i^α(x, u, z)zⁱ_α≥ν1|z|^1+γ−f²(x). (3.6) for a.e. x ∈ Ω and all u, v ∈ R^N, z1, z2 ∈ R^nN. Here L, ν1 are positive constants, 1≤γ0<(n+ 2)/n, 0≤γ < 1,f, f_i^α ∈ L^2,n(Ω),fi ∈ L^σq0,nq0(Ω), σ >2,q0=n/(n+ 2). It is not difficult to see that from assumptions (3.4)–(3.6) follow (3.1)–(3.3) withλ=n.

We can now formulate the main result of this paper.

Theorem 3.2 Letu∈W_loc^1,2(Ω,R^N)be a weak solution to the system (1.1) and suppose that the conditions (1.2), (1.3), (3.4), (3.5) and (3.6) hold. Let further A^αβ_ij ∈L^∞(Ω)∩ L^2,Ψ(Ω), for eachi,j = 1, . . . , N,α,β = 1, . . . , n andΨ be a function satisfying the conditions (2.1) and (2.2) with ζ = n and 0 < ξ ≤ 2.

ThenDu∈ L^2,Φ_loc(Ω,R^nN)withΦ(R) =R^n/2in the case when the functionψhas a form of some power function and Φ(R) =R^λ/2ψ^(r−2)/2r(R) for some r >2 and arbitrary λ < nin another cases.

Remark The conditions (2.1) and (2.2) in Theorem 3.2 are for example sat- isfied with the functionψ(r) = 1/(1 +|lnr|) (see also Proposition 2.2(b)).

4 Some lemmas

In this section we present the results needed for the proof of the main theorem.

InB_R(x)⊂Rⁿ we consider a linear elliptic system

−D_α(A^αβ_ij D_βu^j) = 0 (4.1) with constant coefficients for which (1.3) holds.

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

Z

Bσ

|Du(y)|²dy≤cσ R

nZ

B_R

|Du(y)|²dy, Z

B_σ

|Du(y)−(Du)σ|²dy≤cσ R

ⁿ⁺²Z

B_R

|Du(y)−(Du)R|²dy

hold with a constantc independent of the homotethie.

The following lemma generalizes [7, Lemma 3.1] and is fundamental for prov- ing Theorem 3.2.

Lemma 4.2 Let ψ be a function from the condition (2.1). Further let φ be a nonnegative function on (0, d] and A, B, C, α, β be nonnegative constants.

Suppose that for all 0< σ < R≤d, we have:

φ(σ)≤h Aσ

R ^α

+Bi

φ(R) +C R^βψ(R), (4.2)

φ(d)<∞. (4.3)

Further let the constant 0 < K < 1 exist such that ε = kψ(AK^α−β−ξ + BK^−β−ξ)<1. Thenφ(σ)≤c σ^βψ(σ), for0< σ≤d, where

c= maxn Ckψ

(1−ε)K^β+ξ, sup

r∈[Kd,d]

φ(r) r^βψ(r)

o .

Proof From (4.2) and (4.3), it follows that sup_r∈[σ,d]φ(r)<∞. We set cn= sup

r∈[1/n,d]

φ(r) r^βψ(r).

It is obvious thatc_n≤c₀= sup_r∈(0,d]φ(r)/r^βψ(r).

Whenc0= sup_r∈[Kd,d]φ(r)/r^βψ(r), we have the result. Also c0> sup

r∈[Kd,d]

φ(r) r^βψ(r)

and there exists a sequence{r_n}^∞n=n₀ such that 1/n < r_n< Kd and

φ(rn) r^βnψ(rn)−cn

< cn

n.

Putσ=r_n,R=r_n/K in (4.2) and using (2.2) we get K^ξ

kψ

φ(r_n)

r^βnψ(rn) ≤ φ(r_n)

rn^βψ(^r_Kⁿ) ≤(AK^α−β+BK^−β) φ(^r_Kⁿ)

(^r_Kⁿ)^βψ(^r_Kⁿ)+CK^−β

and thus φ(r_n)

r^βnψ(rn) ≤k_ψ(AK^α−β−ξ+BK^−β−ξ) φ(^r_Kⁿ)

(^r_Kⁿ)^βψ(^r_Kⁿ)+Ck_ψK^−β−ξ. Asr_n/K ∈[1/n, d], we have

φ(^r_Kⁿ)

(^r_Kⁿ)^βψ(^r_Kⁿ) ≤cn

and also

cn−cn

n ≤ φ(rn)

(rn)^βψ(rn) ≤εcn+CkψK^−β−ξ. Then

1−ε−1 n

cn ≤CkψK^−β−ξ.

Forn→ ∞, we get the statement of this lemma.

