A computer assisted proof of the symmetries of least energy nodal solutions on squares
aa r X i v : . [ m a t h . A P ] F e b A COMPUTER ASSISTED PROOF OF THE SYMMETRIES OFLEAST ENERGY NODAL SOLUTIONS ON SQUARES
ARIEL SALORT AND CHRISTOPHE TROESTLER
Abstract.
In this article, we prove that the least energy nodal solutions toLane-Emden equation − ∆ u = | u | p − u with zero Dirichlet or Neumann bound-ary conditions on a square are odd with respect to one diagonal and even withrespect to the other one when p is close to 2. We also show that this symmetrybreaks on rectangles close to squares. Contents
1. Introduction 12. Preliminary results 22.1. Variational formulation 32.2. The limit equation 32.3. Symmetries 43. Reduction to a one-dimensional functional 74. Ingredients of the computer-assisted proof 84.1. Interval arithmetic 84.2. Interval extension of u log | u | Introduction
Given a connected domain Ω ⊆ R N , N > < p < ∗ := NN − (+ ∞ if N = 2), we consider the so-called Lane-Emden problem with Dirichlet boundary Mathematics Subject Classification.
Fondsde la Recherche Fondamentale Collective , Belgium. conditions ( − ∆ u = | u | p − u in Ω ,u = 0 on ∂ Ω . (1.1)In this article we aim on symmetry properties of solutions of (1.1) for values of p close to 2. The literature on this subject is vast. In the early ’70s Ambrosetti andRabinowitz in [2] prove that (1.1) has a positive ground state solution, which, in [8]was shown to inherit all symmetries of convex domains.Later on, in the ’90s, sign-changing solutions with minimum energy (or leastenergy nodal solutions hereafter referred to as l.e.n.s. ) were proved to exists byCastro, Cossio and Neuberger [5]. A natural question is whether l.e.n.s. inherit(some) symmetries of the domain, at least for values of p close to 2. There aremany result on this direction. In [1] Aftalion and Pacella proved that, on a ball, al.e.n.s. cannot be radial moreover, in [3] partial symmetry results were obtained onradial domains. Finally, in [4] symmetries on more general domains were studied,as well as the asymptotic behavior of solutions when p is close to 2.It is worthy to mention that the arguments leading to uniqueness of l.e.n.s. of(1.1) for values of p close to 2, as well as their symmetry properties, strongly dependson the simplicity of λ . Indeed, when λ is simple, uniqueness of the limit solutionfollows by an applying the implicit function theorem . In fact, for p close to 2,l.e.n.s. are unique (up their sign) and possess the same symmetry as eigenfunctionscorresponding to λ . In particular, when a rectangle in the plane is considered, thenodal line is the small median. See [4].However, the approach is different when λ has multiplicity. In this case, forgeneral domains the implicit function theorem cannot be applied (with exceptionof particular configuration, as the case of radial domains, see [3, 4]) and symmetriesof l.e.n.s. for values of p near 2 are less understood. As anticipated some partialresult can be found in [1, 3, 4]. Even, for simple domains as planar squares, manyquestion still remaining open. Particularly, numerical computations carried out in[4] seem to support the following assertion: Conjecture 1.1. If Ω ⊆ R is a square and p is close to , l.e.n.s. are symmetricwith respect to the diagonal and antisymmetric in the orthogonal direction. An argument leading to a theoretical proof of that fact seems not to be clear.The main attention of this article will be focused in proving this conjecture by usinga computer-assisted proof approach.The manuscript is organized as follows. In section 2 we introduce some prelimi-nary results in this context. In Section 3 we sketch the main lines of our arguments.Section 4 is devoted to describe and implement our computer-assisted proof. Finallyin Section 5 we present the main code and some numerical experiments.2.
Preliminary results
In this section we introduce some notations and preliminary results. We firstrecall a variational formulation for sign-changing solutions of (1.1) as well as thelimit equation they fulfill as p approaches 2. Finally we present some abstractresults on symmetries of solutions for values of p close to 2. COMPUTER ASSISTED PROOF OF THE SYMMETRIES OF L.E.N.S. 3
Variational formulation.
In order to study solutions of (1.1), and to avoidfurther normalizations, we consider the following equivalent problem ( − ∆ u = λ | u | p − u in Ω ,u = 0 on ∂ Ω , ( P p )where λ is the second eigenvalue of − ∆ with Dirichlet boundary conditions. Wedenote E the eigenspace related to λ . Weak solutions to ( P p ) are critical pointsof the energy functional J p : H (Ω) → R given by J p ( u ) = 12 Z Ω |∇ u | d x − λ p Z Ω | u | p d x. This functional has Fr´echet derivative given by ∀ v ∈ H (Ω) , J ′ p ( u )[ v ] = Z Ω ∇ u · ∇ v d x − λ Z Ω | u | p − uv d x. We remark that from regularity theory arguments, it can be proved that weaksolutions to ( P p ) are C (Ω) ∩ C (Ω) and hence classical solutions. Moreover, it isstraightforward to see that symmetry properties of solutions to (1.1) and ( P p ) arethe same since u solves (1.1) if and only if λ / (2 − p )2 u solves ( P p ).Clearly the zero function solves ( P p ). In order to obtain non-zero sign changingsolutions we introduce the Nehari manifold N p := (cid:8) u ∈ H (Ω) \ { } : J ′ p ( u )[ u ] = 0 (cid:9) and the nodal Nehari set M p := (cid:8) u ∈ H (Ω) : u ± ∈ N p (cid:9) where as usual u ± := max {± u, } . Given a function u ∈ H (Ω) with u ± = 0, itholds that u ∈ M p if and only if R Ω |∇ u ± | d x = λ R Ω | u ± | p d x .It is worthy to mention that M p contains all sign-changing critical points of J p .If u minimizes J p on this manifold, then u is a nodal solution of ( P p ) usually referredas the least energy nodal solution . The set of solutions is not empty as showed in [5]:there exists at least a l.e.n.s. of ( P p ) having exactly two nodal domains.In particular, we will be interested in the behavior of l.e.n.s. of ( P p ) as p tendsto 2 and the limit equation they satisfy.2.2. The limit equation.
Let us analyze the behavior of solutions of ( P p ) as p →
2. We remark that any family ( u p ) p> of l.e.n.s. of ( P p ) can be proved tobe bounded in H (Ω) and bounded away from 0 (see [4, Lemma 4.1 and Lemma4.4]), which implies that any weak accumulation point u ∗ of ( u p ) p> verifies a limitequation [4, Theorem 4.5]: if u p n ⇀ u ∗ weakly in H (Ω) as p n →
2, then u p n → u ∗ in H (Ω) \ { } and u ∗ fulfills ( − ∆ u ∗ = λ u ∗ in Ω ,u ∗ = 0 on ∂ Ω . (2.1)Moreover, u ∗ minimizes the reduced functional J ∗ over the reduced Nehari manifold N ∗ , where J ∗ : E → R : u
7→ J ∗ ( u ) = λ Z Ω u (1 − log u ) d x, (2.2) N ∗ = { u ∈ E \ { } : J ′∗ ( u )[ u ] = 0 } , SALORT AND TROESTLER and t log t is extended continuously by 0 at t = 0. It is easy to see that(2.3) ∀ v ∈ E , J ′∗ ( u )[ v ] = − λ Z Ω vu log | u | d x. Any nontrivial critical point of J ∗ again belongs to N ∗ . This manifold is compactand such that u ∈ N ∗ if and only if u = 0 and Z Ω u log | u | d x = 0 or equivalently iff J ∗ ( u ) = λ Z Ω u d x. Let us now show that u ∗ satisfies an equivalent minimization problem. This problemwill be the one used in our computer assisted proof. First, by standard argumentsof critical point theory, it follows that(2.4) inf u ∈N ∗ J ∗ ( u ) = inf u ∈ S max t> J ∗ ( tu ) , where S is the L -unit sphere of E . Observe that, for any u ∈ E \ { } , the ray { tu : t > } intersects N ∗ at a single point t ∗ u u , where t ∗ u is given by(2.5) t ∗ u = exp (cid:18) − R Ω u log | u | d x R Ω u d x (cid:19) . In view of the above expression, given u ∈ S , we get J ∗ ( t ∗ u u ) = λ Z Ω ( t ∗ u u ) d x = λ − R Ω u log | u | d x and consequently, since the map t e t is increasing, (2.4) is equivalent to(2.6) minimize S ∗ ( u ) := − Z Ω u log | u | d x on the L -sphere S .This function can be seen as the entropy associated to the density | u | , and accu-mulation points of the minimal energy nodal solutions are multiples of the eigen-functions with minimal entropy.Note that for any u ∈ S and r > S ∗ ( ru ) = r S ∗ ( u ) − r log r so, instead of (2.6), one can equivalently minimize S ∗ on the sphere r S of radius r .Now that the limit functions of l.e.n.s. sequences is characterized, our followingconcern is to identify their symmetries.2.3. Symmetries.
