A SYMMETRY RESULT FOR ELLIPTIC SYSTEMS IN PUNCTURED DOMAINS

. We consider an elliptic system of equations in a punctured bounded domain. We prove that if the domain is convex in one direction and symmetric with respect to the reﬂections induced by the normal hyperplane to such a direction, then the solution is necessarily symmetric under this reﬂection and monotone in the corresponding direction. As a consequence, we prove symmetry results also for a related polyharmonic problem of any order with Navier boundary conditions.

1. Introduction. In this paper we will study general cooperative systems of secondorder elliptic equations in the spirit of Troy [20]. The equations are set in a punctured domain, thus allowing possible singularities; nevertheless, thee structure of the system being (partially) cooperative, we are allowed to use the maximum principle. Exploiting the moving plane method, we will prove that positive solutions in a domain which is symmetric under a given reflection also possess the same type of symmetry, and moreover they are monotone with respect to the direction of symmetry (in particular, if the domain is a punctured ball, the solution is necessarily radial and radially decreasing).
We now introduce the mathematical setting in which we work.
Let now m ∈ N be fixed, we consider the following elliptic system (1.1) We consider classical solutions of (1.1), namely u i ∈ C 2 (Ω \ {0}) ∩ C(Ω \ {0}) for every i = 1, . . . , m. We assume further that f i : R m → R satisfy where the sign assumption on the partial derivatives of f i has to be intended in the L ∞ -sense. As anticipated, the main result of the present paper concerns symmetry properties of the positive solutions of (1.1) and is the following Then the following facts hold true: for every i = 1, . . . , m, (1) u i is symmetric in x 1 , i.e., u i (x 1 , x 2 , . . . , x n ) = u i (−x 1 , x 2 , . . . , x n ) for every x ∈ Ω; (2) u i is decreasing with respect to x 1 on Ω ∩ {x 1 > 0}. Theorem 1.1 extends a classical result by Troy [20] to the case of punctured domains. In particular, when m = 1, our result recovers the classical result in [4]; on the other hand, when m = 2, the system (1.1) finds natural applications in engineering, for instance in the description of "hinged" rigid plates, see e.g. [12]. This leads to consider other examples of polyharmonic operators which naturally appear in the phase separation of a two component system, as described by the Cahn-Hilliard equation (see [5]), and when comparing the pointwise values of a function with its average (see [16]).
As is well-known, the literature concerning symmetry results for elliptic PDEs is extremely wide, and is far from our scopes to present here an exhaustive list of references. We must mention the seminal papers [3,13,18] for the use of the moving planes method in the PDEs setting; we also highlight [4,10,15,17,19] (for symmetry results for singular solutions of scalar semilinear equations in local and non-local setting) and [2,[6][7][8][9]11,20] (for symmetry results for semilinear polyharmonic problems and cooperative elliptic systems).
To continue in the direction traced in the aforementioned papers, it is then natural to apply Theorem 1.1 to a suitable class of polyharmonic problems with Navier boundary conditions, which comes from Pizzetti-type superpositions of polyharmonic operators with appropriate structural assumptions on the coefficients. More precisely, given α 1 , . . . , α m ∈ R, we define α := (α 1 , . . . , α m ) ∈ R m and we consider the characteristic polynomial expansion where s m (α) = 1 (independently on α) and, for every k = 0, . . . , m − 1, we have We stress that, by the Descartes rule of signs, the positivity of s 0 (α), . . . , s m−1 (α) is equivalent to the positivity of all α i ; see also Lemma A.1 for a self-contained proof. Then, we consider the equation In this context, we require f to satisfy the following assumptions: The symmetry and monotonicity result for (1.4) then goes as follows.
We notice that, if one aims to prove the existence of such a solution, some regularity on the boundary ∂Ω of Ω must be required see e.g., [12,Theorem 2.19]. On the other hand, we do not need to take any additional assumption here, since we are assuming a priori that a solution exists and we aim at proving its symmetry and monotonicity properties.