The following lemma is a special case of [3, Lemma 3.4].

Lemma 4.3 ([3, pp. 757-758]) (i) Letu∈W_loc^1,2(Ω,R^N),Du∈L^2,τ(Ω,R^nN), 0≤τ < nand (3.1) be satisfied withf_i ∈L^2q0,µ0q0(Ω),0< µ₀≤n. Then a_i∈L^2q0,λ0(Ω) and for each ballB_R(x)⊂Ω we have

Z

B_R(x)

|ai(x, u, Du)|^2q0dy≤c R^λ0, (4.4) wherec=c(n, L, γ0,diam Ω,kfekL^2q0,µ0q0(Ω,R^N),kDukL²(Ω,R^nN)) andλ0= min{µ0q0, n−(n−τ)q0γ0}.

(ii) Letu∈W_loc^1,2(Ω,R^N)and (3.2) be satisfied withf_i^α∈L^2,λ(Ω),0< λ≤n.

Then, for eachε∈(0,1)and all B_R(x)⊂Ω, Z

B_R(x)

|g^α_i(x, u, Du)|²dy≤c(L)ε Z

B_R(x)

|Du|²dy+c R^λ. (4.5)

wherec=c(n, L, ε, γ,diam Ω,kefekL^2,λ(Ω,R^nN),kDukL²(Ω,R^nN)).

For the proof of (i) can be found in [2, pp. 106-107] and the proof (ii) in [5].

In the following considerations we will use a result about higher integrability of the gradient of a weak solution of the system (1.1).

Proposition 4.4 ([6, p. 138]) Suppose that (1.2), (1.3), (3.1)–(3.3) or (3.4)–

(3.6) are fulfilled and let u∈W_loc^1,2(Ω,R^N) be a weak solutions of (1.1). Then there exists an exponentr >2such thatu∈W_loc^1,r(Ω,R^N). Moreover there exists a constantc=c(ν, ν₁, L,kAkL^∞)andR >e 0such that, for all ballsB_R(x)⊂Ω, R <R, the following inequality is satisfiede

Z

B_R/2(x)

−|Du|^rdy1/r

≤cn Z

B_R(x)

−|Du|²dy1/2

+ Z

B_R(x)

−(|f|^r+|fee|^r)dy1/r

+R Z

B_R(x)

−|fe|^rq0dy1/rq0o .

5 Proof of Theorems

Proof of Theorem 3.1. 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 of the following system

Z

B_R/2(x₀)

(A^αβ_ij )x₀,R/2Dβw^jDαϕⁱdx

= Z

B_R/2(x₀)

(A^αβ_ij )_x0,R/2−A^αβ_ij (x)

D_βu^jD_αϕⁱdx

− Z

B_R/2(x0)

g^α_i(x, u, Du)D_αϕⁱdx+ Z

B_R/2(x0)

a_i(x, u, Du)ϕⁱdx (5.1)

for allϕ∈W₀^1,2(BR/2(x0),R^N). It is known that, under the assumption of this theorem, such solution exists and it is unique for allR < R⁰ (R⁰ is sufficiently small). We can putϕ =w in (5.1) and, using ellipticity, H¨older and Sobolev inequalities, we obtain

ν Z

B_R/2(x₀)

|Dw|²dx≤cZ

B_R/2(x₀)

|Ax₀,R/2−A(x)|²|Du|²dx +

Z

B_R/2(x₀)

|g(x, u, Du)|²dx+ Z

B_R/2(x₀)

|a(x, u, Du)|^2q0dx1/q0

=c(I+II+III).

From Proposition 4.4 withr >2, H¨older inequality (r⁰ =r/(r−2)) and from the fact that, for a BMO-function, allL^r norms, 1≤r <∞are equivalent (see Proposition 2.1(d)) we obtain

I≤cZ

B_R/2(x₀)

|A(x)−A_x0,R/2|^2r0dx^1/r0Z

B_R/2(x₀)

|Du|^rdx^2/r

(5.2)

From the assumptions of this theorem and taking into account the properties of matrixA= (A^αβ_ij ) we can estimate the first term on the right hand side of (5.2)

Z

B_R/2(x0)

|A(x)−A_x0,R/2|^2r0dx≤cZ

B_R/2(x0)

|A(x)−A_x0,R/2|²dx1/2

×

×Z

B_R/2(x0)

|A(x)−A_x0,R/2|^2(2r0−1)dx^1/2

≤ c(n,[A]_2,Ψ,Ω)kAk^2r_L∞⁰⁻(Ω,R^{1 n2N2})Rⁿψ(R). (5.3)

To estimate the last integral in (5.2) we use Proposition 4.4 obtaining Z

B_R/2(x₀)

|Du|^rdx^2/r

≤cn 1 R^n(1−2/r)

