A symmetry result for cooperative elliptic systems with singularities

We obtain symmetry results for solutions of an elliptic system of equation possessing a cooperative structure. The domain in which the problem is set may possess"holes"or"small vacancies"(measured in terms of capacity) along which the solution may diverge. The method of proof relies on the moving plane technique, which needs to be suitably adapted here to take care of the complications arising from the vacancies in the domain and the analytic structure of the elliptic system.


Introduction and main results
The moving plane method was introduced in the pioneer works of Aleksandrov [1,2] in order to characterize spheres as the only closed, smooth and connected surfaces having constant mean curvature. Afterwards, starting from the seminal paper of Serrin [26] concerning the overdetermined torsion problem, Gidas, Ni and Nirenberg [19] and Berestycki and Nirenberg [4] developed further this technique in order to establish some qualitative properties of solutions of elliptic partial differential equations such as symmetry and monotonicity. The method of proof is very elegant, it relies on a beautiful geometric intuition, and its essential ingredient is the appropriate use of the maximum principle in comparing the values of the solution of the equation at two different points after a suitable reflection, which is determined by a hyperplane which gets moved up to a critical position.
In this paper, we exploit the moving plane technique in order to obtain symmetry results in a setting which is not usually comprised by the classical method, since two difficulties will be taken into account. First of all, we will consider the case of general cooperative elliptic systems rather than that of a single equation, for which the moving plane technique has been settled by Troy [29]. This setting is also motivated by equations driven by polyharmonic operators with Navier boundary conditions (which, up to repeated substitutions, can be framed into elliptic systems of second order equations). Moreover, we take into account the case in which the domain presents "holes", or "cuts", or more general vacancies, along which the solution can become singular. This is an extension of our previous work [5] where we were dealing only with singularities made out of a single point, as studied in [6,28] for the case of a single scalar equation.
Of course, one cannot expect a general treatment of these two situations without additional assumptions. Indeed, general elliptic systems do not satisfy the maximum principle and there is no natural order in the vectorial case, making the classical regularity theories fail in such a situation. Moreover, if the vacancies in the domain are too large, they can affect the geometry involved in the reflections and produce singularities that cannot be treated analytically in any convenient way.
To overcome these difficulties, inspired by the recent works [16,25], we will restrict ourselves to the case of cooperative systems, in which an appropriate use of the maximum principle is possible, and consider domain vacancies that are "sufficiently small", in terms of capacities.
The precise mathematical formulation in which we work is the following. Let m ≥ 2 be a fixed natural number. Throughout the present paper, we shall be concerned with second-order cooperative (elliptic) systems of the following form in Ω \ Γ, where . . , f m ∈ Lip(R m ) and, for every i, j ∈ {1, . . . , m} with i = j, the map is non-decreasing on (0, ∞) for every choice of t 1 , . . . , t j−1 , t j+1 , . . . , t n > 0. We refer to Definition 2.2 for the rigorous definition of solution used in this paper. See also Definition 2.1 for the precise meaning of capacity of a set and a detailed explanation of the assumption (H.2). We want to point out that the capacitary assumption (H.2) cannot be removed nor replaced with the request that L n (Γ) = 0, see Remark 2.6.
We are ready to state the main result of this paper. Then, u 1 , . . . , u m are symmetric with respect to the hyperplane Π and increasing in the The proof of Theorem 1.1 is pretty much inspired by [16,25]. The main idea in there relies in proving the symmetry (and monotonicity) of the solution through a clever use of integral estimates. To be more precise, given the function u and its reflection across a given hyperplane, one considers the positive part of their difference and shows that its gradient is actually 0. Passing to elliptic systems this technique becomes more involved because the presence of more equations naturally leads to interactions between the solutions which have to be carefully treated. Indeed, these interaction between the different components of the (vectorial) solution, causes an important loss of information on the single equations. To overcome this difficulty we will implement a sort of bootstrap procedure in which an estimate on a single component is reflected into the next one, thus producing an iterative procedure that eventually leads to a closed formula valid for all the components of the solution. We also want to stress that our result extends our previous result in [5] and it is general enough to cover a bunch of polyharmonic semilinear problems with Navier boundary conditions, even allowing for possibly singular terms.
