Sectional symmetry of solutions of elliptic systems in cylindrical domains

In this paper we prove a kind of rotational symmetry for solutions of semilinear elliptic systems in some bounded cylindrical domains. The symmetry theorems obtained hold for low-Morse index solutions whenever the nonlinearities satisfy some convexity assumptions. These results extend and improve those obtained in \cite{DaPaSys, DaGlPa1, Pa, PaWe}.


Introduction
We consider the Dirichlet problem for a semilinear elliptic system of the type Nirenberg [14], based on the moving planes method, asserts that if Ω is a ball then every positive solution of (1.1) is radial if the nonlinear term f = f (|x|, u) is monotone decreasing with respect to r = |x|. The result of [14] was then extended to systems in [22], [11], [12].
It is well known that the radial symmetry of a solution does not hold, in general, when Ω is an annulus or if sign changing solutions are considered and even if f does not have the right monotonicity with respect to |x| (see for example [17]). Nevertheless when the hypotheses of the theorem of Gidas, Ni and Nirenberg fail another kind of symmetry can be recovered, namely the foliated Schwarz symmetry for solutions of (1.1) in a ball or in annulus having low Morse index and assuming that the nonlinear term has some convexity properties in the U-variable. We refer to Section 2 for the definition of Morse index.
A symmetry result of this type was first proved in [16] in the case m = 1 for solutions having Morse index one and assuming that the nonlinearity f = f (|x|, s) is convex in the second variable. Later it was extended in [18] to solutions having Morse index not larger than the dimension N and assuming that the derivative ∂f ∂s is a convex function in the s-variable. Finally in [6] and [9] the foliated Schwarz symmetry was proved for low Morse index solutions of cooperative elliptic systems, i.e. when m ≥ 2. Let us point out that the extension of the results in [16] and [18] to systems is nontrivial. Indeed the results of [6] could not be proved for any convex nonlinearity F = F (|x|, U) but some additional hypotheses were required.
In this paper we extend the above results by considering more general symmetric domains and not just balls or annulus. As a consequence we will get less symmetry of the solutions, depending also on a tighter bound on their Morse index. Moreover we are able to improve the results in [6] by allowing any convex nonlinearity in (1.1).
To state precisely our results we need some preliminary definitions. The first one concerns the domains we consider.
Let N ≥ 2 and 2 ≤ k ≤ N. If k < N, let us denote by x = (x ′ , x ′′ ) a point in R N , with x ′ ∈ R k , x ′′ ∈ R N −k , and for a bounded domain Ω let us denote by Ω ′′ the set We will consider domains of the following type. DEFINITION 1.1. Assume that N ≥ 2, 2 ≤ k ≤ N. We say that a bounded domain Ω in R N , is k-rotationally symmetric if either k = N and Ω is a ball or an annulus, or 2 ≤ k < N and the sets Ω h = Ω ∩ {x = (x ′ , x ′′ ) ∈ R N : x ′′ = h} are either k-dimensional balls or k-dimensional annulus with the center on (0, h) for every h ∈ Ω ′′ .
In other words we require that the set Ω h , which represents a section of Ω at the level x ′′ = h is either a ball or an annulus in dimension k.
We will call it k-sectional foliated Schwarz symmetry.
When k = N the previous definition coincides with that of foliated Schwarz symmetry. Definition 1.2 just means that the functions x ′ → U(x ′ , h) defined in Ω h are either radial for any h ∈ Ω ′′ , or nonradial but foliated Schwarz symmetric for any h ∈ Ω ′′ , with the same axis of symmetry. In the case k = N −1 the sectional foliated Schwarz symmetry was defined in [7] to study some elliptic problems with nonlinear mixed boundary conditions.
In order to prove symmetry of solutions we also need some symmetry on the nonlinearity. Therefore from now on we assume that Ω is a smooth bounded k-rotationally symmetric domain in R N and we rewrite the system (1.1) as The symmetry results we get are the following (the definition of Morse index and fully coupled systems will be recalled in Section 2). THEOREM 1.1. Let Ω be a k-rotationally symmetric domain in R N , 2 ≤ k ≤ N, and let U ∈ C 2 (Ω; R m ) be a solution of (1.2) with F satisfying (1.3). Assume that i) the system (1.2) is fully coupled along U in Ω ii) for any i = 1, . . . m the scalar function f i (|x ′ |, x ′′ , S) is convex in the variable S = (s 1 , . . . , s m ) ∈ R m .
