On Morrey and Bmo Regularity for Gradients of Weak Solutions to Nonlinear Elliptic Systems with Non-differentiable Coefficients

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 z only 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.


Introduction
In the paper, we consider the problem of interior everywhere regularity of gradients of weak solutions to the nonlinear elliptic system − div a(x, u, Du) = b(x, u, Du), where a : N ) is called a weak solution to (1.1) in Ω if 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 variable z and so we can not suppose any condition of the type g z (x, u, z) → 0 for |z| → ∞.

Notation and definitions
We consider the bounded open set Ω ⊂ R n with points x = (x 1 , . . ., x n ), n ≥ 3, u : Ω → R N , N > 1, u(x) = (u 1 (x), . . ., u N (x)) is a vector-valued function, Du = (D 1 u, . . ., D n u), D α = ∂/∂x α .The meaning of Ω 0 ⊂⊂ Ω is that the closure of Ω 0 is contained in Ω, i.e.Ω 0 ⊂ Ω.For the sake of simplicity we denote by | • | the norm in R n as well as in R N and R nN .If x ∈ R n and r is a positive real number, we write B r (x) = {y ∈ R n : |y − x| < r}, i.e., the open ball in R n with radius r > 0, centered at x and Ω r (x) = Ω ∩ B r (x).Denote by [11]), we use the following Morrey and Campanato spaces.
Proposition 2.2.For a domain Ω ⊂ R n of the class C 0,1 we have the following (a) With the norms u L q,λ and u L q,λ = u L q + [u] L q,λ , L q,λ (Ω, R N ) and L q,λ (Ω, R N ) are Banach spaces.
Then we have for each u ∈ L p,n (Ω, R N ) For more details see [2,7,11,16].Definition 2.3 (see [14]).Let f ∈ BMO(R n ) and where B ρ (x) ranges over the class of the balls of R n of radius ρ.We say that f ∈ We can observe that substituting R n for Ω we obtain the definition of VMO(Ω).Some basic properties of the above-mentioned classes are formulated in [1,14,16].

Main results
Suppose that for almost all x ∈ Ω and all u ∈ R N , z ∈ R nN the following conditions hold: where is nondecreasing on [0, ∞), absolutely continuous on every closed interval of finite length and H(0) = 0.The relationship between t 0 > 0 and s can be expressed through an inequality s ≤ (e + t 0 ) ln(e + t 0 )/t 0 .Now we can state a result for the continuous case.
) be a weak solution to the system (1.1) with (1.2) and the conditions 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: where f ∈ L qq 0 ,λq 0 (Ω), F ∈ L q,λ (Ω), q > 2, ν 1 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].
To obtain L 2,n -regularity for the first derivatives of the weak solution we strengthen the conditions on the coefficients g and b.Namely suppose that 2) with λ = n.Now we can formulate the main result of the paper.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 2q 0 ,nq 0 (Ω) and (3.7) with 0 < s ≤ 1 hold.Let further A ∈ C 0,α (Ω, R nN ) for some α ∈ (0, 1].Then Du ∈ L 2,n loc (Ω, R nN ).

Some lemmas
In this section we present results needed for the proofs of the theorems.In B R (x) ⊂ R n we consider a linear elliptic system (here the summation convention over repeated indices is used) 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.N ) be a weak solution to the system (4.1).Then, for each 0 hold with constants L 1 , L 2 independent of the homothety.
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 1,2 loc (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 1,r loc (Ω, R N ).Moreover there exists a constant c = c(ν, ν 1 , L, A L ∞ ) and R > 0 such that, for all balls B R (x) ⊂ Ω, R < R, the following inequality is satisfied Lemma 4.5 ([17, p. 37]).Let φ : [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 ∈ R n : w(y) > t} then 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.Remark 4.7.The foregoing estimate is optimal because if we put f (t) = 1, t ∈ [a, ∞) then an equality will be achieved.

Proofs of the theorems
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 < R (R ≤ 1 is sufficiently small).We can put ϕ = w in the previous equation and, using ellipticity, Hölder and Sobolev inequalities, we get Now we obtain where From the assumption (3.2) (taking into account (3.3) and the comments below it), putting m R (t) = m y ∈ B R (x 0 ) : |Du| 2 > t , we can estimate II as follows.
We can estimate III by means of Lemma 4.3 (with η = 0) and we have III ≤ cR λ 0 /q 0 .( Together we have (5.7) The function v = u − w ∈ W 1,2 (B R/2 , R N ) is the solution to the system and from Lemma 4.1 we have, for 0 By means of (5.7) and the last estimate we obtain, for all 0 < σ ≤ R, the following estimate: where the constants c 1 and c 2 only depend on the above-mentioned parameters and λ Further we can choose k < 1 such that E 1 k n−λ < 1/2.It is obvious (the coefficients A are continuous) that the constants R 0 > 0 and t For all 0 < σ ≤ R ≤ min{d 0 , R 0 } the assumptions of Lemma 4.2 are satisfied and therefore If min{d 0 , R 0 } < diam Ω 0 , it is easy to check that for min{d 0 , R 0 } ≤ σ ≤ diam Ω 0 we have and thus we get If λ = λ the Theorem is proved.If λ < λ the previous procedure can be repeated with η = λ in Lemma 4.3.It is clear that after a finite number of steps (since λ increases in each step as it follows from Lemma 4.3) we obtain λ = λ.
Proof of Theorem 3.2.Using the same procedure as in the foregoing proof we get the inequality (5.1).The terms I, II and III we can estimate as follows.
From assumption (3.5) (taking into account (3.3) and the comments below it) we can estimate II as follows.

Lemma 4 . 6 .
Let f : [a, ∞) → R be a nonnegative function which is integrable on [a, b] for all a < b < ∞ and ) dt < ∞ is satisfied.Let g : [a, ∞)→ R be an arbitrary nonnegative, non-increasing and integrable function.Then By means of Lemma 4.5 and Lemma 4.6 we get give us II ≤ c 1 ln s (e + t 0 ) B R |Du| 2 dx + t 0 ln s (e + t 2 0 )