The literature concerning symmetry results for elliptic PDEs is pretty wide and this makes it hard for us to present here an exhaustive list of references. We already mentioned the seminal papers [4,19,26] for the introduction and the use of the moving planes method in the elliptic PDEs setting. More recently, there has been an increasing interest in the study of elliptic PDE's (in bounded domains Ω ⊂ R n ) allowing for possible singularities, namely PDE's of the form with γ > 0. In this perspective, we want to mention [13], which is one of the first contributions dealing with singular nonlinearities, and then the more recent series of papers [7][8][9][10].
To the best of our knowledge, one of the first papers dealing with symmetry of positive solutions of elliptic PDE's in domains with holes given by a single point, dates back to [28], which was then extended to slightly more general operators and sets in [6]. The same kind of result, but with a necessary and delicate modification of the technique involved, can be also obtained in presence of a bigger hole. In this direction, we refer to [16,25] where the authors allow (respectively) for a hole given by a n − 2-dimensional smooth manifold and a set of null capacity. Their ideas have also been successfully applied in the non-local setting, see [22]. Let us now spend a few words concerning the case of (cooperative) elliptic systems, which can also include the case of higher order polyharmonic PDE's with Navier boundary conditions. The first result aiming at extending the results in [19] to the vectorial case is contained in [29]. Subsequently, there has been an impressive amount of contributions dealing with the validity of maximum principles (see e.g. [15,27]). Let us finally mention [3,5,11,12,14,18] (for symmetry results for semilinear polyharmonic problems and cooperative elliptic systems with or without singularities).
The paper is organized as follows. In Section 2 we fix the notation used throughout the paper and we recall and prove a few technical results needed for the proof of Theorem 1.1, which is the content of the final Section 3.

Notations and auxiliary results
The aim of this section is to introduce the relevant notations we shall need in the sequel, and to state some auxiliary results on which we shall base the proof of Theorem 1.1. To begin with, we briefly review in this remark the precise meaning of assumption (H.2) (in the meaningful case n ≥ 3). We then say that E has vanishing 2-capacity (and we write Cap 2 (E) = 0) if for every open set U ⊆ R n .
We recall that it can be easily proved that a compact set E ⊆ R n has vanishing 2-capacity if and only if there exists a bounded open neighborhood U 0 of E such that For a demonstration of this fact we refer, e.g., to [21,Lemma 2.9].
We now specify what we mean by a solution of the system in (1.1).
for every i = 1, . . . , m; (2) for every i ∈ {1, . . . , m} one has (3) for every i ∈ {1, . . . , m} one has u i > 0 a.e. on Ω and u i ≡ 0 on ∂Ω. In this paper, if U ⊆ R n is an arbitrary open set, the space H 1 0 (U ) is intended as the closure of C ∞ 0 (U, R) (or, equivalently, of Lip(U ) ∩ C 0 (U, R)) with respect to the norm 3. We point out that, on account of assumption (H.3), the right-hand side of any equation of the system in (1.1) is locally bounded; as a consequence, if U = (u 1 , . . . , u m ) is a solution of this system of PDEs, from standard elliptic regularity we infer that u 1 , . . . , u m ∈ C 1,α loc (Ω \ Γ; R), for every 0 < α < 1.
As a consequence, by condition (3) in Definition 2.2 we have u i > 0 for every x ∈ Ω.
We are now ready to set the standing notation needed to perform the moving plane technique. If Ω ⊆ R n satisfies assumption (H.1), we set Moreover, for every fixed λ ∈ R, we define and we denote by R λ the symmetry with respect to the hyperplane Π λ := {x 1 = λ}, i.e., We explicitly notice that, since Ω is open, then the same is true of Ω λ := R λ (Ω); furthermore, since Ω is convex, we clearly have that Σ λ is convex and Σ λ ⊆ Ω ∩ Ω λ . We collect in the next Lemma 2.4 some topological facts we shall need in the sequel.  Let then x 0 = y 0 ∈ U be fixed, and let O 0 ⊆ R n be an open neighborhood of E such that x 0 , y 0 / ∈ O 0 . Moreover, let ρ > 0 be so small that We claim that there exist a point x ∈ B(x 0 , ρ) ⊆ U such that (2.4) the segment [x, y 0 ] joining x to y 0 does not intersect E.