If m(U) ≤ k, where m(U) is the Morse index of U, then U is ksectional foliated Schwarz symmetric, and if the functions x ′ → U(x ′ , h), h ∈ Ω ′′ , are not radial then they are strictly decreasing in the angular variable.
This theorem not only extends the results in [6] to k-rotationally symmetric domains but also improves the result of [6] for the case k = N since it only requires that the components f i of the nonlinearity are convex without further assumptions.
The next theorem concerns the case when the nonlinearity has convex first derivatives.
Let Ω be a k-rotationally symmetric in R N , 2 ≤ k ≤ N, and let U ∈ C 2 (Ω; R m ) be a solution of (1.2). Assume that: i) the system (1.2) is fully coupled along U in Ω ii) for any i, j = 1, . . . m the function ∂f i ∂s j (|x ′ |, x ′′ , S) is convex in S = (s 1 , . . . , s m ).
If m(U) ≤ k − 1 then a solution U is k-sectionally foliated Schwarz symmetric and if the functions x ′ → U(x ′ , h), h ∈ Ω ′′ , are not radial then they are strictly decreasing in the angular variable.
The previous theorem extends to k-rotationally symmetric domains the result in [9] and we provide a different proof which also simplify the one given in [9]. Note that in Theorem 1.2 the bound on the Morse index m(U) ≤ k − 1 is stricter than in Theorem 1.1. When m = 1, i.e. in the scalar case, it is possible to improve it to m(U) ≤ k (adapting the proof in [18] for k = N). However in the vectorial case serious difficulties arise when m(U) = k, which prevent to use the same approach, though we believe that the symmetry result should be true also in this case.
It is interesting to see in the previous theorems how the Morse index of a solution is related to the "dimension" of the sectional symmetry of the domain. Remark 1.1. In the particular case of stable solutions (see Section 2 for the definition) we get the radial symmetry on each section Ω h without requiring any convexity on F . This can be proved easily as in the proof of Theorem 1.5 of [15].
We will deduce from the proof of Theorem 1.1 and Theorem 1.2 that for nonradial Morse index one solutions the following condition holds.
for any i = 1, . . . , m, with (r, ϑ) as in Definition 1.2. In particular if m = 2 then (1.4) implies that The paper is organized as follows. In Section 2 we recall suitable versions of weak and strong maximum principles as well as comparison principles for systems. Moreover we state some results from the spectral theory for an eigenvalue problem related to a symmetrized version of the system (1.1). Finally we define the Morse index. In Section 3 we give some sufficient conditions for ksectional foliated Schwarz symmetry and prove Theorem 1.1, Theorem 1.2 and Corollary 1.1.

Spectral theory for linear elliptic systems.
Let Ω be a bounded domain in R N , N ≥ 2, and D a m × m matrix with bounded entries: We consider the linear elliptic system Before going on we fix some notations and definitions.
It is well known that either condition (2.3) or conditions (2.3) and (2.4) together are needed in the proofs of maximum principles for systems (see [11], [13], [21] and the references therein). In particular if both are fulfilled the strong maximum principle holds as it is stated in the next theorem (see [8], [11], [13], [21] for the proof).
, Ω satisfies the interior sphere condition at P ∈ ∂Ω and u k (P ) = 0 then ∂u k ∂ν (P ) < 0, where ν is the unit exterior normal vector at P .
(2) if in addition (2.4) holds, then the same alternative holds for , Ω satisfies the interior sphere condition at P ∈ ∂Ω and U(P ) = 0 then ∂U ∂ν (P ) < 0, where ν is the unit exterior normal vector at P .
Together with the bilinear form (2.8) we consider the quadratic form for Ψ = (ψ 1 , . . . , ψ m ) ∈ H 1 0 (Ω). Sometimes we will also write Q(Ψ; Ω) instead of Q(Ψ) specifying the domain. It is easy to see that this quadratic form coincides with the quadratic form B C associated to the symmetric linear operator −∆ + C where Let us observe that if the matrix D is cooperative, respectively fully coupled, so is the associate matrix D.