Uniqueness and symmetry of l.e.n.s. of ( P p ) for p close to 2 arethe main subjects in this article. We recall some results that can be found in theliterature. When studying symmetries there are two possibles scenarios dependingof the simplicity of λ .When dim E = 1, that is, the non-degenerate case , by using the implicit functiontheorem , it follows that ( P p ) has unique solution up its sign for p ≈
2. If we consideran eigenfunction u of λ normalized such that k∇ u k L (Ω) = 1, it is proved in [4,Theorem 2] that for values of p close to 2 and any reflection R such that R (Ω) = Ω,the l.e.n.s. of ( P p ) possesses the same symmetry or antisymmetry as u with respectto R . For instance, when Ω is a rectangle, since λ is simple, l.e.n.s. to ( P p ), with p ≈
2, are odd with respect to the small median of the rectangle and so their nodalline is that median.When λ is not simple, the situation becomes more delicate. For some par-ticular shapes of the domain Ω, the implicit theorem method can be applied in COMPUTER ASSISTED PROOF OF THE SYMMETRIES OF L.E.N.S. 5 order to prove uniqueness of l.e.n.s. as p →
2. When Ω is a ball, uniqueness canbe guaranteed up to the action of the rotation group because it can generate E from a one-dimensional subspace. However, for general domains, that method can-not be applied and must be replaced by a Lyapunov-Schmidt-type reduction (seeTheorem 2.2).Regarding symmetries, in the particular case of the ball, exploiting the struc-ture of the zeroes of second eigenfunctions, some results about symmetry can beobtained. In [3], it is proved that for p close to 2, l.e.n.s. of ( P p ) are radially sym-metric with respect to N − p close to 2, l.e.n.s. are anti-symmetric with respect to the orthogonaldirection. In particular the nodal line is the diameter.For general domains, symmetries of l.e.n.s. to ( P p ) for p close to 2 are addressedby the following abstract result [4, Theorem 3.6]. Consider a family of groups( G α ) α ∈ E acting on H (Ω) in a such way that, for every α ∈ E , g ∈ G α , p closeto 2, and u ∈ H (Ω), the following holds:(i) g ( E ) = E , (ii) g ( E ⊥ ) = E ⊥ , (iii) gα = α, (iv) J p ( gu ) = J p ( u ) . Then, for all
M >
0, if p is close to 2, any l.e.n.s. u p ∈ { u ∈ B M : P E ( u ) / ∈ B /M } of ( P p ) is invariant under the isotropy group G α p = { g ∈ G : gα p = α p } of α p = P E u p , the orthogonal projection of u p on E .As an example, if G α describes the symmetry (or antisymmetry) of α , for p close to 2, u p possesses the same symmetries as its orthogonal projection α p . Forinstance, when Ω is a square, for p close to 2, l.e.n.s. of ( P p ) are odd with respectto the center of the square.Moreover, as we anticipated, in [4, Conjecture 5.4] numerical computations sug-gest that the nodal line of u ∗ is a diagonal of the square, u ∗ is antisymmetric withrespect to the diagonal and symmetric with respect to the other diagonal. It seemsnatural that these symmetries extend to u p for p close to 2. This is the content ofour main result. Theorem 2.1. If Ω ⊆ R is a square and p is close to , l.e.n.s. are symmetricwith respect to a diagonal and antisymmetric in the orthogonal direction. Unfortunately, Theorem 3.6 of [4] is not powerful enough for this purpose. In-deed, the computer assisted proof below will characterize the symmetries of u ∗ ∈ N ∗ but that does not readily implies that P E u p = u ∗ , so it is not clear that u p enjoysthe same symmetries as u ∗ . This rest of this section is devoted to establish thatthe symmetries of u ∗ are inherited by u p for p ≈ Theorem 2.2 (Lyapunov-Schmidt reduction) . Assume the functional J ∗ definedby (2.2) is of class C on E \ { } and ( p, u )
7→ J ′′ p ( u ) is continuous on { } × ( E \ { } ) . For any non-degenerate critical point u ∗ ∈ E \ { } of J ∗ , there existsa neighborhood U ∗ of u ∗ in H (Ω) and a continuous curve γ : [2 , ε [ → H (Ω) , ε > , such that γ (2) = u ∗ and ∀ p ∈ ]2 , ε [ , ∀ u ∈ U ∗ , u solves ( P p ) ⇐⇒ u = γ ( p ) SALORT AND TROESTLER
Proof.
Let us split any u ∈ H (Ω) as v + w where v ∈ E and w ∈ E ⊥ . Problem ( P p )is equivalent to finding ( v, w ) ∈ E × E ⊥ such that ( G ( p, v, w ) := J ′ p ( v + w ) (cid:12)(cid:12) E ⊥ = 0 , J ′ p ( v + w ) (cid:12)(cid:12) E = 0 . Using Lebesgue’s Dominated Convergence Theorem and the following inequality (cid:12)(cid:12) | u | p − log | u | (cid:12)(cid:12) C δ max { , | u | p − δ } (where C δ does not depend on p ∈ [2 , ∗ ]nor u ∈ R ), one proves that [2 , ∗ [ × H (Ω) → ( H (Ω)) ′ : ( p, u )
7→ J ′ p ( u ) is ofclass C and so G ∈ C . Note that G (2 , v,
0) = 0 for any v ∈ E and the map ∂ w G (2 , v,
0) : E ⊥ → ( E ⊥ ) ′ is given by ∂ w G (2 , v, w ] = w Z Ω ∇ w ∇ w − λ Z Ω w w . Clearly ∂ w G (2 , v,
0) is a compact perturbation of an invertible linear map, hencea Fredholm operator of index 0. Moreover, it is easy to see that it is one-to-one. Therefore, it is invertible and the Implicit Function Theorem implies thatthere exists ε > ρ > W a neighborhood of 0 in E ⊥ , and a function ω :[2 , ε [ × B ρ ( u ∗ ) → W of class C , where B ρ ( u ∗ ) is the open ball of center u ∗ andradius ρ in E , such that ∀ p ∈ [2 , ε [ , ∀ v ∈ B ρ ( u ∗ ) , ∀ w ∈ W, G ( p, v, w ) = 0 ⇔ w = ω ( p, v ) . Let
H ∈ C (cid:0) [2 , ε [ × B ρ ( u ∗ ); E ′ (cid:1) be defined by H ( p, v ) := J ′ p (cid:0) v + ω ( p, v ) (cid:1)(cid:12)(cid:12) E .Clearly, for all v ∈ B ρ ( u ∗ ), ω (2 , v ) = 0 and so H (2 , v ) = 0. Let us define K ( p, v ) := H ( p, v ) p − p > ,∂ p H (2 , v ) if p = 2 . Using the mean value theorem and the fact that ( p, u )
7→ J ′ p ( u ) is C , it is easy tosee that K is continuous. A direct computation shows that ∂ p H (2 , v ) = J ′∗ ( v ) + J ′′ ( v )[ ∂ p ω (2 , v ) , · ] (cid:12)(cid:12) E = J ′∗ ( v )where the last equality results from the fact that J ′′ ( v )[ v , v ] = R ∇ v ∇ v − λ R v v = 0 when v ∈ E . As J ∗ is assumed to be C on E \ { } , ∂ v K existsand is continuous on [2 , ε [ × B ρ ( u ∗ ) (taking ρ smaller if necessary to ensure0 / ∈ B ρ ( u ∗ )). Moreover, as u ∗ is non-degenerate, ∂ v K (2 , u ∗ ) is invertible. TheImplicit Function Theorem thus guarantees that there is a continuous function˜ γ : [2 , ε [ → B ρ ( u ∗ ) (taking ε and ρ smaller if needed) such that ∀ p ∈ [2 , ε [ , ∀ v ∈ B ρ ( u ∗ ) , K ( p, v ) = 0 ⇔ v = ˜ γ ( p ) . Taking U ∗ := B ρ ( u ∗ ) × W and γ ( p ) = ˜ γ ( p ) + ω ( p, ˜ γ ( p )) concludes the proof. (cid:3) Proposition 2.3.
