Geometric inequalities and symmetry results for elliptic systems

We obtain some Poincar\'{e} type formulas, that we use, together with the level set analysis, to detect the one-dimensional symmetry of monotone and stable solutions of possibly degenerate elliptic systems of the form {eqnarray*} {{array}{ll} div(a(|\nabla u|) \nabla u) = F_1(u, v), div(b(|\nabla v|) \nabla v) = F_2(u, v), {array}. {eqnarray*} where $F\in C^{1,1}_{loc}(\R^2)$. Our setting is very general, and it comprises, as a particular case, a conjecture of De Giorgi for phase separations in $\R^2$.

1. Introduction. In this paper we consider a class of quasilinear (possibly degenerate) elliptic systems in R n . We prove that, under suitable assumptions, the solutions have one-dimensional symmetry, showing that the results obtained in [1,2,8] hold in a more general setting.
In [1] the following problem has been studied:    ∆u = uv 2 , ∆v = vu 2 , u, v > 0. (1.1) The authors proved the existence, symmetry and nondegeneracy of the solution to problem (1.1) in R; in particular, they showed that entire solutions are reflectionally symmetric, namely that there exists x 0 such that u(x − x 0 ) = v(x − x 0 ). Moreover, they estabilished a result that may be considered the analogue of a famous conjecture of De Giorgi for problem (1.1) in dimension 2, that is they proved that monotone solutions of (1.1) in R 2 have one-dimensional symmetry under the additional growth condition In [2] the authors proved that the above mentioned one-dimensional symmetry still holds in R 2 when the monotonicity condition is replaced by the stability of the solutions (which is a weaker assumption). Moreover, they showed that there exist solutions to (1.1) which do not satisfy the growth condition (1.2), by constructing solutions with polynomial growth.
Moreover, we mention the paper [14], where the author proved that, for any n ≥ 2, a solution to (1.1) which is a local minimizer and satisfies the growth condition (1.2) has one-dimensional symmetry.
These assumptions may look rather technical at a first glance, but they are the standard conditions that comprise as particular cases the classical elliptic degenerate and nonlinear operators, such as the p-Laplacian and the mean curvature operator.
In order to state our main result, we give the definition of monotone and stable solution.
Definition 1.1. We say that a solution (u, v) of (1.3) satisfies a monotonicity condition if u n > 0, v n < 0. (1.8) (1.9) In our general framework, since F 1 and F 2 may not be everywhere differentiable, the integral in (1.9) may not be well defined. Therefore it is convenient to introduce the sets D := (t, s) ∈ R 2 : F 11 (t, s), F 12 (t, s), F 22 (t, s) exist , and N := R 2 \ D. It is known that the set N is Borel and with zero Lebesgue measure (1.10) (see pages 81-82 in [4]). Moreover, we consider the sets So we say that (u, v) is a stable solution to (1.3), if for any φ, ψ ∈ C ∞ 0 (R n ), (1.11) Of course, (1.11) reduces to (1.9) when F is in C 2 loc (R 2 ). Then, we state our symmetry result. For this, we denote by ℑ(u, v) the image of the map (u, v) : , and ∇u, ∇v ∈ W 1,2 loc (R n ). Suppose that either (A2) holds or that {∇u = 0} = ∅, and that either (B2) holds or that {∇v = 0} = ∅.
Assume that either the monotonicity condition (1.8) holds, and (1.14) then (u, v) has one-dimensional symmetry, in the sense that there exist u, v : Moreover, if we assume in addition that either the monotonicity condition (1.8) holds, and there exists a non-empty Notice also that one can consider a more general function F such that F 12 (u, v) = 0, that is a system with two independent equations, and there is no reason why u and v should have one-dimensional symmetry with the same vector.
We notice that, as paradigmatic examples satisfying the assumptions of Theorem 1.3, one may take the p-Laplacian, with p ∈ (1, +∞) if {∇u = 0} = ∅ and any p ∈ [2, +∞) if {∇u = 0} = ∅ (in this case, for instance, a(t) = t p−2 ) or the mean curvature operator (in this case, a(t) = (1 + t 2 ) −1/2 ). Moreover, we observe that Theorem 1.3 holds even if a and b are two different functions satisfying the hypotheses (e.g., one can take a to be of p-Laplacian type and b of mean curvature type).