Thus, let us review some results for a symmetric linear operator −∆ + C, with C such that (2.13) c ij ∈ L ∞ (Ω) , c ij = c ji a.e. in Ω Let us consider the bilinear form (2.14) Using the theory of compact selfadjoint operators we get that there exists a sequence {λ j } = {λ j (−∆ + C)} of eigenvalues, with −∞ < λ 1 ≤ λ 2 ≤ . . . , lim j→+∞ λ j = +∞, and a corresponding sequence of eigenfunctions {W j } which weakly solve the systems Moreover by (scalar) elliptic regularity theory applied iteratively to each equation, the eigenfunctions W j belong at least to C 1 (Ω; R m ) and the eigenvalues can be given a variational formulation. We refer to [6], [8] for the construction of the sequences λ j and {W j } as well as for the proof of the following theorem, which gives some variational properties of eigenvalues and eigenfunctions.
and B as in (2.14). Then the following properties hold, where H k denotes a k-dimensional subspace of H 1 0 (Ω) and the orthogonality conditions V ⊥W k or V ⊥H k stand for the orthogonality in L 2 (Ω). i) iii) if W ∈ H 1 0 (Ω), W = 0, and R(W ) = λ 1 , then W is an eigenfunction corresponding to λ 1 . iv) if the system is fully coupled in Ω, then the first eigenfunction does not change sign in Ω and the first eigenvalue is simple, i.e. up to scalar multiplication there is only one eigenfunction corresponding to the first eigenvalue. v) if X = (C 1 c (Ω)) m and we denote by X k a k-dimensional subspace of X then  The bilinear form corresponding to the symmetric operator will be denoted by B s (U, Φ), i.e. B s is as (2.14).
As already remarked, the quadratic form (2.11) corresponding to the linear operator −∆+D coincides with that associated to the symmetric linear operator −∆ + C. DEFINITION 2.2. We say that the maximum principle holds for the Let us denote by λ As an almost immediate consequence we get a quick proof of the following "classical" and "small measure" forms of the weak maximum principle (see [5], [13], [19], [21]).
Hence λ (Ω ′ ) ≤ 0, since the system is cooperative (and symmetric), there exists a corresponding nontrivial nonnegative first eigenfunction Φ 1 ≥ 0, Φ ≡ 0, and the maximum principle does not hold, since −∆Φ 1 However the converse of Theorem 2.3 is not true for general nonsymmetric systems, since there is an equivalence between the validity of the maximum principle for the operator −∆+D and the positivity of another eigenvalue, the principal eigenvalueλ 1 , (we recall below the definition given in [3]), and the inequalityλ 1 (Ω ′ ) ≥ λ Let us recall some of the properties of the principal eigenvalue. We refer to [5] for the proofs of items i) -iii), as well as for references on the subject, and to [6], [8] for the proof of iv). i) there exists a positive eigenfunction Ψ 1 ∈ W 2,N loc (Ω ′ ; R m ) which satisfies Moreover the principal eigenvalue is simple, i.e. any function that satisfy (2.18) must be a multiple of Ψ 1 ii) the maximum principle holds for the operator −∆ + D in Ω ′ if and only ifλ 1 (

Comparison principles for semilinear elliptic systems.
Let us consider a semilinear elliptic system of the type is the conjugate exponent of the critical Sobolev exponent 2 * = 2N N −2 ) and (2.20) . Moreover we say that U satisfies in a weak sense the inequality for any ϕ ∈ H 1 0 (Ω) with ϕ ≥ 0 in Ω. This is equivalent to require that As a consequence of Theorem 2.4 and Theorem 2.1 the following comparison principles hold (see [8] for the proof). if Ω ′ ⊆ Ω is a bounded subdomain of Ω, meas N ([u > v] ∩ Ω ′ ) < δ and

THEOREM 2.7 (Strong Comparison Principle for systems).
Let Ω be a (bounded or unbounded) domain in R N , and let U, V ∈ C 1 (Ω) weakly satisfy : Ω × R m → R m is a C 1 function and (2.24) holds.
(1) For every i ∈ {1, . . . , m} the following holds: either where the interior sphere condition is satisfied then ∂u i ∂s (x 0 ) < ∂v i ∂s (x 0 ) for any inward directional derivative.