Under the same assumptions as in Theorem 2.2, if u ∗ is aminimum of J ∗ over N ∗ and gu ∗ = g ∗ (where gu ( x ) := u ( g − x ) ) for some g ∈ O ( N ) such that g Ω = Ω and J p ( gu ) = J p ( u ) for all u ∈ H (Ω) and p , then, for p closeto , gu p = u p .Proof. If the conclusion does not hold, there exists a sequence ( u p n ) of l.e.n.s.to ( P p n ) such that ∀ n, gu p n = u p n . Going if necessary to a subsequence, u p n ⇀u ∗ ∈ E and thus u p n → u ∗ [4, Theorem 4.5]. In particular, for n large enough,( p n , u p n ) ∈ [2 , ε [ × U ∗ and so u p n = γ ( p n ). But gu p n is also a solution to ( P p n ) COMPUTER ASSISTED PROOF OF THE SYMMETRIES OF L.E.N.S. 7 and gu p n → gu ∗ = u ∗ . Therefore, for n large enough, gu p n = γ ( p n ) = u p n which isa contradiction. (cid:3) Reduction to a one-dimensional functional
We sketch our strategy to prove Theorem 2.1. As we remarked before, a sequence( u p ) p> of l.e.n.s. of ( P p ) converges weakly in H (Ω) as p n → u ∗ verifying the limit equation (2.1) and solves the minimization entropy problem (2.6).Let Ω ⊆ R be a square. Without loss of generality we can consider Ω = ]0 , .In this case, a basis of eigenfunctions of E is given by(3.1) ϕ ( x, y ) = sin( πx ) sin(2 πy ) , ϕ ( x, y ) = sin(2 πx ) sin( πy ) . Moreover, it easy to check that these functions are orthogonal in L (Ω), which allowus to parametrize S as S = (cid:8) u ∈ E : k u k L (Ω) = (cid:9) = (cid:8) u θ ( x, y ) := ϕ ( x, y ) cos θ − ϕ ( x, y ) sin θ : θ ∈ [0 , π [ (cid:9) . (3.2)Hence, instead of (2.6)–(2.7), we can restrict ourselves to study the minimizationproblem(3.3) min θ ∈ [0 , π ] S ∗ ( u θ ) . Since u θ + π/ ( x, y ) = u θ ( y, − x ), u π/ − θ ( x, y ) = u π/ θ ( y, x ) and S ∗ is invariantunder the group of symmetries of the square (the dihedral group of order 8), wehave that the function g ( θ ) : [0 , π ] → R given by(3.4) g ( θ ) := S ∗ ( u θ ) = − Z Ω f (cid:0) u θ ( x, y ) (cid:1) d x d y, with f ( t ) = ( t log | t | if t = 0 , t = 0 , is π -periodic and g ( π − θ ) = g ( π + θ ) (see Fig. 1). θ π/ π/ π π/ π . . S ∗ ( u θ ) Figure 1.
Graph of θ
7→ S ∗ ( u θ ). Observe that u π/ ∈ S is even with respect to a diagonal and odd with respect tothe second one. Because of the symmetries of g , it is clear that θ = π/ g . If we prove that the minimal energy in (3.3) is achieved at θ = π/ J ∗ is non-degenerate, Theorem 2.3 willassert that l.e.n.s. to ( P p ) would enjoy the symmetries of Conjecture 1.1. SALORT AND TROESTLER Ingredients of the computer-assisted proof
In this section we prove that the function θ g ( θ ) reaches its minimum at θ = π . Throughout this section Ω stands for the square ]0 , .The arguments in the proof can be sketched roughly as follows. We designan adaptive integration method integ which allows to integrate a function of twovariables on Ω. By applying it to ( x, y ) f ( u θ ( x, y )), we obtain a function entropy which estimates g . With these tools, our task is reduced to discard subintervals of[0 , π ] which are guaranteed not to contain the minimum of g . This is performedby the function exclude . The outcome is a small interval around π/ g in orderto guarantee that it contains a unique critical point. Consequently π/ machine numbers , to guarantee rigorous re-sults when performing computations, instead of considering values , we will consider intervals containing the true values. In the next subsection, for the reader’s con-venience, we recall some basic facts about computing with floating numbers andinterval arithmetic.4.1. Interval arithmetic.
When implementations are made in digital electronicdevices, due to the limited space and speed, only a finite subset of real numbersis available. For the present considerations, we will use the set of double precisionfloating point numbers that we will denote F . Elementary arithmetic operations(+, − , · , / ) do not necessarily return an element in F even when their operandsare in F . The standard IEEE-754 mandates that their implementation returns avalue in F closest to the exact result. This yields small rounding errors whichmay propagate badly along the computations. A standard example is to evaluate f ( x, y ) = 333 . y + x (11 x y − y − y −
2) + 5 . y on ( x, y ) = (77617 , f and x , y belong to F , when all computationsare performed with double precision , the result is − . ... · while the correctvalue is −
2. Rounding errors may also affect the constants in the program: forexample 0 . / ∈ F (because it has an infinite binary expansion) and π / ∈ F arerounded in computer memory. Errors also come from the necessary approximation(think of the truncation of an infinite sum) performed by the algorithm computingfunctions such as sin, cos, log,...In order to account for all these errors along computations performed on a com-puter, values a ∈ R will be represented by an interval[ a ] = [ a, a ] = { x ∈ R : a x a } (where [ a ] must read as a single symbol) such that a ∈ [ a ]. The set of intervals withendpoints in R will be denoted IR . The width of [ a ] is a − a and denoted width([ a ]).We also define low([ a, a ]) := a and high([ a, a ]) := a . Vectors a ∈ R N will berepresented as [ a ] = (cid:0) [ a ] , . . . , [ a N ]) ∈ ( IR ) N standing for the box [ a ] × · · · × [ a N ]. COMPUTER ASSISTED PROOF OF THE SYMMETRIES OF L.E.N.S. 9
Elementary arithmetic operations (+, − , · , / ) and in general any function f : R N → R M must be extended to an interval function [ f ] : ( IR ) N → ( IR ) M so thatthe following containment property holds: ∀ [ a ] ∈ Dom[ f ] , f ([ a ]) = (cid:8) f ( x ) : x ∈ [ a ] ∩ Dom f (cid:9) ⊆ [ f ] (cid:0) [ a ] (cid:1) . This means that all exact values f ( x ) for x ranging in the interval [ a ] are containedin the interval returned by the function [ f ] on the operand [ a ]. In this case, we willcall [ f ] an interval extension of f . Composition of interval extensions are againinterval extensions.When implemented on a computer, interval extensions of f : R N → R M willnaturally be functions [ f ] defined on (a subset of) ( IF ) N and returning valuesin ( IF ) M where IF denotes the set of intervals with endpoints in F . Roundingerrors of arithmetic operators will be handled by using directed rounding modes ofthe processor: for example, the interval extension of the addition if [ a ] ∈ IF and[ b ] ∈ IF will be given by (cid:2) ↓ ( a + b ) , ↑ ( a + b ) (cid:3) where ↓ ( . . . ) (resp. ↑ ( . . . )) means thatthe outcome of operations within the braces if rounded towards −∞ (resp. + ∞ ).For approximations made to evaluate functions, a theoretical error bound must bederived in terms of computable quantities which will then be used to provide an(over)estimation of the error on the computer.We refer the reader interested in more details about interval analysis to [9, 10].4.2. Interval extension of u log | u | . One of the basic functions to implementfor our task is u u log | u | . The naive extension is [ u ] [ u ] · [log] (cid:12)(cid:12) [ u ] (cid:12)(cid:12) butit is unsuitable. Indeed if 0 ∈ [ u ], the logarithm interval extension will returnan unbounded interval (of the form [ −∞ , a ], which is possible because ±∞ ∈ F )and so will subsequent computations. Thus a more precise interval extension ofthis function is necessary. Because this function is even, one can suppose w.l.o.g.that [ u ] ⊆ [0 , + ∞ ]. In order to derive interval bounds, one has to distinguishsub-intervals based on the monotonicity of u u log u (which is decreasing on[0 , e − / ] and increasing after) and the sign of log u . Thus one has first to computean interval estimate [ m, m ] = [exp] (cid:0) [ − . , − . (cid:1) of the location of the minimum.Then, for an interval of the form [0 , u ] with 0 < u < m , the algorithm will returnthe interval [ ↓ ( ↑ ( u ) · log u ) ,
0] (one has to round up the square because log u < u ]. This more clever interval extensionreturns tight bounds for the function and, in particular, returns bounded intervalseven when 0 ∈ [ u ] (see Fig. 2). Naive −∞ Tight bounds − . . . Figure 2.