To prove Theorem 1.3 we borrow a large number of ideas from [5] and [6], and exploit some techniques of [12,13]. In particular, we will show that a formula proved in [12,13] and its extension obtained in [6] for elliptic equations still hold for systems (see Corollaries 3.3 and 4.4). Since this formula bounds a weighted L 2 -norm of any test function by a weighted L 2 -norm of its gradient, we may see it as a weighted Poincaré type inequality. Such a formula is geometric in spirit, since it bounds tangential gradients and curvatures of level sets of monotone and stable solutions in terms of suitable energy integrals.
Our result extends the one obtained in [8], where the authors studied problem (1.3) in the case a = b = Id, and use this kind of geometric Poincaré inequality to show that in R 2 any stable solution has a one-dimensional symmetry. Of course in our setting several technical and conceptual complications arise due to the possible degeneracy of the operators considered and to the nonlinear dependence on the gradient terms.
Moreover, as a particular case, Theorem 1.3 comprises a conjecture of De Giorgi for phase separations in R 2 (see the end of Section 7).
We refer the reader to [7] for a recent review on the conjecture of De Giorgi and related topics.
The paper is organized as follows. In Section 2 we collect some preliminary material. Sections 3 and 4 are devoted to show that some geometric Poincaré type inequalities hold for monotone and stable solutions to (1.3) respectively. In Section 5 we develop the level set analysis. In Section 6 we provide the proof of Theorem 1.3, by using the results obtained in the previous sections. Finally, in Section 7, we give an application of Theorem 1.3, namely we prove that a conjecture of De Giorgi holds in R 2 for systems like (1.3), and in particular for phase separations.
2. Some useful results. In this section we collect some results that we will use in the sequel.
Then, for any j = 1, . . . , n, Proof. First of all, we observe that  (x)). In the same way one shows (2.5).
A direct computation also shows that on {∇u = 0} As a consequence, Reasoning in the same way, we conclude also that . We use the above observations to obtain that We observe that in the proof of Lemma 2.2 it is sufficient to assume that ∇u, ∇v ∈ W 1,1 loc (R n ). Since such a generality is not needed here, we assumed, for simplicity, ∇u, ∇v ∈ W 1,2 loc (R n ) in order to use the above result in the sequel. Let us notice that (2.3) means that, for any φ, ψ ∈ C ∞ 0 (R n ), and for any j = 1, . . . , n, Since the integrals in (2.6) may not be well defined, recalling the definitions of the sets D, N , N uv , D uv given in the Introduction and using (1.10) we can say In the sequel we will need to use (2.7) for a less regular test functions. To do this, we prove the following: Under the assumptions of Lemma 2.2, we have that (2.7) holds for any j = 1, . . . , n, any φ, ψ ∈ W 1,2 0 (B) and any ball B ⊂ R n . Proof. Let us prove the first equality in (2.7). Given φ ∈ W 1,2 0 (B), we consider a sequence of functions φ k ∈ C ∞ 0 (B) which converge to φ in W 1,2 0 (B). Let m u and M u (respectively m v and M v ) be the minimun and the maximum of |∇u| (respectively |∇v|) on the closure of B. Moreover, let In the same way, one has that also K B < +∞. Now, since the assumptions of Lemma 2.2 hold, we deduce from (2.7) , which tends to zero as k tends to infinity, because of the assumptions on u, v. The latter consideration and (2.8) give the first equality in (2.7). Reasoning in a similar way, we obtain also the second equality in (2.7).
We will now consider the tangential gradient with respect to a regular level set. Given w ∈ C 1 (R n ), we define the level set of w at x as (2.9) If ∇w(x) = 0, L w,x is a hypersurface near x and one can consider the projection of any vector onto the tangent plane: in particular, the tangential gradient, which 8 SERENA DIPIERRO will be denoted as ∇ Lw,x , is the projection of the gradient. This means that, given f ∈ C 1 (B r (x)), for r > 0, the tangential gradient is (2.10) We will use the following lemma (see Lemma 2.3 in [6] for a simple proof): Given y ∈ L w,x ∩ {∇w = 0}, let k 1,w (y), . . . , k n−1,w (y) denote the principal curvatures of L w,x at y.