(2) If moreover U ∈ C 1 (Ω; R m ) is a solution of (2.19) and the system is fully coupled along U in Ω (i.e. also (2.25) with Ω ′ = Ω holds) then either U ≡ V in Ω or U < V in Ω (i.e. the same alternative holds for any component u i ). In the latter case assume that U, V ∈ C 1 (Ω) and let x 0 ∈ ∂Ω a point where U(x 0 ) = V (x 0 ) and the interior sphere condition is satisfied. Then ∂U ∂s (x 0 ) < ∂V ∂s (x 0 ) for any inward directional derivative. i) Let U ∈ H 1 0 (Ω)∩L ∞ (Ω) be a weak solution of (1.1). We say that U is linearized stable (or has zero Morse index) if the quadratic form The crucial, simple remark that allowed to extend some of the symmetry results known for equations to the case of systems in [6] and [9], is that the quadratic form associated to the linearized operator at a solution U, i.e. to the linear operator which in general is not selfadjoint, coincides with the quadratic form corresponding to the selfadjoint operator where J t F is the transpose of the matrix J F . Therefore the symmetric eigenvalues of L, i.e. the eigenvalues of L s U , as defined in Section 2.2 can be exploited to study the symmetry of the solution U, using the information on its Morse index.
Let us define, with a little abuse of notations, (3.2) S k−1 = {e ∈ S N −1 : e · e j = 0 , j = k + 1, . . . , N} A sufficient condition for the k-sectional foliated Schwarz symmetry is the following.
. Assume that the system is fully coupled along U in Ω and that ∀ e ∈ S k−1 Then U is k-sectionally foliated Schwarz symmetric.
The proof is similar to the one given, for the case k = N, in [6], with some obvious change.
Let us consider a pair of orthogonal directions η 1 , η 2 ∈ S k−1 , the polar coordinates (ρ, ϑ) in the plane spanned by them and the corresponding cylindrical coordinates (ρ, ϑ,ỹ), withỹ ∈ R N −2 . Then we define for U ∈ C 2 (Ω; R m ) the angular derivative (trivially extended if ρ = 0) which solves the linearized system and, if e ∈ span (η 1 , η 2 ) and U ≡ U σ(e) in Ω(e), also the system Using the properties of the principal eigenvalue and of the corresponding eigenfunction we deduce, as in the case k = N, (see [6], [8], [9]), the following sufficient conditions for the k-sectionally foliated Schwarz symmetry.

THEOREM 3.2 (Sufficient conditions for sectional FSS-Sistems).
Let Ω be a k-rotationally symmetric domain in R N , 2 ≤ k ≤ N, and Then U is k-sectionally foliated Schwarz symmetric provided one of the following conditions holds: i) there exists a direction e ∈ S k−1 such that U ≡ U σ(e) in Ω(e) and the principal eigenvalueλ 1 (Ω(e)) of the linearized operator in Ω(e) is nonnegative. ii) there exists a direction e ∈ S k−1 such that either U < U σ(e) or U > U σ(e) in Ω(e) Remark 3.1. Let us observe that in Theorem 3.2 it is the nonnegativity of the principal eigenvalue the crucial hypothesis, while the information we get in the sequel will concern the symmetric eigenvalues of the linearized system. Therefore in the proofs that follow there will be an interplay and a comparison between the principal eigenvalue and the first symmetric eigenvalue in the cap Ω(e).
If U is a solution of (1.2), e ∈ S k−1 and the system is fully coupled along U in Ω, then the difference W = W e = U − U σ(e) = (w 1 , . . . , w m ) satisfies a linear system in Ω, which is fully coupled in Ω and Ω(e): LEMMA 3.1. The following assertions hold.
i) Assume that U ∈ C 1 (Ω; R m ) is a solution of (1.2) and that the system is fully coupled along U in Ω. Let us define for any Then for any e ∈ S k−1 the function W e = U − U σ(e) satisfies in Ω(e) the linear system which is fully coupled in Ω(e). ii) If Ψ = (ψ 1 , . . . , ψ m ) ∈ H 1 0 (Ω(e)) let Q e (Ψ; Ω(e)) denote the quadratic form associated to the system (3.8) in Ω(e), i.e. Let us set V = U σ(e) . For any i = 1, . . . , m we have that As a consequence W e satisfies (3.8). Moreover if i = j then b e ij (x) ≤ 0 by (2.24) , so that the linear system (3.8) is weakly coupled.
If U ∈ C 1 (Ω; R m ) is a solution of (1.2) and the system is fully coupled along U then the linear system associated to the matrix B e is fully coupled in Ω. Indeed if i 0 = j 0 and Since B e is symmetric with respect to the reflection σ e , (3.8) is fully coupled in Ω(e) as well and i) is proved.