We now briefly describe how Theorem 1.1 can be used to prove Theorem 1.2. First of all, if α = (α 1 , . . . , α m ) ∈ R m , we set (1.6) with this notation, the 2m-th order boundary vale problem (1.4) is equivalent to the following system (1.7) We stress that (1.7) is a particular case of (1.1) with In view of this, we spend few words on the relation between (f1) and (f2). The request f (0) ≥ 0 is closely related to the fact that in (1.4) we asked only u 1 > 0 and so the positivity of all the other components has to be proved. Indeed, condition (f2) together with the weak maximum principle in punctured domains (see [4, Lemma 2.1]), the lower-boundedness of the u i 's and the standard strong maximum principle, yields the positivity of the components u 1 , . . . , u m of U in Ω \ {0}. If we ask immediately for u i > 0 for every i = 1, . . . , m, we can then avoid the extra assumption f (0) ≥ 0, and then (f2) becomes a particular instance of (f1). Finally, a couple of comments on the regularity assumption of f in (f2). When m = 1, in [4] the analogue of Theorem 1.2 (in the case α 1 = . . . = α m = 0) is proved under the weaker assumption that f is only locally Lipschitz-continuous (and possibly depending on the spatial variable x). In our case, we instead assume a global Lipschitz assumption in (f1), since boundedness issues become more involved when m ≥ 2 (roughly speaking, for the case of systems, the positivity of one component in a subdomain does not imply the positivity of the other components). For a similar reason, we also assume the bound inf Ω\{0} u i > −∞. (1.8) Indeed, when m = 1 dealing with positive solutions implies immediately the former bound. On the other hand, for m ≥ 2, while this is still true for u 1 , this is not automatically inherited by the other components of the system. We think that it is an interesting open problem to further investigate whether either the global Lipschitz regularity assumption or the bound from below of the u j = (−∆) j u can be relaxed as in [4]. The paper is organized as follows. After stating some notation, in Section 2 we present the main technical results, related to suitable versions of the maximum principle for cooperative systems; then, Theorem 1.1 will be proved in Section 3 obtaining the symmetry result by the moving plane and reflection methods.
2. Notation, assumptions and preliminary results. We introduce some notation and the standing assumptions used along the paper. For a function U : Ω → R m , U = (u 1 , . . . , u m ), we say that U ≥ 0 if u i ≥ 0 for every i = 1, . . . , m.
The notation for the moving plane technique is as in the paper of Serrin [18], and it goes as follows. Given a point x ∈ R n , we denote by (x 1 , . . . , x n ) its components; A SYMMETRY RESULT FOR ELLIPTIC SYSTEMS IN PUNCTURED DOMAINS 5 moreover, when more practical, we equivalently write x = (x 1 , x ) ∈ R × R n−1 . For a given unit vector e ∈ R n and for λ ∈ R, we define the hyperplane From now on, without loss of generality we assume that e = e 1 , i.e. the normal to T λ is parallel to the x 1 -direction. To simplify the readability, we further assume that sup Now, for every λ ∈ (0, 1) we define We stress that (2.1) may lead to points that do not belong to Ω: for example, we have 0 λ = (2λ, 0, . . . , 0) ∈ Ω for λ > 1 2 , in view of (A1). Proceeding further with the notation, given any λ ∈ R, we introduce the possibly empty set Σ λ := {x ∈ Ω : x 1 > λ} and its reflection with respect to T λ , Since Ω ⊂ R n is bounded, by (A1) we have that T λ does not touch Ω for λ > 1; moreover, T 1 touches Ω and, for every λ ∈ (0, 1), the hyperplane T λ cuts off from Ω the portion Σ λ . At the beginning of this process, the reflection Σ λ of Σ λ will be contained in Ω.
Then, there exists δ = δ(n, diam(Ω)) > 0 such that if |Ω| < δ, U ≥ 0 in Ω The following technical result can be seen as a slight variation of Lemma 2.1 and as an extension of [4, Proposition 2.1] to the case of (special) cooperative systems.
(2.6) Moreover, we claim that if r > 0 is small enough. Indeed, since a ij ∈ L ∞ (Ω) for any i, j, we can define Then, a direct computation shows that as |x| → 0 .
