Nonlinear Analysis Maximum principles for some quasilinear elliptic systems

a α,βi,j ( x,y ). In this paper, we assume that off-diagonal coefficients a α,βi,j , α ̸ = β , have support in some staircase set along the diagonal in the y α ,y β plane.


a b s t r a c t
We give maximum principles for solutions u : Ω → R N to a class of quasilinear elliptic systems whose prototype is where α ∈ {1, . . . , N } is the equation index and Ω is an open, bounded subset of R n . We assume that coefficients a α,β i,j (x, y) are measurable with respect to x, continuous with respect to y ∈ R N , bounded and elliptic. In vectorial problems, when trying to bound the solution by means of the boundary data, we need to bypass De Giorgi's counterexample by means of some additional structure assumptions on the coefficients a α,β

Introduction
We consider the system of N equations where α ∈ {1, . . . , N } is the equation index and Ω is an open, bounded subset of R n , with n ≥ 2. Moreover, u : Ω ⊂ R n → R N and u = (u 1 , . . . , u N ). Coefficients a α,β i,j (x, y) are measurable with respect to x, continuous with respect to y ∈ R N , bounded and elliptic.
In scalar case (N = 1) a maximum principle is available as In vectorial case (N ≥ 2), in general, we cannot expect to bound u inside Ω by means of its boundary values: in [1] it is shown that it can be constructed a system of linear equations with measurable bounded coefficients a α,β i,j (x) whose solution is the function u(x) = x/|x| γ defined in the unit ball centred at the origin with a suitable γ > 1. Then u(x) = x on the boundary of Ω but u blows up inside Ω ; see also [12,2], the surveys [9,10] and the recent paper [11]. So, the main effort is finding (additional) structure assumptions on coefficients a α,β i,j (x, u) that keep away De Giorgi's counterexample and that allow for regularity. A simple case of such a structure is the case in which off-diagonal coefficients a α,β i,j , with β ̸ = α, are identically zero; in such a case, the system (1.1) is (almost) decoupled, since D j u α = ∂u α ∂x j appears only in the α row of the system see [12]. In this case, the α row of the system (1.1) becomes decoupled, like (1.3), when x ∈ {u α > θ α }; then we get sup see [12,3] and [7]. Actually, what matters is the possibility to suitably control from below the quantity for a suitable positive constant ν α and a nonnegative constant M α . Such an assumption guarantees the following estimate for a suitable positive constant c depending only on n and the measure of Ω , see [3].
Since we do not know the solution u, condition (1.7) is replaced by the following stronger version for all y ∈ R N and all p ∈ R N ×n . Let us mention that a kind of maximum modulus principle can be found in [8] under the following structure assumption: there exist numbers λ > 0, L ≥ 0 and two nonnegative functions d(x), g(x), such that is fulfilled for all δ ∈ ]0, 1[ and all (x, y, p), with |y| > L.
Note that [8] and [3] deal with systems having a nonlinear dependence on the gradient Du. Going back to condition (1.9) we stress that such an assumption requires that, when we are above the level θ α , the row α has such a good behaviour. In the present paper, we study a different situation: off-diagonal coefficients a α,β i,j (that are responsible of the appearance of the other components Du β ) appear above every level θ α but their support is contained in a sequence of squares, see Fig. 1.
Such a condition turned out to be useful when proving existence of solutions to elliptic systems with a right-hand side which is a measure, see [4] and [5].
Under the assumption that the support of the off-diagonal coefficients a α,β i,j have such r-staircase shape, we consider the maximum of the boundary values of all components We are able to provide a minimum principle as well. The previous result is true when the support of off-diagonal coefficients a α,β i,j is contained in the squares along the diagonal y α = y β . If we allow the support  to stay also in the other diagonal y α = −y β , so that the support is contained in a crossed r-staircase set as in Fig. 3, then we can prove the maximum modulus estimate where and int [t] denotes the integer part of t, that is, the largest integer less than or equal to t. We are also able to give other results assuming a different shape of the support of the off-diagonal coefficients; they apply only to solutions satisfying suitable compatibility conditions. The paper is organized as follows: in Section 2 we list all the required assumptions and we state the main theorems whose proofs appear in Section 3. An example of a system verifying our assumptions but neither (1.9) nor (1.10) is given in Section 4.

Assumptions and results
Assume Ω is an open bounded subset of R n , with n ≥ 2. For the sake of brevity, [k] denotes the set {1, . . . , k} when k ≥ 1 is an integer. Consider the system of N ≥ 2 equations , and all α, β ∈ [N ], we require that a α,β i,j : Ω ×R N → R satisfies the following conditions: for almost all x ∈ Ω and for all y ∈ R N ; (A 2 ) (ellipticity of all the coefficients) for some positive constant m > 0, we have (see Fig. 1).
We say that a function u : Ω → R N is a weak solution of the system (2.1), if u ∈ W 1,2 ( Ω , R N ) and

Remark 1. Suppose assumptions (A) hold. Then conditions of Leray-Lions
Theorem (see [6]) are satisfied and we get existence of weak solutions to (2.2) provided a boundary datum u * ∈ W 1,2 (Ω , R N ) has been fixed.
In the aforementioned setting, weak solutions enjoy the following maximum principle.
We now modify the assumption on the support of the off-diagonal coefficients. Namely, we assume (see Fig. 3). If we denote by (Ã) the set of assumptions (A 0 ), (A 1 ), (A 2 ), (Ã 3 ) we can prove the following companion of Theorem 1. Now, we can give other results assuming a different shape of support for the off-diagonal coefficients. However, they apply only to solutions satisfying suitable compatibility conditions. We assume, (see Fig. 4).
Now we have to select a suitable value for L. We take L = S * given by (2.10) and we get (2.9). This ends the proof of Theorem 2. □ Proof of Theorem 3. Let us consider the test function φ such that Since (2.12) holds true, then φ α ∈ W 1,2 0 (Ω ) and Using the test function φ in the weak form (2.2) of system (2.1), we have when β ̸ = α. Note that (3.5) holds true when β = α as well. Then, ∫ Now, as in Theorem 1, we can use the ellipticity assumption (A 3 ) with ξ α i = D i φ α and we get |Dφ α | = 0; since φ α ∈ W 1,2 0 (Ω ), we can use Poincaré inequality and we obtain max{0, This ends the proof of Theorem 3. □ Proof of Theorem 4. Letũ = −u. Thenũ ∈ W 1,2 (Ω ; R N ) and since u satisfies (2.2) thenũ verifies We observe that new coefficients defined by (

An example
In this section we provide an example satisfying our structural conditions (A) but verifying neither (1.10) nor (1.9). We fix n = 3, N = 2 and we set where δ ij is Kronecker delta, w(y) ≥ 0 is a bounded, continuous function, with support contained in the r-staircase set, see Fig. 1. Furthermore, we require that w = 5 in the centre of every square of the sector y 1 ≥ r, y 2 ≥ r, that is w((h + 1/2)r, (h + 1/2)r) = 5, for every h ∈ N. Note that Observe that the coefficients defined in (4.1) readily satisfy conditions (A).
We want to show that the coefficients defined in (4.1) do not satisfy (4.3). To this purpose we choose and we replace in (4.3). According to the aforementioned definitions, we can rewrite We select y to be the centre of the squares of the sector y 1 ≥ r, y 2 ≥ r: with h ∈ N and h ≥ int + 1 in such a way that |y| > L and w(y) = w(y 1 , y 1 ) = 5.