To get (3.10) it is enough to multiply the i-th equation of the system for w i and integrate. Instead, multiplying the i-th equation of (3.8) for i.e. (3.11) in the case of the positive part.
For the negative part we proceed analogously multiplying the i-th equation of (3.8) for w − i and integrating. We get i.e. (3.11) in the case of the negative part.
Remark 3.2. Note that the inequalities in (3.11) could be strict. Indeed the products w + i w − j could be not identically zero if i = j, and therefore Q(W e ) does not coincide in general with Q((W e ) + ) + Q((W e ) − ), as it happens in the scalar case.

Nonlinearities having convex components.
We will prove Theorem 1.1 by several auxiliary results. LEMMA 3.2. Assume that U is a solution of (1.2) and that the hypotheses i)-ii) of Theorem 1.1 hold. Then for any direction e ∈ S k−1 where Q U is the quadratic form defined in (2.28) and W e is as in Lemma 3.1.
Proof. For any i = 1, . . . , m we have Testing the equation with w + i we obtain (3.12) where ∇ stands for the gradient of f i with respect to the variables S = (s 1 , . . . , s m ). Moreover because ∂f i ∂s j ≥ 0 by the weak coupling assumption. By (3.12), taking into account the previous inequalities, we get Thus, summing on i = 1, . . . , m, we obtain (3.13) i.e. Q U ((W e ) + ; Ω(e)) ≤ 0.
Proof. The assertion is immediate if the Morse index of the solution satisfies m(u) ≤ 1. Indeed in this case for any direction e at least one among λ e 1 and λ −e 1 must be nonnegative. Indeed if this would not be the case then the quadratic form Q U (Ψ) = Ω (|∇Ψ| 2 − C(x) (Ψ, Ψ)) dx would be negative definite on the 2-dimensional space spanned by the trivial extensions of the eigenfunctions Φ e 1 and Φ −e 1 and hence m(u) ≥ 2. So let us assume that 2 ≤ j = m(u) ≤ k. Denote by Φ k the L 2 (Ω) normalized eigenfunctions of the operator L U = −∆ − C in Ω, with Φ 1 positive in Ω, and for any direction e ∈ S k−1 let us consider the function where Φ e 1 is the first positive L 2 -normalyzed eigenfunction in Ω(e), as in (3.14).
Proof of Theorem 1.1. By Lemma 3.3 there exists a direction e ∈ S k−1 such that the first symmetric eigenvalue λ s 1 (L U , Ω(e)) of the linearized operator is nonnegative, so that the principal eigenvalueλ 1 (Ω(e)) is nonnegative as well. Moreover by Lemma 3.2 we have that Q U ((W e ) + ) ≤ 0, so that either (W e ) + ≡ 0, or λ s 1 (L U , Ω(e)) = 0 and (W e ) + is the positive first symmetric eigenfunction in Ω(e). In any case either U ≤ U σ(e) or U ≥ U σ(e) in Ω(e) holds. Thus, by the strong maximum principle, either U ≡ U σ(e) in Ω(e), and the principal eigenvalueλ 1 (Ω(e)) is nonnegative, or U < U σ(e) in Ω(e) or U > U σ(e) in Ω(e). Hence, by Theorem 3.2 U is foliated Schwarz symmetric.
Indeed since U < U σ(g) for any direction g between e and e ′ , we have that 0 is the principal eigenvalue of the system satisfied by U − U σ(g) , namely (3.8), with coefficients As g → e ′ , where e ′ is the symmetry position, the coefficients b ij approach the coefficients of the linearized system, namely c ij = − ∂f i ∂s j , so by continuityλ 1 (Ω(e ′ )) =λ 1 (Ω(−e ′ )) = 0.

Nonlinearities with convex derivatives.