Monotone solutions.
Recalling the definition of monotone solution given in (1.8), in this section we obtain some geometric inequalities.
Proof. By Lemma 2.3, we have that u n satisfies (2.7). We use φ 2 un as test function in the first equality in (2.7): since (2.1) holds. This implies the first inequality in (3.1). Using ψ 2 vn as test function in the second equality in (2.7), and reasoning as above, we obtain also the second inequality in (3.1).
In the subsequents Theorem 3.2 and Corollary 3.3 we obtain some inequalities which involve the principal curvature of the level sets and the tangential gradient of the solution. For any x ∈ Ω let L u,x and L v,x denote the level set of u and v respectively at x, according to (2.9).

3)
for any locally Lipschitz functions φ, ψ : Ω → R whose supports are compact and contained in Ω.
For any x ∈ Ω let L u,x and L v,x denote the level set of u and v respectively at x, according to (2.9).
Proof. By summing up the inequalities in (3.2) and (3.3), we have that, for any ϕ as in the corollary, which gives the conclusion, since F 12 (u, v) ≥ 0.
4. Stable solutions. In this section we obtain some geometric inequalities for stable solutions of (1.3). Since we will use the stability condition (1.11) with a less regular test functions, we need to state the following: Suppose that either (A2) holds or that {∇u = 0} = ∅, and that either (B2) holds or that {∇v = 0} = ∅. Then, the stability condition (1.11) holds for any φ, ψ ∈ W 1,2 0 (B), and any ball B ⊂ R n . Proof. As in the proof of Lemma 2.3, we introduce m u , M u , m v , M v , K A , K B , and notice that, under the hypotheses of Lemma 4.1, K A , K B < +∞. Moreover, given φ, ψ ∈ W 1,2 0 (B), we consider two sequences φ k , ψ k ∈ C ∞ 0 (B) which converge to φ, ψ respectively in W 1,2 0 (B). Therefore, , which tends to zero as k tends to infinity.
Similarly, one obtains , which again tends to zero. Moreover, one has that, as k tends to infinity, , which converges to zero as k tends to infinity. This concludes the proof.
We prove next that, under suitable assumptions, a monotone solution of (1.3) is also stable.  Let (u, v) be a weak solution of (1.3), with u, v ∈ C 2 (R n ), and ∇u, ∇v ∈ W 1,2 loc (R n ). Suppose that the monotonicity condition (1.8) holds, and that F 12 (u, v) ≥ 0. Then (u, v) is a stable solution.
Proof. By summing up the inequalities in (3.1), we have where we have used the monotonicity condition, the fact that F 12 (u, v) ≥ 0, together with This concludes the proof.
In the subsequents Theorem 4.3 and Corollay 4.4, we prove that a formula obtained in [12,13] and its extension obtained in [6] hold also for a system of the form (1.3). These formulas relate the stability of the system with the principal curvatures of the corresponding level sets and with the tangential gradient of the solution. For any x ∈ Ω let L u,x and L v,x denote the level set of u and v respectively at x, according to (2.9).

GEOMETRIC INEQUALITIES AND SYMMETRY RESULTS FOR ELLIPTIC SYSTEMS 15
Then, for any locally Lipschitz function ϕ : Ω → R whose support is compact and contained in Ω.
For any x ∈ Ω let L u,x and L v,x denote the level set of u and v respectively at x, according to (2.9).