Taking this claim for granted for a moment, we are able to complete the demonstration of this assertion: in fact, if x ∈ B(x 0 , ρ) is as in (2.4), the polygonal connects x 0 to y 0 and it is contained in U \E (this is a straightforward consequence of (2.3), (2.4) and of the fact that, by assumption, U is convex).
We now turn to prove the above claim. To this end, we argue by contradiction and we assume that, for every fixed x ∈ B(x 0 , ρ), there exists t = t x ∈ (0, 1) such that If u ∈ C ∞ 0 (O 0 , R) is any smooth function satisfying u ≥ 1 on E, by combining (2.3) with (2.5) we obtain the following estimate (note that x / ∈ O 0 ⊃ supp(u)): by Hölder's inequality, and setting κ 0 : Due to the arbitrariness of x ∈ B(x 0 , ρ), we are entitled to integrate both sides of (2.6) on B(x 0 , ρ) with respect to x: this gives (with ω n := |B(0, 1)|) Since the function u was arbitrary, the above estimate implies that , but this is in contradiction with (2.2). Thus, (2.4) holds.
(2) If n = 2, from the convexity of Ω (and the fact that, by assumption, λ > a Ω ) it readily follows that γ λ consists exactly of two points; as a consequence, If, instead, n ≥ 3, we claim that Taking this claim for granted for a moment, we are able to complete the proof of the statement: indeed, on account of (2.7), it is readily seen that the Hausdorff dimension of γ λ is precisely n − 2; as a consequence, we have (see, e.g., [17]) We then turn to prove (2.7). To this end, let ξ ∈ γ λ be fixed. Since Ω is an open set of class C ∞ (see assumption (H.1)), there exist an index i ∈ {1, . . . , n}, a number ρ > 0 and Moreover, since Ω is convex and λ > a Ω , it is quite easy to recognize that θ is either convex or concave on B(ξ ′ , ρ) and that, setting As a consequence, if we introduce the R 2 -valued function (b) the Jacobian matrix of α at ξ has full rank; Gathering together all these facts, we conclude that γ λ is a smooth manifold of dimension n − 2, and the proof is finally complete.
Remark 2.5. We explicitly observe that, on account of Lemma 2.4-(1), we have that In fact, since Γ fulfills (H.2), we have that R λ (Γ) is compact and Cap 2 (R λ (Γ)) = 0 (for every n ≥ 2); moreover, as Ω is convex, the same is true of Σ λ = Ω ∩ {x 1 < λ}. Actually, (2.8) can be proved in a more direct (and simpler) way by observing that In fact, since R λ (Γ) has vanishing 2-capacity, it is well-known that where H dim (R λ (Γ)) stands for the Hausdorff dimension of R λ (Γ) in R n (see, e.g., [21]); as a consequence, there necessarily exists (at least) one point By combining (2.9) with (2.10) it is very easy to recognize that, if Let now Γ ⊆ Ω satisfy assumption (H. 2), and let f 1 , . . . , f m be as in assumption (H.3). If U = (u 1 , . . . , u m ) : Ω → R m is any solution of the elliptic system (1.1) (according to Definition 2.2), we then introduce the following functions (defined on Ω λ \ R λ (Γ)): On account of Remark 2.3, we clearly have (for every 0 < α < 1) We explicitly notice that, since U λ is not of class C 2 , by saying that u n solve the (system of) PDEs in (2.13) we mean, precisely, that Owing to the classical weak and strong maximum principles, it is readily seen that u > 0 on O = Ω \ Γ; moreover, since u is continuous up to O and since x → e x 1 is not even in x 1 , we infer that u cannot be symmetric with respect to the hyperplane Π = {x 1 = 0}.

