Butterfly support for off diagonal coefficients and boundedness of solutions to quasilinear elliptic systems

: We consider quasilinear elliptic systems in divergence form. In general, we cannot expect that weak solutions are locally bounded because of De Giorgi’s counterexample. Here we assume that off-diagonal coefficients have a "butterfly support": this allows us to prove local boundedness of weak solutions.

On the coe cients a α,β i,j (x, y) we set the usual conditions, that is they are measurable with respect to x, continuous with respect to y, bounded and elliptic. When N = , that is in the case of one single equation, the celebrated De Giorgi-Nash-Moser theorem ensures that weak solutions u ∈ W , (Ω) are locally bounded and even Hölder continuous, see section 2.1 in [27].
But in the vectorial case N ≥ , the aforementioned result is no longer true due to the De Giorgi's counterexample, see [6], section 3 in [27] and the recent paper [29]; see also [32] and [20].
So it arises the question of nding additional structural restrictions on the coe cients a α,β i,j that keep away De Giorgi's counterexample and allow for local boundedness of weak solutions u, see Section 3.9 in [28].
In the present work we assume a condition on the support of o -diagonal coe cients: there exists L ∈ [ , +∞) such that ∀ L ≥ L , when α ≠ β, (a α,β i,j (x, y) ≠ and y α > L) ⇒ y β > L, (1.2) (see Figure 1 and note that the support has the shape of a butter y in the plane y β − y α ).
Under such a restriction we are able to prove local boundedness of weak solutions. All the necessary assumptions and the result will be listed in section 2 while proofs will be performed in section 3.
It is worth to stress out that systems with special structure have been studied in [33], [26] and o -diagonal coe cients with a particular support have been successfully used when proving maximum principles in [21], L ∞ -regularity in [22], when obtaining existence for measure data problems in [23], [24] and, for the degenerate case, in [7].
Higher integrability has been studied as well in [10] when o -diagonal coe cients are small and have staircase support and in [11] when o -diagonal coe cients are proportional to diagonal ones.
Let us mention as well that when the ratio between the largest and the smallest eigenvalues of a α,β i,j is close to 1, then regularity of u is studied at page 183 of [12]; see also [31], [18], [17], [19].
Let us also say that proving boundedness for weak solutions could be an important tool for getting fractional di erentiability, see the estimate after (4.15) in [8]. In the present paper we deal with local boundedness of solutions. If the reader is interested in regularity up to a rough boundary it could be worth looking at [25].

Assumptions and Result
Assume Ω is an open bounded subset of R n , with n ≥ . Consider the system of N ≥ equations for almost all x ∈ Ω, for all y ∈ R N and for all ξ ∈ R N×n ; (see Figure 1).
where α, β = , and i, j = , . . . , n with n ≥ and N = . In this case we have c = , ν = / and we can pick for instance L = .
We say that a function u : Ω → R N is a weak solution of the system (2.1), if u ∈ W , Ω, R N and for every α = , ..., N and for every r, R with < r < R and

where c is the constant involved in assumption (A ), ν is given in (A ) and L appears in (A ).
Remark 2.4. The present local L ∞ -regularity result improves on [22] since assumption (A ) allows o diagonal coe cients to have a larger support than in [22]. [7] when proving the existence of at least one globally bounded solution to a (possibly) degenerate problem with zero boundary value problem. In the present work we prove local boundedness of every solution to a non degenerate system regardless of boundary values.

Proof of the result
The proof of Theorem 2.3 will be performed in several steps

where c is the constant involved in assumption (A ), ν is given in (A ) and L appears in (A ).
Proof of Lemma 3.1 Let u ∈ W , Ω, R N be a weak solution of system (2.1). Let η : R n → R be the standard Then Using this test function in the weak formulation (2.2) of system (2.1), we have when β ≠ α and L ≥ L . It is worthwhile to note that (3.2) holds true when α = β as well; then where we used the inequality ab ≤ ϵa + b /ϵ, provided ϵ > . Merging (3.5), (3.4) and (3.6) into (3.3) we get We choose ϵ = ν/( cn N ) and we have Using the properties of the cut o function η we deduce Note that this ends the proof of Lemma 3.1.

STEP 2. Sup estimate for general vectorial functions
In the next Lemma we state and prove a general result that holds true for some general vectorial function v ∈ W ,p (Ω, R N ). Eventually, we will use such a result with v = (|u |, ..., |u N |) and p = .

Proof of Lemma 3.2 Let us consider balls
Then, using Hölder inequality, Sobolev embedding and the properties of the cut-o function, where c = [(n − )p/(n − p)] p . Now we sum upon α from to N obtaining . (3.11) In order to control |Dv α | p we use our assumption (3.8) with s = (r + r )/ and t = r : we get . (3.12) We want to estimate |A β L,r | by means of (v β − L) p . We are able to do that for a lower levelL. Indeed, for L >L ≥ L , we have Note that − (p/p * ) = p/n. (3.14) Inserting (3.14) and (3.13) into (3.12) we deduce (3.17) Now we x < r < R, with B(x , R) ⊂ Ω, and we take the following sequence of radii for i = , , , ...; then ρ = R and ρ i − ρ i+ = (R − r)/ i+ > , so ρ i strictly decreases and r < ρ i ≤ R.
Let us x a level d ≥ L and we take the following sequence of levels (3.19) for i = , , , ...; then k = d and k i+ − k i = d/ i+ > , so k i strictly increases and L ≤ d ≤ k i < d. We can use (3.17) with levels L = k i+ > k i =L and radii r = ρ i+ < ρ i = r : . (3.20) Let us set then (3.20) can be written as follows We would like to get lim i→∞ J i = ; (3.23) this is true provided , (3.24) as Lemma 7.1 says at page 220 in [13]. Let us try to check (3.24): we rst rewrite it as follows ; (3.25) we keep in mind that k = d and ρ = R; so, (3.25) can be written in the following way Using (3.27), we get the following su cient condition when checking (3.26): Then, we x d verifying (3.28) and L ≤ d; then (3.24) is satis ed and (3.23) holds true. We keep in mind that r < ρ i and k i < d, so we can use (3.16) with r = r < ρ i , L = d andL = k i :  (3.1). In [4], [1] and [3] only one component of u is modi ed and a Caccioppoli's inequality without the summation on α is proved. Moreover, the Caccioppoli's inequality proved in [4] and [1] has an exponent p * on the right-hand side in contrast with the same p that we have on both sides of (3.8), see also [30], [9], [2], [5], [14], [15], [16].

Remark 3.4.
In [22] it is used max{u α − L; } in the test function φ, see Figure 2 (left), while in the present paper we use G L (u α ) instead, see Figure 2 (right). Such a function G L (u α ) allows us to deal with support larger than in [22] for o diagonal coe cients.

Con ict of interest statement.
Authors state no con ict of interest.