5.
Level set analysis. We recall here the geometric analysis performed in Subsection 2.4 in [6]. In order to make this paper self-contained, we include the proofs in full detail. We consider connected components of the level sets (in the inherited topology).
Proof. Since any connected components of L w,x ∩ {∇w = 0} is a regular hypersurface, any two points in it may be joined by a C 1 path. We notice that, if thanks to (2.10). As a consequence, if  in R n for which z ∈ U 2 and L w,x ∩ U 2 is a hypersurface. Since M ⊆ π, we have that L w,x ∩ U 2 ⊆ π, hence L w,x ∩ U 2 is open in the topology of π. Then, z ∈ L w,x ∩ U 1 ∩ U 2 , which is an open set in π. This proves (5.3).

GEOMETRIC INEQUALITIES AND SYMMETRY RESULTS FOR ELLIPTIC SYSTEMS 19
Also, M is closed in R n and so M = M ∩ π is closed in π.
Hence, M is open and closed in π.
Then, L is a flat hyperplane.
Proof. We use a standard differential geometry argument (see, for instance, page 311 in [11]). Since the principal curvatures vanish identically, the normal of L is constant, thence L is contained in a hyperplane. Then, the claim follows from Corollary 5.3.
has zero principal curvatures at all points (5.4) and that, for any x ∈ {∇w = 0}, Then, w possesses one-dimensional symmetry, in the sense that there exists w : R → R and ω ∈ S n−1 in such a way that w(x) = w(ω · x), for any x ∈ R n .
Proof. If ∇w(x) = 0 for any x ∈ R n , the one-dimensional symmetry is trivial.
If ∇w(x) = 0, then the connected component of L w,x ∩ {∇w = 0} passing through x is a hyperplane, thanks to Lemma 5.4. We notice that all these hyperplanes are parallel, since connected components cannot intersect. Moreover, w is constant on these hyperplanes, because each of them lies on a level set.
On the other hand, w is also constant on any other possible hyperplane parallel to the ones of the above family, since the gradient vanishes identically there.
From this, the one-dimensional symmetry of w follows by noticing that w only depends on the orthogonal direction with respect to the above family of hyperplanes.
Now, we chose conveniently ϕ in (6.1). For any R > 1, we define the function ϕ R as We denote by Therefore, by using ϕ R in (6.1), we have
Since we know that (u, v) has a one dimensional symmetry, this implies that ω u = ω v . This concludes the proof of Theorem 1.3.
7. An application. In this section, we use the result stated in Theorem 1.3 to obtain a proof of a conjecture of De Giorgi for the system (1.3) in R 2 .
Then (u, v) has one-dimensional symmetry, in the sense that there exist u, v : R → R and ω u , ω v ∈ S n−1 in such a way that (u(x), v(x)) = (u(ω u · x), v(ω v · x)), for any x ∈ R n .

GEOMETRIC INEQUALITIES AND SYMMETRY RESULTS FOR ELLIPTIC SYSTEMS 23
Moreover, if we assume in addition that either the monotonicity condition (1.8) holds, and there exists a non-empty open set Ω ′ ⊆ R n such that F 12 (u(x), v(x)) > 0 for any x ∈ Ω ′ , or (u, v) is stable, and there exist two open intervals I u , I v ⊂ R such that (I u × I v ) ∩ ℑ(u, v) = ∅ and F 12 (u, v) > 0 for any (u, v) ∈ I u × I v , then (u, v) has one-dimensional symmetry, and ω u = ω v .
Notice that, as a particular case of (1.3), we can consider the following system, which arises in phase separation for multiple states Bose-Einstein condensates:    ∆u = uv 2 , ∆v = vu 2 , u, v > 0. (7.1) In fact, in this case, the operators in (1.3) reduce to the standard Laplacian and F (u, v) = 1 2 u 2 v 2 . Under the assumptions of Theorem 7.1 (notice that F 12 (u, v) = 2uv > 0), one has that the monotone solutions of (7.1) have one-dimensional symmetry. This result has been proved in [1].