Since n ≥ 2 and a ∈ (0, 1), if r 1 is sufficiently small we obtain  Moreover, let U = (u 1 , . . . , u m ) ∈ C 2 (Ω; R m ) be such that If there exists x 0 ∈ Ω such that u k (x 0 ) = 0 (for some k ∈ {1, . . . , m}), then Proof. We consider the R m -valued function V ∈ C 2 (Ω; R m ) defined by where α > 0 is a suitable real constant which will be chosen later on. For every fixed k ∈ {1, . . . , m} and every x ∈ Ω we have thus, owing to the first inequality in (2.12) we get From this, by combining the second inequality in (2.12) with (2.11) we obtain where L α := ∆ + 2α∂ x1 . Now, since the function a kk is bounded on Ω (by assumption), it is possible to choose α > 0 in such a way that α 2 + a kk ≥ 0 on Ω; with this choice of α we then get (see also (2.12)) We are ready to conclude: the operator L α being uniformly elliptic in Ω, the classical Strong Maximum Principle holds for L α ; as a consequence, since v k = e −αx1 u k ≤ 0 on Ω, if u k (x 0 ) = 0 (for some x 0 ∈ Ω) we have v k (x 0 ) = 0, and hence v k ≡ 0 on Ω. This obviously implies that u k ≡ 0 on Ω, and the proof is complete.

S. BIAGI, E. VALDINOCI AND E. VECCHI
Finally, in order to make the paper self-contained, we state and prove a Hopf Lemma for weakly cooperative ellitic systems. Moreover, let U = (u 1 , . . . , u m ) ∈ C 2 (Ω; R m ) be such that (2.14) If there exists x 0 ∈ ∂Ω such that u k (x 0 ) = 0 (for some k ∈ {1, . . . , m}) and if Ω satisfy an interior ball condition at x 0 , then provided that the outer normal derivative does exist.
Proof. We consider once again the R m -valued map V ∈ C 2 (Ω; R m ) defined by V := e −αx1 U on Ω.
If k ∈ {1, . . . , m} is arbitrarily fixed and if α > 0 is chosen in such a way that α 2 + a kk ≥ 0 on Ω (remind that, by assumption a kk ∈ L ∞ (Ω)), by arguing exactly as in the proof of Theorem 2.3 we infer that where L α := ∆ + 2α ∂ x1 . Since the operator L α is uniformly elliptic in Ω, the classical Hopf Lemma holds for L α (see, e.g., [14, Lemma 3.4]); as a consequence, since v k = e −αx1 u k < 0 on Ω (see (2.14)), if there exists a point x 0 ∈ ∂Ω such that v k (x 0 ) = 0 (and if the outer normal derivative of u k at x 0 exists) we have This obviously implies the desired (2.15), and the proof is complete.
3. Proof of Theorem 1.1. In the present setting, we can now perform the proof of Theorem 1.1. For simplicity we separate the proof of the first claim in Theorem 1.1, which is the core of the argument, from the proof of the second claim, which is mostly straightforward.
Proof of Theorem 1.1 -(1). First of all we observe that, for every fixed λ ∈ (0, 1), the functions v We then observe that, since λ is strictly positive, the reflection of ∂Σ λ ∩ ∂Ω with respect to the hyperplane T λ is entirely contained in Ω (remind the structural assumptions satisfied by Ω). As a consequence, by (3.1) and (3.2), we derive that (for We explicitly point out that, if λ = 1/2, we have that 0 λ / ∈ ∂Σ λ . Gathering all these facts we see that, for every fixed λ ∈ (0, 1), the R m -valued map V λ satisfies on Σ λ \ {0 λ } the following elliptic system of PDEs: where A(x; λ) is given by and, by virtue of assumption (f1), it holds that Furthermore, since u 1 , . . . , u m are positive and continuous on Ω \ {0} and, for every fixed λ ∈ (0, 1), the set Σ λ is compactly contained in Ω \ {0}, we also have It is immediate to check that the system in (3.4) is (weakly) cooperative and satisfies the assumptions of Lemma 2.1. This implies that, for λ very close to 1, V λ ≥ 0 in Σ λ (note that, if λ ∼ 1, then 0 λ / ∈ Ω). Moreover, by Theorem 2.3 we have We can then define We list below some useful properties of µ.
(iv) If µ > 0, then V µ > 0 on Σ µ \ {0 µ }. Indeed, by the previous point (iii) we know that V µ ≥ 0 on Σ µ \ {0 µ }; moreover, assuming that µ > 0, we know that V µ satisfies (3.4) on the same set (which is open and connected); we are then entitled to apply the strong maximum principle in Theorem 2.3, which ensures that The goal is to show that µ = 0. (3.9) Indeed, if this is the case, by the above (iii) we have (for x ∈ Σ 0 = Ω ∩ {x 1 > 0}) Thus, by applying this result to the function Ω x → W (x) := U (−x 1 , . . . , x n ) (which solves the same system of PDEs in (1.1)), we conclude that, for every x ∈ Ω ∩ {x 1 > 0}, (3.10) and this proves that U is symmetric in x 1 , as desired.