The proof of Theorem 1.2 follows the scheme of the proof of Theorem 1.1, and it is based upon the following results. ∂f i ∂s j (|x|, U(x)) + ∂f i ∂s j (|x|, U σ(e) (x)) Then the linear system with matrix B e,s is fully coupled in Ω and Ω(e) for any e ∈ S N −1 . Moreover for any i, j = 1, . . . , m and x ∈ Ω it holds Finally for the quadratic forms Q e and Q e,s associated to the matrices B e and B e,s we have that (3.19) 0 ≥ Q e ( (W e ) ± ; Ω(e) ) = Proof. By hypothesis ii) of Theorem 1.2 we get This implies (3.18) and hence the full coupling of the system with matrix B e,s , since, by Lemma 3.1, the system with matrix B e is fully coupled. From (3.11) and (3.18), since w ± k ≥ 0, we get (3.19) in the case of the positive part of W e . Analogously we get the corresponding inequality for the negative part of W e . ∂f i ∂s j (|x|, U(x)) + ∂f i ∂s j (|x|, U σ(e) (x)) ψ i ψ j dx Then there exists a direction e ∈ S k−1 such that Q e,s (Ψ; Ω(e)) ≥ 0 ∀ Ψ ∈ C 1 c (Ω(e); R m ) Equivalently the first symmetric eigenvalue λ s 1 (L e,s , Ω(e)) of the operator L e,s (V ) = −∆V + B e,s V in Ω(e) is nonnegative (and hence also the principal eigenvalueλ 1 (L e,s , Ω(e)) is nonnegative).
Proof. Let us assume that 1 ≤ j = m(U) ≤ k − 1 and let Φ 1 , . . . , Φ j be mutually orthogonal eigenfunctions corresponding to the negative symmetric eigenvalues λ s 1 (L U , Ω), . . . , λ s j (L U , Ω) of the linearized operator L U (V ) = −∆V − J F (x, U)V in Ω . For any e ∈ S k−1 let φ e,s be the first positive L 2 -normalized eigenfunction of the symmetric system associated to the linear operator L e,s in Ω(e). We observe that φ e,s is uniquely determined since the corresponding system is fully coupled in Ω(e). Let Φ e,s be the odd extension of Φ e,s to Ω, and let us observe that Φ −e,s = −Φ e,s , because B e,s is symmetric with respect to the reflection σ e . The mapping e → Φ e,s is a continuous odd function from S k−1 to H 1 0 (Ω ∪ Γ), therefore the mapping h : S k−1 → R j defined by h(e) = (Φ e,s , Φ 1 ) L 2 (Ω) , . . . , (Φ e,s , Φ j ) L 2 (Ω) is an odd continuous mapping, and since j ≤ k − 1, by the Borsuk-Ulam Theorem it must have a zero. This means that there exists a direction e ∈ S k−1 such that Φ e,s is orthogonal to all the eigenfunctions Φ 1 , . . . , Φ j . This implies that Q U (Φ e,s ; Ω) ≥ 0, because m(U) = j, and since Φ e,s is an odd function, we obtain that 0 ≤ Q U (Φ e,s ; Ω) = Q e,s (Φ e,s , Ω) = 2Q e,s (φ e,s , Ω(e)) = 2λ s 1 (L e,s , Ω(e)) Proof of Theorem 1.2. By Lemma 3.5 there exists a direction e such that the first symmetric eigenvalue λ s 1 (L e,s , Ω(e)) of the operator L e,s (V ) = −∆V +B e,s V in Ω(e) is nonnegative, and hence also the principal eigenvalueλ 1 (L e,s , Ω(e)) is nonnegative.
Since Q e,s ( (W e ) ± ; Ω(e) ) ≤ 0 by Lemma 3.4, we have two alternatives. The first one is that (W e ) + and (W e ) − both vanish, in which case W e ≡ 0 in Ω(e), and this implies in turn that L e,s = L U . Then U is symmetric and the principal eigenvalueλ 1 (L U , Ω(e)) is nonnegative, so that the hypothesis i) of Theorem 3.2 holds and we get that U is foliated Schwarz symmetric. The second alternative is that one among (W e ) + and (W e ) − does not vanish and λ s 1 (L e,s , Ω(e)) = 0. Then either (W e ) + or (W e ) − is a first symmetric eigenfunction of the operator L e,s (V ) in Ω(e). If (W e ) + is a first symmetric eigenfunction of the operator L e,s (V ) = −∆V + B e,s V in Ω(e) then it is positive in Ω(e), i.e. U > U σe in Ω(e). In the case when (W e ) − is the first symmetric eigenfunction we get the reversed inequality. Then, by the sufficient condition ii) given by Theorem 3.2, u is foliated Schwarz symmetric.
The proof in the case when the hypotheses of Theorem 1.2 hold is the same.