Evaluation of u u log | u | . Integration.
In order to compute S ∗ , an integration on the square Ω must beperformed. There is a vast literature about integration methods, both for the one-variable and multivariate cases, including for interval arithmetic (see for example [7,11, 11] and the references therein). For accurate evaluation of integrals in highdimensions, it is recommended to avoid tensor products of one-dimensional rules(because they require a number of function evaluations that is exponential in thedimension) and instead turn to rules such as the one devised by Smolyak [12]. Inthis paper however, the dimension is small and the fact that error bounds of tensorproduct formula do not depend on mixed derivatives is interesting. Our integrandis also not smooth everywhere.In this subsection, we briefly describe the adaptive verified integration schemethat we are using. Given an interval extension [ f ] of a function f : Ω → R , we wantto return an interval [ I Ω ] such that [ I Ω ] ∋ I Ω := 1 | Ω | Z Ω f ( x, y ) d x d y. To compute [ I Ω ], a rule that evaluates [ f ] at various points and estimates the errorin terms of the variation (on Ω) of f and its derivatives is used. Such a rule returnsan interval [ I ]. If we are happy with the width of [ I ], we take [ I Ω ] = [ I ] andstop. If not, we split Ω into four squares of equal sizes Ω = Ω ∪ Ω ∪ Ω ∪ Ω andrecursively apply this procedure to have [ I Ω i ], i = 1 , . . . ,
4. Then [ I Ω ] is defined by(4.1) [ I Ω ] := 14 X i =1 [ I Ω i ] ∋ X i =1 | Ω i || Ω | I Ω i = I Ω . Since (4.1) implies width([ I Ω ]) max i =1 ,..., width([ I Ω i ]), a natural stopping cri-teria for the recursion is that width([ I Ω i ]) ε where ε is a desired tolerance.4.3.1. Basic integration rule.
The simplest way to estimate I Ω is to use the follow-ing simple fact (related to the mean value theorem for integrals): if f is integrableand ∀ ξ ∈ Ω , m f ( ξ ) m , then1 | Ω | Z Ω f ( ξ ) d ξ ∈ [ m, m ] . If Ω ⊆ [ x ] × [ y ], this is in particular true for [ m, m ] = [ f ]([ x ] , [ y ]). Thus, one takes[ I ] := [ f ] (cid:0) [ x ] , [ y ] (cid:1) . The associated adaptive procedure does not perform very well however. Indeed,we expect the width of [ f ] (cid:0) [ x ] , [ y ] (cid:1) to be of size k∇ f k L ∞ (Ω) diam Ω where diam Ωis the diameter of Ω. Thus, if h denotes the diameter of the small squares obtainedat depth d of the recursion, for f ∈ C (Ω), width([ I Ω ]) = O ( h ). In terms of thenumber n of points at which the function needs to be evaluated and in terms of thedepth d of the recursion, this gives:(4.2) width([ I Ω ]) = O ( h ) = O (cid:16) √ n (cid:17) = O (cid:16) d (cid:17) . This is quite slow. For example, integrating the analytic function ϕ (definedin (3.1)) using a recursion depth of 14 must perform at least 268 468 225 functionevaluations for a final precision of about 10 − . This is pictured in Fig. 3. Clearly, We choose to compute the integral mean because the weights in the sum approximating theintegral are independent of the size of Ω.
COMPUTER ASSISTED PROOF OF THE SYMMETRIES OF L.E.N.S. 11 this is not practical as the procedure to determine a small neighborhood of theminumum requires many evaluations of S ∗ with a good precision. n width 1 10 10 − − − − − B a s i c r u l e M i dp o i n t r u l e S i m p s o n ’ s r u l e Figure 3.
Errors of adaptive integration rules.
Midpoint rule.
A simple rule with error behaving as O ( h ) is obtained bytaking the tensor product of two one-dimensional midpoint rules. It is well known(see e.g., [6, p. 54]) that the error of the one-dimensional midpoint rule for f ∈ C ([ a, a + h ]; R ) is given by1 h Z a + ha f ( x ) d x − f ( a + h ) = 124 h f ′′ ( ξ ) for some ξ ∈ ] a, a + h [ . Using Fubini’s theorem, it is not difficult to derive [6, Section 5.6] the followingerror bound for integrating a function f ∈ C ( P ; R ) over a rectangle P = [ a , a + h ] × [ a , a + h ]:(4.3) (cid:12)(cid:12)(cid:12)(cid:12) | P | Z P f ( x, y ) d( x, y ) − f (cid:0) a + h , a + h (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0) h k ∂ x f k ∞ + h k ∂ y f k ∞ (cid:1) where the L ∞ -norms are taken over P . Thus1 | P | Z P f ∈ [ I P ] := [ f ] (cid:0) [ x mid ] , [ y mid ] (cid:1) + [ e ]where [ x mid ] (resp. [ y mid ]) is an interval containing a + h (resp. a + h ) and[ e ] := [ − e, e ] with e being the right hand side of (4.3) where the derivatives boundsare obtained using interval arithmetic and all arithmetic operations are rounded up.In view of (4.3) and using the same notations as for (4.2), one has for this rulethat width[ I P ] = O ( h ) = O (cid:16) n (cid:17) = O (cid:16) d (cid:17) . The convergence is twice faster than the one of the basic rule. In particular, achiev-ing a precision of 10 − is no longer prohibitive (see Fig. 3 for a graphical illustrationof that fact for the function ϕ ).4.3.3. Simpson’s rule.
To locate the minimum of (3.4) reasonably fast, we evaluate S ∗ ( u θ ) at θ = π/ − . A higher order integration rule istherefore desirable. The one we use is the tensor product of two Simpson’s rules.Recall that the error of the one-dimensional Simpson’s rule applied to a function f ∈ C ([ a, a + h ]; R ) is given by error [6, p. 288]:1 h Z a + ha f ( x ) d x − (cid:0) f ( a ) + 4 f ( a + h ) + f ( a + h ) (cid:1) = − h f (4) ( ξ ) for some ξ ∈ ] a, a + h [. As above, this yields a two-dimensional rule on a rectangle P :Simpson( f ) = 136 (cid:0) f + f + f + f + 4( f + f + f + f ) + 16 f (cid:1) where f ij := f ( a + ih / , a + jh / (cid:12)(cid:12)(cid:12)(cid:12) | P | Z P f − Simpson( f ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:0) h k ∂ x f k ∞ + h k ∂ y f k ∞ (cid:1) . From this, an interval version is easily derived with convergence speed width[ I P ] = O ( h ) = O (cid:0) n (cid:1) = O (cid:0) d (cid:1) .4.3.4. Integrating a non-smooth function.