Case I: We note that, in this case, 0 µ / ∈ Σ µ , so that there is no effect of the singularity. If δ > 0 is as in the statement of Lemma 2.1, we choose a compact set We claim that there exits a small 0 > 0, which we may assume to be smaller than µ − 1/2, with the following property: for every ∈ [0, 0 ], one has |Σ µ− \ K| < δ, (3.11) and V µ− > 0, in K.
(3.12) While (3.11) follows by continuity, the claim in (3.12) can be proved by arguing as follows: we set ρ := dist(K, 0 µ ) > 0 and we consider the compact set It is very easy to see that, if R λ is as in (2.1) and 0 < ρ/4, then for every x ∈ K and every ∈ [0, 0 ]; as a consequence, since U is uniformly continuous on K , by shrinking 0 we get for every i = 1, . . . , and every ∈ [0, 0 ]. In particular, this implies that Now, since V µ− satisfies the system of PDEs on account of (3.11) and (3.13) we are entitled to apply Lemma 2.1, which gives By combining this last fact with (3.12) we then get As a consequence, taking into account (3.4) and (3.6), we infer from Theorem 2.3 that which contradicts the definition of µ. This excludes the case µ ∈ (1/2, 1).
Case II: In this case, we need to distinguish two possible sub-cases. If the 0 1/2 does not lie in Ω, then there is no singularity involved and we can argue exactly as in Case I. We then have to rule out just the case 0 1/2 ∈ ∂Ω. Let us consider a positive constant ρ > 0 sufficiently small and let us define the set D ρ ⊂ Σ 1/2 D ρ := y ∈ Σ 1/2 : dist(y, ∂Σ 1/2 ) ≥ ρ .
Since D ρ ⊂ Σ 1/2 , we have V 1/2 > 0 in D ρ ; We then define the set A ρ ⊂ Ω as By definition, D ρ ∩ A ρ = ∅; moreover, it follows from the above (iv) that Now, by arguing as in the proof of (3.12), we can show that there exists 0 = 0 (ρ) > 0 such that, for any ∈ [0, 0 ], we have 0 1/2− ∈ Ω ∩ B ρ/2 (0 1/2 ) and (3.14) as a consequence, since (3.14) obviously implies that and, by (3.4), the map V 1/2− satisfies the system of PDEs by eventually shrinking ρ we are entitled to apply Lemma 2.1, which gives Gathering together (3.14) and (3.15), for every ∈ [0, 0 ] we then get We now proceed in this second step by showing that, setting we are then entitled to apply Lemma 2.2, which gives as claimed. By combining this last fact with (3.16) we obtain thus, again by taking into account (3.4) and (3.6), we deduce from Theorem 2.3 that This proves that even the case µ = 1 2 is not possible.
Case III: To prove that also this case is not possible, we argue essentially as in the previous Case II. First of all, given any ρ > 0 such that dist(0 µ , ∂Σ µ ) > ρ, we define Since K ρ is compact and, by (iv), the function V µ is continuous and strictly positive on From this, by arguing as in Case II, we infer the existence of a small 0 = 0 (ρ) > 0 such that, for every ∈ [0, 0 ], the point 0 µ− lies in D ρ and In particular, since B ρ/2 (0 µ ) lies in the interior of D ρ , we get for every ∈ [0, 0 ]. As a consequence, since V µ− solves the system of PDEs if ρ has been chosen in such a way that |Σ µ− 0 \ D ρ | is sufficiently small, we can apply the weak maximum principle in Lemma 2.1 and obtain Gathering together (3.18) and (3.17) we conclude that, for every 0 ≤ ≤ 0 , . We now turn to prove that, for every fixed ∈ [0, 0 ], we have V µ− ≥ 0 throughout the punctured ball B ρ/2 (0 µ ) \ {0 µ−ε }, so that (see the above (3.18)) To this end, we argue as in the previous case: indeed, it suffices to apply Lemma 2.2, with A(x; µ − ε) as in (3.5), on B ρ/2 (0 µ ). We are now ready to conclude: by virtue of the above (3.19) and since, by (3.4), we have v µ−ε i ≡ 0 on ∂Σ µ−ε \ {0 µ−ε } (for any i = 1, . . . , m), we derive from the strong maximum principle in Theorem 2.3 that The proof of (3.9) is therefore complete.