Example 2:
In Euclidean space R 2 , let Ω := B(0, 1) and let Γ : Moreover, for every fixed n ≥ 2, we consider the (closed) rectangle and we choose a function ϕ n ∈ Lip(R n ) such that Finally, we define Ω n := Ω \ R n . Since Ω n is regular for the Dirichlet problem for ∆, it is possible to find a unique function Furthermore, by the classical weak and strong maximum principles we have (2.15) 0 ≤ u n ≤ 2 on Ω n and u n > 0 on Ω n .
We claim that the sequence {u n } n has a cluster point u 0 which is a solution of (1.1) (with m = 1 and f ≡ 0) but which is not symmetric with respect to the hyperplane {x 1 = 0}.
To prove the claim we first observe that, if k ∈ N is arbitrarily fixed and if there exists a natural n k ≥ 2 such that O k ⊆ Ω n for every n ≥ n k . As a consequence, since {u n } n≥n k is a sequence of harmonic functions in O k which is uniformly bounded on O k , there exists a harmonic function u 0k on O k such that (up to a sub-sequence) lim n→∞ u n = u 0k , uniformly on every compact set of O k .
From this, by exploiting a suitable Cantor diagonal argument, it is then possible to find a sub-sequence {u n j } j of {u n } n and a harmonic function u 0 on Ω \ Γ such that lim j→∞ u n j = u 0 , uniformly on every compact set of Ω \ Γ.
In particular, since u n ≡ 0 on ∂Ω and u n > 0 on Ω n for every n ∈ N, we infer that Let now n ≥ 2 be arbitrarily fixed, let P n := (−1/n, 0) and let n , it follows from classical results (see, e.g., Theorem 4.11 in [20]) that where C is a suitable positive constant which is independent of n. From this, by letting n → ∞ (and reminding that u n j → u 0 as j → ∞ point-wise on Ω \ Γ) we get As a consequence, we infer that On the other hand, if Q n := (1/n, 0) and if by arguing exactly as before we get where C ′ is another positive constant which is independent of n. From this, by letting n → ∞ and by taking the limit as x → 0 with x 1 > 0, we obtain Gathering together (2.16) and (2.17) we readily see that u 0 cannot be symmetric with respect to the hyperplane {x 1 = 0}; moreover, since u 0 is harmonic and non-negative on Ω \ Γ, by the strong maximum principle we conclude that u 0 > 0 on Ω \ Γ. Summing up, u 0 is a solution of (1.1) (with m = 1 and f ≡ 0) which is not symmetric with respect to the hyperplane {x 1 = 0}. Note that, even if |Γ| = 0, the set Γ cannot have vanishing 2-capacity: in fact, its Hausdorff dimension is strictly greater than n − 2 = 0.
After these preliminaries, we continue this section by constructing two sequences of functions which shall play a fundamental rôle in the proof of Theorem 1.1. In order to do this, we exploit some ideas contained in [16] (see, precisely, Section 2).
First of all we observe that, if λ ∈ (a Ω , 0) is arbitrarily fixed, on account of Lemma 2.4-(2) we have Cap 2 (R λ (Γ)) = 0 (both in the case n = 2 and in the case n ≥ 3); as a consequence, if O ⊆ R n is any open neighborhood of R λ (Γ), we have On account of (2.18), for any k ∈ N it is possible to find a function ψ k ∈ C ∞ 0 (R n , R) (also depending on the fixed λ) such that Starting from the sequence { ψ k } k∈N , we then define Clearly, {ψ k } k∈N ⊆ Lip(R n ) and, for every fixed k ∈ N, one has Furthermore, since ∇ψ k = (T ′ • ψ k ) · ∇ ψ k a.e. on R n , we also have Arguing analogously, we construct a second sequence of functions {φ h } h∈N such that, for every h ∈ N, the function φ h is identically 0 near the set To this we first remind that, by Lemma 2.4-(2), we have Cap 2 (γ λ ) = 0; as a consequence, for every open neighborhood V ⊆ R n of γ λ one has On account of this last fact, in correspondence to every natural h it is possible to construct a function φ h ∈ C ∞ 0 (R n , R) (also depending on the fixed λ) such that Starting from the sequence { φ h } h∈N , we define (as above) where T is as in (2.19).