Z

B_R(x)

|Du|²dy +Z

BR(x)

(|f|^r+|fee|^r)dy2/r

+R^2(1−2/r)Z

BR(x)

|fe|^rq0dy2/rq0o

≤c 1 R^n(1−2/r)

Z

B_R(x)

|Du|²dy+R^2λ/r+R^{2(r−2+λ)/r)} ,

(5.4) wherec=c(r,kfkL^r,λ(Ω),ke

fekL^r,λ(Ω),kfekL^rq0,λq0(Ω)). From (5.2), (5.3) and (5.4) we obtain

I≤c

ψ^1/r0(R) Z

BR(x0)

|Du|²dx+ (R^2λ/r+R^{2(r−2+λ)/r)})R^n/r0ψ^1/r0(R)

≤c

ψ^1/r0(R) Z

B_R(x₀)

|Du|²dx+ R^{n−2(n−λ)/r}ψ^1/r0(R) ,

wherec=c(n, r,[A]_2,Ψ,Ω,kAk_L∞(Ω,R^n2N2),kfkL^r,λ(Ω),kfeekL^r,λ(Ω),kfekL^rq0,λq0(Ω)).

We can estimate II and III by means of Lemma 4.3 (withτ = 0) and we have ν²

Z

B_R/2(x0)

|Dw|²dx≤cn

(ε+ψ^1/r0(R)) Z

BR(x0)

|Du|²dx+R^µo

, (5.5) where µ= min{n, n−2(n−λ)/r, n+ 2−nγ₀}.

The functionv=u−w∈W^1,2(B_R/2(x₀),R^N) is the solution of the system Z

B_R/2(x₀)

(A^αβ_ij )_x0,R/2D_βv^jD_αϕⁱdx= 0, ∀ϕ∈W₀^1,2(B_R/2(x₀),R^N). (5.6)

From Lemma 4.1 we have, for 0< σ≤R/2, Z

Bσ(x0)

|Dv|²dx≤cσ R

nZ

B_R/2(x0)

|Dv|²dx.

By means of (5.5) and the last estimate we obtain, for all 0 < σ ≤ R and ε∈(0,1), the following estimate

Z

Bσ(x0)

|Du|²dx≤c1

hσ R

n

+ε+ψ^1/r0(R)iZ

BR(x0)

|Du|²dx+c2R^µ, where the constantsc1andc2only depend on the above-mentioned parameters.

Now, in a way analogous to that from [5], we obtain the result.

Proof of Theorem 3.2. By Theorem 3.1, Du ∈ L^2,λ_loc(Ω,R^nN) for arbi- trary λ < n. Let B_R/2(x0) ⊂ BR(x0) ⊂ Ω be an arbitrary ball and let

w∈W₀^1,2(B_R/2(x0),R^N) be a solution of the following system Z

B_R/2(x0)

(A^αβ_ij )_x0,R/2Dβw^jDαϕⁱdx

= Z

B_R/2(x₀)

(A^αβ_ij )x₀,R/2−A^αβ_ij (x)

Dβu^jDαϕⁱdx

− Z

B_R/2(x₀)

g_i^α(x, u, Du)−(g^α_i(x, u, Du))_x0,R/2

D_αϕⁱdx

+ Z

B_R/2(x₀)

a_i(x, u, Du)ϕⁱdx

(5.7)

for allϕ∈W₀^1,2(B_R/2(x₀),R^N). It is known that, under the assumption of The- orem 3.2, such solution exists and, it is unique for allR < R⁰ (R⁰ is sufficiently small,R⁰≤1). We can putϕ=win (5.7) and using the ellipticity, the H¨older and the Sobolev inequalities, we obtain

ν² Z

B_R/2(x₀)

|Dw|²dx≤cZ

B_R/2(x₀)

|Ax₀,R/2−A(x)|²|Du|²dx +

Z

B_R/2(x₀)

|g_i^α(x, u, Du)−(g^α_i(x, u, Du))_x0,R/2|²dx +

Z

B_R/2(x₀)

|a(x, u, Du)|^2q0dx^1/q0

=c(I+II+III).

(5.8) The estimate of I is analogous to that in Theorem 3.1 and we have

I≤c ψ^1/r0(R) Z

BR(x0)

|Du|²dx+c R^2λ/r+R^{2(r−2+λ)/r)}

R^n/r0ψ^1/r0(R)

≤cZ

B_R(x₀)

|Du|²dx+c R^{n−2(n−λ)/r}

ψ^1/r0(R)

≤c R^λ+R^{n−2(n−λ)/r}

ψ^1/r0(R)≤c R^λψ^1/r0(R),

wherec=c(n, r,kAk_L∞(Ω,R^n2N2),kfkL^r,λ(Ω),kefekL^r,λ(Ω),kfekL^rq0,λq0(Ω)).