For the purpose of integrating u θ log | u θ | ,the midpoint or Simpson’s rules cannot be used over the whole Ω. Indeed, whenever u θ = 0, the second and fourth derivatives of the function are unbounded. Thus,the following algorithm to estimate | Ω | R Ω f ( x, y ) d( x, y ) is used (illustrated for themidpoint rule, a similar one is used for the Simpson’s rule). mean integ( [ f ] , [ ∂ f ] , [ x left ] , [ x right ] , [ y left ] , [ y right ] , d, tol) Evaluate the derivatives [ ∂ xx f ], [ ∂ xy f ] and [ ∂ yy f ] on the box [ x ] × [ y ] := (cid:2) low([ x left ]) , high([ x right ]) (cid:3) × (cid:2) low([ y left ]) , high([ y right ]) (cid:3) and compute e using the right hand side of (4.3), rounding all operations towards + ∞ .let [ x mid ] (resp. [ y mid ]) be the middle points of [ x left ] and [ x right ] (resp.[ y left ] and [ y right ]).If e tol , then let [ I ] be defined by [ f ] (cid:0) [ x mid ] , [ y mid ] (cid:1) + [ − e, e ],otherwise set [ I ] := [ f ] (cid:0) [ x ] , [ y ] (cid:1) .If d ∨ e tol , then return [ I ];else, call recursively mean integ on the four sub-squares (see Fig. 4)with d := d −
1. This will return [ I ] , . . . , [ I ]. Finally, in accordancewith (4.1), return (cid:0) [ I ] + · · · + [ I ] (cid:1) / tol is used as an indicationof when to stop the recursion, the final accuracy being actually given by the widthof the returned interval. Fig. 5 depicts how this algorithm subdivides the square[0 , to compute R [0 , u θ log | u θ | with θ = π/ .
05 withdepth d = 5. A sub-square is colored whenever the higher order formula was usedon it.The function integ( [ f ] , [ ∂ f ] , d, tol) mentioned at the beginning of section 4 issimply defined as mean integ( [ f ] , [ ∂ f ] , [0 , , [1 , , [0 , , [1 , , d, tol) .[ x left ][ y left ] [ x mid ][ y mid ] [ x right ][ y right ] Figure 4.
Square subdivisions.
Figure 5.
Using the appropriate integrationrule with midpoint rule (left) and Simpson’srule (right).
COMPUTER ASSISTED PROOF OF THE SYMMETRIES OF L.E.N.S. 13
Now that verified integration is available, it is easy to define entropy( θ, d, tol) that estimates g ( θ ). Indeed it suffices to pass the function ( x, y )
7→ − ( u θ ( x, y )) · log | u θ ( x, y ) | and its second or fourth derivatives (which are straightforward to com-pute as everything is explicit) to integ .4.4. Locating the minimum.
As we described before, our task is reduced to min-imize the entropy function S ∗ ( u θ ) on the sphere S . Due to symmetry considerations,it is enough to prove that the function g ( θ ) = − R Ω u θ log | u θ | d x d y defined on [0 , π ]attains its minimum at θ = π , where u θ is defined in (3.2). The symmetries alsoimply that π is a critical point of g .In this section, we determine a “small” interval around π/ m, m ] :=[ g ] (cid:0) [ π ] / (cid:1) ∈ IF to have bounds on the value g ( π/ π ] ∈ IF denotes a smallinterval containing π . Then, starting from the left of [0 , π/ x ] is discarded if the lower bound of [ g ] (cid:0) [ x ] (cid:1) (which contains all g ( ξ ) for ξ ∈ [ x ]) isgreater than m . This simple idea leads to the following algorithm. exlude( [ g ] , x, x, step , n )if n then return [ x, x ] elselet x next = min { x + step , x } inlet [ y, y ] = [ g ] (cid:0) [ x, x ] (cid:1) inif y > m then return exclude( [ g ] , x next , x, step , n − )else return exclude( [ g ] , x, x, step , n − ) Remark that n is the number of iterations to be performed. In practice, the depth(and hence the precision) with which the integral in [ g ] is computed is graduallyincreased as x gets closer to π/
4. This function is called with x = 0 and x =high([ π ] /
4) to obtain the location of the minimum. This is illustrated on Fig. 6 for n = 80 where the gray rectangles are [ x ] × [ g ] (cid:0) [ x ] (cid:1) with [ x ] excluded. g ( θ ) = S ∗ ( u θ ) θ π . . Figure 6.
The exclusion procedure for locating the minimum of g ( θ ). Analysis of concavity.
The final step is to analyze the concavity of g in theinterval I returned by exlude to guarantee that θ = π is the unique critical pointin I , hence the only minimum. Proposition 4.1.
The functional J ∗ : E → R defined by (2.2) is of class C on E \ { } and its second derivative ∂ u J ∗ at u ∈ E \ { } is the bilinear map given by (4.5) ∂ u J ∗ ( u )[ u , u ] = − λ Z Ω u u (1 + log | u | ) d x d y. The function g : R → R defined by (3.4) with u θ being given in (3.2) is C and (4.6) g ′′ ( θ ) = − (cid:18)
14 + g ( θ ) + Z Ω | u ′ θ | log | u θ | d x d y (cid:19) where ′ = ∂∂θ . Proof. 1.