Clearly, {φ h } h∈N ⊆ Lip(R n ) and, for every fixed h ∈ N, one has Lemma 2.7. Let λ ∈ (a Ω , 0) be such that R λ (Γ) ∩ Ω = ∅, and let {φ h } h∈N be the sequence defined in (2.22). Moreover, let g ∈ C 1 (Σ λ , R) ∩ C(Σ λ , R) be such that Then, the sequence of functions {ϕ h } h∈N defined by (here, g + = max{g, 0} is the positive part of g) satisfies the following properties: (iii) for every h ∈ N, and a.e. on Ω ∪ Ω λ , one has and Then, the (double) sequence of functions {ϕ h,k } h,k∈N defined by satisfies the following properties: (iii) for every h ∈ N, and a.e. on Ω ∪ Ω λ , one has . In particular, ϕ h,k ∈ Lip(Σ λ ) and ϕ h,k ≡ 0 on ∂Σ λ , so that ϕ h,k ∈ H 1 0 (Σ λ ). We also have the following regularity result for the solutions of (1.1), which can be demonstrated by arguing essentially as in the proof of [16, Lemma 3.2]. Lemma 2.9. Let λ ∈ (a Ω , 0) and i ∈ {1, . . . , m} be fixed. Then, where ϕ h is as in Lemma 2.7, where ϕ h,k is as in Lemma 2.8, i . Finally, we prove a technical lemma which will be used in the proof of Theorem 1.1.
Lemma 2.10. Let n ≥ 2 and let U ⊆ R n be an open and bounded set with Lipschitz boundary. There exists a real constant Θ = Θ n > 0, independent of U , such that Proof. We first prove (2.30) for a function v ∈ C ∞ 0 (U ) (not identically vanishing on U ). Since, in particular, we can think of v as a function belonging to C ∞ 0 (R n ), by applying the Nash inequality (see, e.g., [23]) and Hölder's inequality we get v 1+2/n where Θ > 0 is a real constant only depending on the dimension n. As a consequence, since we have assumed that v ≡ 0 on U , we obtain v L 2 (U ) ≤ Θ |U | 1/n ∇v L 2 (U ) .
The proof of (2.30) for a general u ∈ H 1 0 (U ) follows by a density argument.

Proof of Theorem 1.1
In the present section we give the proof of our Theorem 1.1. In doing this, we take for granted all the notations introduced in the preceding sections.
Our aim is to demonstrate that I = ∅ and that λ 0 = 0. From now on, in order to ease the readability, we split the proof into some different steps.
Step I: In this step we prove that I = ∅ and that λ 0 > a Ω . We fix t 0 ∈ (a Ω , 0) such that R t 0 (Γ) ⊂ Ω c . Necessarily, we have that R t (Γ) ⊂ Ω c for every t ∈ (a Ω , t 0 ). Now, for every i = 1, . . . , m we consider the function ϕ i,h : Ω → R defined as (2.22). By density, we can use ϕ i,h as a test function, finding By Fatou Lemma, sending h → 0 + we get By Hölder inequality on every term on the right hand side, we get From this, by using (2.30) (on every term on the right hand side), for every t ∈ (a Ω , t 0 ) and every index i ∈ {1, . . . , m} we get where we have introduced the notation (repeatedly used in the sequel) θ n (Σ t ) := Θ |Σ t | 1/n (with Θ > 0 is as in Lemma 2.10).
We now aim at proving the following assertion: for every fixed k ∈ {1, . . . , m − 1} there exist t k ∈ (a Ω , t 0 ) and a real constant C k = C k (m, c f ) > 0 such that for all 1 ≤ i ≤ k and every t ∈ (a Ω , t k ).
To prove (3.8) we argue by (finite) induction and we start with k = 1. By (3.7) we have Since θ n (Σ t ) → 0 as t → a Ω , it is possible to find t 1 ∈ (a Ω , t 0 ) such that . for every t ∈ (a Ω , t 1 ).