Further, we estimate the second integral on the right hand side of (5.8).

From the assumption (3.5) and by means of Young inequality, we obtain II≤

Z

B_R/2(x₀)

− Z

B_R/2(x₀)

|g_i^α(x, u(x), Du(x))−g_i^α(y, u(y), Du(y))|²dy dx

≤c Z

B_R/2(x0)

|efe−(efe)_x0,R/2|²dx+ Z

B_R/2(x0)

|Du−(Du)_x0,R/2|^2γdx

≤c ε

Z

BR(x0)

|Du−(Du)x₀,R|²dx+c(ε, γ,ke

fek²L^2,n(Ω,R^nN))Rⁿ ,

whereε∈(0,1) is arbitrary.

We can estimate III by means of Lemma 4.3 (withτ =λ,µ0=n) and, using the estimate I, II, we have

ν² Z

B_R/2(x0)

|Dw|²dx≤c ε Z

BR(x0)

|Du(y)−(Du)x₀,R|²dy +c Rⁿ+R^λψ^1/r0(R) +R^{n+2−(n−λ)γ0}

≤c ε Z

BR(x0)

|Du(y)−(Du)x₀,R|²dy+c Φ²(R), (5.9)

where Φis defined in the formulation of Theorem 3.2.

The functionv=u−w∈W^1,2(BR/2(x0),R^N) is the solution of the system Z

B_R/2(x₀)

(A^αβ_ij )_x0,R/2D_βv^jD_αϕⁱdx= 0, ∀ϕ∈W₀^1,2(B_R/2(x₀),R^N).

From Lemma 4.1, we have, for 0< σ≤R/2 Z

B_σ(x₀)

|Dv−(Dv)_x0,σ|²dx≤cσ R

ⁿ⁺²Z

B_R/2(x₀)

|Dv−(Dv)_x0,R/2|²dx. (5.10) By means of (5.9) and (5.10) we obtain for all 0 < σ≤ R and ε∈ (0,1), the following estimate

Z

Bσ(x0)

|Du(x)−(Du)x₀,σ|²dx

≤c1

σ R

ⁿ⁺² +ε

Z

BR(x0)

|Du(x)−(Du)x₀,R|²dx+c2Φ²(R), where the constantsc₁andc₂only depend on the above-mentioned parameters.

Now from Lemma 4.2 we get the result in the following manner. In the case Φ(R) = R^n/2, the result is obvious. In other cases if we put φ(R) = R

BR(x0)|Du(x)−(Du)_x0,R|²dx,α=n+ 2,β=λ,A=c₁,B =c₁εandC=c₂, we can choose 0< K <1 such thatAk_ψK^{n+2−λ−ξ} <1/2. It is obvious that a constantε >0 exists such thatBk_ψK^−λ−ξ=c₁εk_ψK^−λ−ξ<1/2 and then, for all 0< σ≤R < R₀, R < R₀, the assumptions of Lemma 4.2 are satisfied and therefore

Z

BR(x0)

|Du(x)−(Du)x₀,R|²dx≤c Φ²(R).

From this follows thatDu∈ L^2,Φ_loc(Ω,R^N).

Remark In [2] for a linear system and in [3] for a nonlinear system (1.1), (1.2), it is proved that the gradient of solutionDu∈BM O(Ω₀,R^nN), Ω₀⊂⊂Ω in a situation where the coefficients A^αβ_ij ∈ C^0,γ(Ω), γ ∈ (0,1). Taking into account that for Ψ(R) =R^γ+n/2 we have L^2,Ψ =C^0,γ, one may prove by the method used in the proof of Theorem 3.2 (which is different from the methods in [2] and [3]) the above results too.

Remark In [1] the local BMO-regularity for the gradient of weak solutions of linear elliptic systems is proved. This result was obtained using the global BMO-regularity result and theL^p-regularity result of gradient for all 1< p <∞. Using the global BMO-regularity result from [1] and Theorem 3.2 one may obtain the local BMO-regularity of the gradient too.

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Univ. Carolinae, Vol. 27, No. 4, 1986, pp. 755-764

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Josef Danˇeˇcek

Department of Mathematics, Faculty of Civil Engineering, Ziˇˇzkova 17, 60200 Brno, Czech Republic

e-mail: [email protected]

Eugen Viszus

Department of Mathematical Analysis, Faculty of Mathematics and Physics, Comenius University, Mlynsk´a dolina,

84248 Bratislava, Slovak Republic e-mail: [email protected]