The first derivative of ∂ u J ∗ exists and is given by (2.3). It is sufficientto show that the map ( t, θ ) ∂ u J ∗ ( tu θ ) is C . Indeed, first notice that S :]0 , + ∞ [ × R → E \ { } : ( t, θ ) tu θ is surjective. Second, one readily shows that,for all ( t, θ ), the differential ∂ ( t,θ ) S ( t, θ ) is an one-to-one linear map so the inversefunction theorem says that S is locally invertible as a C -map which proves thedifferentiability of ∂ u J ∗ on E \{ } . Since ∂ u J ∗ ( tu )[ v ] = t ∂ u J ∗ ( u )[ v ] − λ t log t R uv ,the derivative w.r.t. t clearly exists and is continuous. It remains to show that thederivative w.r.t. θ exists, is given by(4.7) ∂ θ (cid:0) ∂ u J ∗ ( tu θ ) (cid:1) [ v ] = − λ Z Ω v tu ′ θ (1 + log | tu θ | ) d x d y and is continuous. Given the above formula for ∂ u J ∗ ( tu ), it is enough to prove (4.7)for t = 1. From the form of both partial derivatives as well as ∂ ( t,θ ) S , one readilydeduces (4.5). Remark that the function f : R → R defined in (3.4) is C and u θ and u ′ θ are bounded functions of ( x, y ). Thus Lebesgue dominated convergence Theoremimplies that g is continuously differentiable and g ′ ( θ ) = − Z Ω ∂ t f ( u θ ) u ′ θ d x d y = − Z Ω u ′ θ u θ (1 + 2 log | u θ | ) d x d y. A direct computation shows that R Ω u ′ θ u θ = 0, so g ′ may be rewritten as(4.8) g ′ ( θ ) = − Z Ω u ′ θ u θ log | u θ | d x d y. Notice this is essentially ∂ u J ∗ ( u θ )[ u ′ θ ] so, if we establish (4.7), (4.6) will easily followby the chain rule and the facts: u ′′ θ = − u θ and k u ′ θ k L (Ω) = 1 / We now establish (4.7) for t = 1. Using the mean value Theorem, the differentialquotient may be written as ∂ u J ∗ ( u θ + ε )[ v ] − ∂ u J ∗ ( u θ )[ v ] ε = − λ Z Ω vu ′ θ ε (1 + log | u θ ε | ) d x d y for some θ ε ∈ ] θ, θ + ε [, so the theorem will be proved if we show that the map(4.9) θ Z Ω ϕ ( θ, x, y ) d x d y, where ϕ ( θ, x, y ) := v u ′ θ (1 + log | u θ | ) , is well defined and continuous for all v ∈ E . To prove this, the Lebesgue dominatedconvergence Theorem cannot be applied directly because, as θ varies, the nodal lines { ( x, y ) : u θ ( x, y ) = 0 } cover a non-zero measure set making any dominating functionnon-integrable. COMPUTER ASSISTED PROOF OF THE SYMMETRIES OF L.E.N.S. 15
We will show the continuity of (4.9) for θ ∈ [0 , π/
4] as, thanks to symmetries (seep. 7), the remaining range of θ is very similar. First note that u θ may be writtenas u θ ( x, y ) = z ( x, y ) w θ ( x, y ) with z ( x, y ) := 2 sin( πx ) sin( πy ) and w θ ( x, y ) := cos θ cos( πy ) − sin θ cos( πx ) . The nodal line of u θ is the subset of Ω where w θ vanishes. For θ ∈ [0 , π/ r θ : [0 , → [0 ,
1] : x r θ ( x ) defined implicitly by(4.10) ∀ x ∈ [0 , , cos θ cos( πr θ ( x )) − sin θ cos( πx ) = 0 . Let us split Ω as the union of Ω + and Ω − whereΩ + := { ( x, y ) : x ∈ ]0 ,
1[ and 0 < y r θ ( x ) } , Ω − := { ( x, y ) : x ∈ ]0 ,
1[ and r θ ( x ) < y < } . Thanks to the oddness of u θ and u ′ θ w.r.t. to the center of Ω, we only have toconsider Ω + , the case of Ω − being similar. One has Z Ω + ϕ ( θ, x, y ) d y d x = Z Z r θ ( x )0 ϕ ( θ, x, y ) d y d x = Z Z ϕ ( θ, x, sr θ ( x )) r θ ( x ) d s d x. If we find an integrable function (independent of θ ) dominating ϕ ( θ, x, sr θ ( x )) r θ ( x ),the proof will be done. Note that, because k u ′ θ k ∞ (cid:12)(cid:12) ϕ ( θ, x, sr θ ( x )) r θ ( x ) (cid:12)(cid:12) k v k ∞ (cid:0) (cid:12)(cid:12) log | u θ ( x, sr θ ( x )) | (cid:12)(cid:12)(cid:1) so it suffices to find an integrable function dominating (cid:12)(cid:12) log | u θ ( x, sr θ ( x )) | (cid:12)(cid:12) . Now,since u θ > + and u θ
2, it remains to find a lower bound of u θ whoselogarithm is integrable. Observe that log u θ = log z + log w θ . Since z ( x, y ) > x (1 − x ) y (1 − y ), z in integrable on Ω, whence on Ω + . For w θ , using (4.10), onehas w θ ( x, sr θ ( x )) = cos( θ ) (cid:0) cos( πsr θ ( x )) − cos( πr θ ( x )) (cid:1) = cos( θ ) πr θ ( x ) Z s sin( πσr θ ( x )) d σ > cos( θ ) πr θ ( x ) Z s π σr θ ( x )(1 − σr θ ( x )) d σ (as σr θ ∈ [0 , θ ) π r θ ( x )(1 − s ) (cid:16) s − s + s r θ ( x ) (cid:17) > cos( θ ) π r θ ( x )(1 − s ) (cid:16) s − s + s (cid:17) (as r θ > cos( θ ) π r θ ( x )(1 − s ) (for s ∈ [0 , > √ π min { x , } (1 − s ) (as θ ∈ [0 , π ]).The logarithm of this last bound is easily seen to be an integrable function of( x, s ) ∈ ]0 , , thereby concluding the proof. (cid:3) In view of (4.6), to check the concavity of g , the following condition must beverified:(4.11) g ′′ ( θ ) > ⇔ h ( θ ) > + g ( θ ) , where h ( θ ) := − Z Ω | u ′ θ | log | u θ | d x d y. Because the integrand of h is singular whenever u θ = 0, when evaluated withinterval arithmetic, [log] will return intervals of the form [ −∞ , y ] ∈ IF which willcontaminate the sum computing the integral. The important remark however isthat the singularity helps h to be large and so does not need to be controlled.Interval extensions [ h ] of h will return intervals of the form [ y, + ∞ ] and, to checkcondition (4.11), one will verify that y > high (cid:0) g ]([ θ ]) (cid:1) .4.6. Non-degeneracy.Proposition 4.2. If θ ∗ is a non-degenerate critical point of g , then u ∗ = t ∗ u θ ∗ u θ ∗ ,where t ∗ u > is defined by (2.5) , is a non-degenerate critical point of J ∗ .Proof. Let θ ∗ be a critical point of g . On one hand, by definition (2.5) of t ∗ u θ ∗ , onehas ∂ u J ∗ ( u ∗ )[ u θ ∗ ] = 0. On the other hand, equations (4.8) and R u ′ θ ∗ u θ ∗ = 0 implythat ∂ u J ∗ ( u θ ∗ )[ u ′ θ ∗ ] = 0. Since ∂ u J ∗ ( u θ ∗ ) vanishes in two orthogonal directions, u ∗ is a critical point of J ∗ .In view of proposition 4.1, ∂ u J ∗ ( u ∗ )[ u ∗ , v ] = ∂ u J ∗ ( u ∗ )[ u ∗ , v ] − ∂ u J ∗ ( u ∗ )[ v ] = − λ Z Ω u ∗ v. In particular, ∂ u J ∗ ( u ∗ )[ u ∗ , u ∗ ] = − λ R Ω u ∗ < ∂ u J ∗ ( u ∗ )[ u ∗ , u ′ θ ∗ ] = 0 (recall-ing that R u θ ∗ u ′ θ ∗ = 0). Using again the definition (2.5) of t ∗ u θ ∗ , one obtains ∂ u J ∗ ( u ∗ )[ u ′ θ ∗ , u ′ θ ∗ ] = λ g ′′ ( θ ∗ ) = 0 . As a consequence, the bilinear form ∂ u J ∗ ( u ∗ ) is non-degenerate. (cid:3) Consequently, our main result is a consequence of the following proposition.
Proposition 4.3.
Let
Ω = ]0 , . Then the minimization problem (3.3) is solvedby θ = π/ and the minimum is non-degenerate. Therefore the minimum of J ∗ on N ∗ is achieved by a multiple of u π/ and it is a non-degenerate critical point. Proof: putting it all together
In this section, we finally run the machinery described above on a computerand give the numerical results. We will sometimes write intervals as, for example,[1 . , . ] instead of [1 . , . θ min ] ∋ π where the entropy function (3.4) attainsits minimum, an small interval [ π ] / π/ g ] (cid:0) [ π ] / (cid:1) is estimated with good precision by calling entropy( [ π ] / , depth, tol) with depth = 13 and tol = 10 − . The result is[ g ] (cid:0) [ π ] / (cid:1) = [0 . , . ] . Then exclude with n = 100 is run on [ g ] and outputs the desired interval [ θ min ]:[ θ min ] = [0 . , . ] . Finally, in order to verify that π is the unique critical point in [ θ min ], we analyze thesecond derivative of (3.4) in [ θ min ] by checking condition (4.11). Note that, if thiscondition is fulfilled, our task is done, otherwise we have to refine the parameters COMPUTER ASSISTED PROOF OF THE SYMMETRIES OF L.E.N.S. 17 to obtain a tighter interval [ θ min ]. We evaluate [ h ] by using integ with a depth d = 8 and tolerance tol = 10 − and get:[ h ] (cid:0) [ θ min ] (cid:1) = [0 . , + ∞ ] . Computing the right hand side of (4.11) yields: + [ g ] (cid:0) [ θ min ] (cid:1) = [0 . , . . Clearly, these results show that condition (4.11) is satisfied, thereby proving The-orem 2.1. 6.