As a consequence, we obtain which is precisely (3.8) for i = 1 (with C 1 = 4c 0 ). Let us now suppose that (3.8) holds for a certain index k ∈ {1, . . . , m − 2} and, by shrinking t k if necessary, let us also assume that θ n (Σ t ) < 1 for all t ∈ (a Ω , t k ). Owing to (3.7) (with i = k + 1), we then have by (3.8), which we are assuming to hold for the index k We now perform a backward induction argument to show that, as a consequence of the validity of (3.8) for the index k, the following fact holds: for every fixed j ∈ {1, . . . , k}, it is possible to find a real constant C j = C j (m, k, c f ) > 0 such that for all 1 ≤ j ≤ k and every t ∈ (a Ω , t k ).
For j = k, (3.10) follows immediately from (3.8) by taking i = k (with C k := C k ). We then suppose the existence of an index j ∈ {2, . . . , k} such that (3.10) holds for every j ≤ r ≤ k, and we exploit once again (3.8) since (3.10) holds for j ≤ r ≤ k, and θ n (Σ t ) < 1 so that (3.10) holds true also for j − 1. By the Induction Principle, we then conclude that estimate (3.10) is valid for every j = 1, . . . , k, as claimed. With (3.10) at hand, we now continue the estimate (3.9): reminding that, by the choice of t k , we have θ n (Σ t ) < 1 for every t ∈ (a Ω , t k ), we have where M k = M k (m, c f ) > 0 is a suitable As a consequence, we obtain Finally, since θ n (Σ t ) → 0 as t → a Ω , we infer the existence oft ∈ (a Ω , t 0 ) such that for every t ∈ (a Ω ,t); from this, we obviously derive the estimate (valid for t ∈ (a Ω ,t)) Taking as t k+1 := min{t k ,t}, and setting C k+1 := max{C k , 4M k }, we then obtain for all 1 ≤ i ≤ k + 1 and every t ∈ (a Ω , t k+1 ), so that (3.8) holds true also for k +1. By the Induction Principle, we conclude that estimate (3.8) is valid for every k = 1, . . . , m − 2, as claimed. Now we have established (3.8), we are able to complete the proof this step. In fact, since the cited (3.8) holds true for k = m − 1, a (finite) backward induction argument shows the existence of a real constant C m = C m (c 0 ) > 0 such that for all 1 ≤ j ≤ m − 1 and every t ∈ (a Ω , t m−1 ); gathering together (3.12) and (3.7) (with i = m), for any t ∈ (a Ω , t m−1 ) we get Since θ n (Σ t ) → 0 as t → a Ω , there exists τ 0 ∈ (a Ω , t m−1 ) such that , for every t ∈ (a Ω , τ 0 ); as a consequence, we obtain ∇(w (t) m ) L 2 (Σt) = 0, for every t ∈ (a Ω , τ 0 ). On account of (3.8), this proves that m ) L 2 (Σt) = 0, for every t ∈ (a Ω , τ 0 ); as a consequence, by Lemma 2.10 (and since W t is continuous on Σ t ) we get i ≤ 0 on Σ t (for every i = 1, . . . , m and every t ∈ (a Ω , τ 0 )). We finally claim that, by the Strong Maximum Principle for C 1 -subsolutions, we have (3.14) i on Σ t , for every i = 1, . . . , m and every t ∈ (a Ω , τ 0 ). Indeed, let i ∈ {1, . . . , m} and t ∈ (a Ω , τ 0 ) be arbitrarily fixed. Clearly, the set Σ t is (open and) connected; moreover, since the (vector-valued) map W t = U − U t solves (3.2) and c ij (·; t) ≥ 0 for every j = i, we have −∆w We explicitly point out that the above inequality has to be intended in the weak sense of distributions on Σ t : this means, precisely, that From this, taking into account (3.5) we get −∆w i ≤ 0, and c f − c ii (·; t) ≥ 0 on Σ t . Gathering together all these facts, we can invoke the Strong Maximum Principle for C 1 -subsolution (see, e.g., [20]), ensuring that either w (t) Since, by (3.2), we know that the function w (t) t is (strictly) negative on the set ∂Σ t \ Π t (notice t < τ 0 < 0), we then conclude that (3.14) holds true.
Finally, on account of (3.14) (and taking into account the very definition of I), we see that (a Ω , τ 0 ) ⊆ I, whence I = ∅, and that λ 0 = sup I ≥ τ 0 > a Ω .
Moreover, w