Symmetry breaking on rectangles
In this section we prove some results regarding the symmetry breaking of leastenergy nodal solutions on rectangles using a computer-assisted proof approach.6.1.
Perturbations of the Laplacian.
The theoretical results introduced in Sec-tion 2 can be extended to a more general class of operators involving perturbationsof the Laplacian. Indeed, one can consider ( − div( A n ∇ u ) = λ ,n | u | p n − u in Ω ,u = 0 on ∂ Ω , ( P n )where, for all n , A n ∈ C (cid:0) Ω , Sym( N ) (cid:1) , with Sym( N ) denoting the space of symmet-ric N × N matrices, and(6.1) k A n − k ∞ → (cid:13)(cid:13)(cid:13)(cid:13) A n − ε n − A ′ (cid:13)(cid:13)(cid:13)(cid:13) ∞ → ε n ) ⊆ ]0 , + ∞ [ converging to 0 and some A ′ ∈ C (cid:0) Ω , Sym( N ) (cid:1) .Note that if A n is a function of ε , the second requirement basically expresses that A ′ is the derivative of A w.r.t. ε at ε = 0 in the L ∞ -topology. The symbol λ ,n standsfor the second eigenvalue of the operator u
7→ − div( A n ∇ u ) defined on H (Ω)(well defined for n large enough). The min-max characterization of eigenvalues andthe above assumptions imply that | λ ,n − λ | = O ( ε n ). So, possibly passing to asubsequence, we can assume that there is a λ ′ such that(6.2) λ ,n − λ ε n → λ ′ ∈ R . We also consider ( p n ) ⊆ ]2 , ∗ [ such that p n →
2. Going if necessary to a subse-quence, we can assume that(6.3) p n − ε n → γ ∈ [0 , + ∞ ] . Solutions to ( P n ) are critical points of the energy functional J n ( u ) := 12 Z Ω A n ∇ u · ∇ u d x − λ ,n p n Z Ω | u | p n d x. In this case, the reduced functional J ∗ ,γ : E → R is given by(6.4) J ∗ ,γ ( u ) = 12 Z Ω A ′ ∇ u · ∇ u d x − λ ′ Z Ω u d x + γ λ Z Ω u (1 − log u ) d x and the reduced Nehari manifold is N ∗ ,γ = (cid:8) u ∈ E : J ′∗ ,γ ( u )[ u ] = 0 and u = 0 (cid:9) .The results of section 2.2 extend as follows to the present case. Theorem 6.1.
Assume (6.1) , (6.2) , (6.3) with γ ∈ [0 , + ∞ [ , and p n → . Thereexists r ∗ > such that any sequence ( u n ) of l.e.n.s. to ( P n ) possesses a subsequencethat converges in H (Ω) to a function u ∗ ∈ E \ B r ∗ which is a critical point of J ∗ ,γ .If γ = 0 , u ∗ minimizes J ∗ ,γ on N ∗ ,γ . If γ = 0 , let us define the linear subspace (6.5) E := (cid:8) u ∈ E (cid:12)(cid:12) J ′∗ , ( u ) = 0 (cid:9) and assume further that (6.6) ∀ n, E ⊆ ker (cid:0) u
7→ − div( A n ∇ u ) − λ ,n u (cid:1) . Then u ∗ ∈ E is a critical point of J ∗ | E where J ∗ is defined by (2.2) .Proof. The proof is inspired by [4] but extended to a slightly more general settingsince we do not assume A is a differentiable function of p (which is important forTheorem 6.4 below). The parts that consist in following [4] with straightforwardadaptations will be terse. Let N n (resp. M n ) denote the Nehari manifold (resp.nodal Nehari set) of J n .First let us show that ( u n ) is bounded. Take v n a second eigenfunction ofthe operator u
7→ − div( A n ∇ u ) such that k v n k = 1. Let ˆ v n := t + n v + n − t − n v − n ,for some suitable t ± n ∈ ]0 , + ∞ [, be its projection on M n . Because u n minimizes J n on M n , one has J n ( u n ) J n (ˆ v n ). Because u n , ˆ v n ∈ N n , that translates to R A n ∇ u n · ∇ u n R A n ∇ ˆ v n · ∇ ˆ v n and so ( u n ) is bounded if we show that ( t ± n ) are.The quantities t ± n ∈ ]0 , + ∞ [ are explicitly given by(6.7) t ± n = (cid:18) R A n ∇ v ± n · ∇ v ± n λ ,n R | v ± n | p n (cid:19) / ( p n − . Taking the logarithm and using log ξ ξ −
1, one sees it is enough to show that Z Ω A n ∇ v n · ∇ v ± n − λ ,n Z Ω | v ± n | p n = O ( p n − v n is a second eigenfunction, − λ ,n Z Ω (cid:0) | v n | p n − v n − v n (cid:1) v ± n = O ( p n − . Noting that, for all v ∈ R and 2 < p q , one has (cid:12)(cid:12)(cid:12)(cid:12) | v | p − v − vp − (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z | v | s ( p − v log | v | d s (cid:12)(cid:12)(cid:12)(cid:12) max { , | v | q − } | v | (cid:12)(cid:12) log | v | (cid:12)(cid:12) , one easily concludes by using log | v | = O ( | v | δ ) and Sobolev’s embedding.Thus the sequence ( u n ) is bounded in H and, going to a subsequence if neces-sary, we can assume that u n ⇀ u ∗ for some u ∗ . Because u n ∈ N n , Rellich’s theoremimplies that u n → u ∗ . Passing to the limit on the weak formulation of ( P n ) yields u ∗ ∈ E .Now let us take v ∈ E . Given that u n is a critical point of J n , one gets Z Ω ( A n − ) ∇ u n · ∇ v + λ Z Ω u n v − λ ,n Z Ω | u n | p n − u n v = 0 . Dividing by ε n yields Z Ω A n − ε n ∇ u n · ∇ v − λ ,n − λ ε n Z Ω u n v − p n − ε n λ ,n Z Ω | u n | p n − u n − u n p n − v = 0 COMPUTER ASSISTED PROOF OF THE SYMMETRIES OF L.E.N.S. 19 and passing to the limit n → ∞ shows that u ∗ is a critical point of J ∗ ,γ . If γ = 0,this means that u ∗ ∈ E . In this case, let us take an arbitrary v ∈ E and, using(6.6), rewrite p n − J ′ n ( u n )[ v ] = 0 as − λ ,n Z Ω | u n | p n − u n − u n p n − v d x = 0 . Passing to the limit n → ∞ yields J ′∗ ( u ∗ )[ v ] = 0, thereby proving that u ∗ is acritical point of J ∗ | E .For the existence of r ∗ , we will follow [4, Lemma 4.4]. For conciseness, let uswrite k u k n for R Ω A n ∇ u · ∇ u . Since k u k n = (1 + o (1)) k u k , we can obtain the lowerbound for the norm k·k n . Let us consider ˆ u n := t n (cid:0) (1 − r n ) u + n − r n u − n (cid:1) where r n ∈ [0 ,
1] will be chosen later and t n (depending on r n ) is the scaling ensuringˆ u n ∈ N n . Because u n ∈ M n , it achieves the maximum of J n along the curveobtained by varying r n in [0 , J n (ˆ u n ) J n ( u n ) i.e., k ˆ u n k n k u n k n . Thus it is enough to show that there exists r > k ˆ u n k n > r for all n large. Now, let us chose r n so that ˆ u n is L -orthogonal to the first eigenfunction of u
7→ − div( A n ∇ u ). Therefore, one has(6.8) λ ,n k ˆ u n k L k ˆ u n k n . Let us take q ∈ ]2 , ∗ [ larger than all p n . H¨older interpolation inequality implies:(6.9) k ˆ u n k p n L pn k ˆ u n k − θ n ) L k ˆ u n k qθ n L q with p n = (1 − θ n )2 + θ n q. Recalling that ˆ u n ∈ N n , one also has k ˆ u n k n = λ ,n k ˆ u n k p n L pn . Combining this with (6.9), Sobolev embedding, and (6.8) yields λ − ,n k ˆ u n k n k ˆ u n k − θ n ) L (cid:0) C k ˆ u n k n (cid:1) qθ n (cid:0) λ − ,n k ˆ u n k n (cid:1) − θ n (cid:0) C k ˆ u n k n (cid:1) qθ n for a constant C > n . This inequality may be rewritten as (cid:0) λ − ,n C − q (cid:1) / ( q − k ˆ u n k n . As λ ,n → λ , this provides the sought lower bound.Finally, if γ > J ∗ ,γ has a mountain pass structure and we can take v ∗ a criticalpoint minimizing J ∗ ,γ on N ∗ ,γ . Let us take w the solution to ( − ∆ w − λ w = div( A ′ ∇ v ∗ ) + λ ′ v ∗ + γλ v ∗ log | v ∗ | in Ω ,w ∈ E ⊥ . This is possible because the right hand side is orthogonal to E . Then take v n := v ∗ + ε n w and ˆ v n := t + n v + n − t − n v − n ∈ M n for some t ± n > Z Ω A n ∇ v ± n · ∇ v ± n − λ ,n Z Ω | v ± n | p n = o ( ε n ) = o ( p n − t ± n →
1. Thus ˆ v n → v ∗ . Because u n is a l.e.n.s., one gets J n ( u n ) J n (ˆ v n ) . Because u n , ˆ v n ∈ N n , one can rewrite both hand sides as J n ( u ) = (cid:16) − p n (cid:17) λ ,n Z Ω | u | p n . Dividing by ε n and passing to the limit yields γλ R u ∗ γλ R v ∗ . Since wealready know from the above that u ∗ is a nontrivial critical point of J ∗ ,γ , whence u ∗ ∈ N ∗ ,γ , and J ∗ ,γ ( u ) = γλ R u whenever u ∈ N ∗ ,γ , this concludes the proof. (cid:3) As in section 2.2, when γ = 0, the minimum of J ∗ ,γ over N ∗ ,γ can be found byseeking a minimum of S ∗ ,γ : S → R , S ∗ ,γ ( u ) = Z Ω A ′ ∇ u · ∇ u d x − γλ Z Ω u log | u | d x and projecting it on N ∗ ,γ by scaling. One can equivalently consider S ∗ ,γ on anysphere r S with r > Lyapunov-Schmidt reduction.
For the Lyapunov-Schmidt reduction, weneed to consider a problem that depends smoothly on parameters. Let A be a map R → C (cid:0) Ω , Sym( N ) (cid:1) : ε A ε such that A = differentiable in a neighborhood of 0 for the L ∞ -topology. We will look at thefunctional J ε,γ : H (Ω) → R defined by J ε,γ ( u ) = 12 Z Ω A ε ∇ u · ∇ u d x − λ ( ε ) p Z Ω | u | p d x where p = 2 + γε and λ ( ε ) is the second eigenvalue of the operator u
7→ − div( A ε ∇ u ) with Dirichletboundary conditions on Ω. Theorem 6.2.
Assume that R → C (cid:0) Ω , Sym( N ) (cid:1) : ε A ε and ε λ ( ε ) aredifferentiable in a neighborhood of and that there is ¯ γ > and a continuous map [0 , ¯ γ ] → E \ { } : γ u γ such that, for all γ ∈ ]0 , ¯ γ ] , u γ is a non-degenerate critical point of J ∗ ,γ : E → R with A ′ = ( ∂ ε A ε ) | ε =0 and λ ′ = ∂ ε λ (0) . Let E be defined by (6.5) and assumefurther that (6.10) for all ε > small, E ⊆ ker (cid:0) u
7→ − div( A ε ∇ u ) − λ ( ε ) u (cid:1) and u ∈ E is a non-degenerate critical point of J ∗ | E . Then there exists ¯ ε > , ρ > , and a continuous function σ : [0 , ¯ ε ] × [0 , ¯ γ ] → H (Ω) : ( ε, γ ) σ ( ε, γ ) suchthat(1) for all γ ∈ [0 , ¯ γ ] , σ (0 , γ ) = u γ ,(2) for all ε ∈ [0 , ¯ ε ] , σ ( ε,
0) = u ,(3) for all ε ∈ ]0 , ¯ ε ] , γ ∈ ]0 , ¯ γ ] and u ∈ H (Ω) such that k u − u γ k ρ , one has u is a critical point of J ε,γ ⇔ u = σ ( ε, γ ) . Remark . Passing to the limit γ → J ′∗ ,γ ( u γ ) = 0 yields J ′∗ , ( u ) = 0 i.e., u ∈ E . Since J ′∗ , vanishes on E , u is necessarily a degenerate critical pointof J ′∗ , (if E is non-trivial). Now, taking v ∈ E , one gets 0 = γ J ′∗ ,γ ( u γ )[ v ] = J ′∗ ( u γ )[ v ] and, passing to the limit γ → u is a critical point of J ∗ | E .Note that, if dim E = 1, the non-degeneracy assumption is trivially satisfied (themap t
7→ J ∗ ( tu ) is explicit). COMPUTER ASSISTED PROOF OF THE SYMMETRIES OF L.E.N.S. 21
Proof.
As the beginning of the proof is similar to the one of Theorem 2.2, wewill sketch it. First one defines G ( ε, γ, v, w ) := J ′ ε,γ ( v + w ) | E ⊥ and shows that ∂ w G (0 , γ, v,
0) : E ⊥ → ( E ⊥ ) ′ is invertible. So the solutions to G ( ε, γ, v, w ) = 0 in aneighborhood of (0 , γ ∗ , v ∗ ,
0) are given by w = ω ( ε, γ, v ). Gluing these functions ina compact range of ( γ ∗ , v ∗ ), we obtain ω : [0 , ¯ ε ] × [0 , ¯ γ ] × ¯ B R → E ⊥ of class C for some ¯ ε > W a neighborhood of 0 in E ⊥ (here ¯ B R denotes the closed ballof radius R in E ) such that G ( ε, γ, v, w ) = 0 ⇔ w = ω ( ε, γ, v ) whenever ( ε, γ, v, w ) ∈ [0 , ¯ ε ] × [0 , ¯ γ ] × ¯ B R × W. We take R large enough so that the curve { u γ | γ ∈ [0 , ¯ γ ] } lies inside B R . Next wedefine H ( ε, γ, v ) := J ′ ε,γ ( v + ω ( ε, γ, v )) | E . Noting that ω (0 , γ, v ) = 0, we see that H (0 , γ, v ) = 0. So we define K : [0 , ¯ ε ] × [0 , ¯ γ ] × ¯ B R → E ′ by K ( ε, γ, v ) := ( H ( ε, γ, v ) /ε if ε > ,∂ ε H (0 , γ, v ) if ε = 0 . The map K is continuous and K (0 , γ, v ) = ∂ ε H (0 , γ, v ) = J ′∗ ,γ ( v ) = J ′∗ , ( v ) + γ J ′∗ ( v )with A ′ = ( ∂ ε A ε ) | ε =0 and λ ′ = ( ∂ ε λ ( ε )) | ε =0 . Since u γ ∗ is a non-degenerate criticalpoint of J ∗ ,γ ∗ (which is C by proposition 4.1), the Implicit Function Theoremimplies that that the solutions to K ( ε, γ, v ) = 0 in a neighborhood of (0 , γ ∗ , u γ ∗ )are given by v = ˜ σ ( ε, γ ). Again using compactness to glue these maps and possiblytaking ¯ ε smaller, we obtain a continuous map˜ σ : [0 , ¯ ε ] × [ γ , ¯ γ ] → E : ( ε, γ ) ˜ σ ( ε, γ ) , where γ > K ( ε, γ, v ) = 0 ⇔ v = ˜ σ ( ε, γ ) whenever ( ε, γ ) ∈ [0 , ¯ ε ] × [ γ , ¯ γ ] and k v − u γ k ˜ ρ for some ˜ ρ > γ = 0. Note that K (0 , , · ) is the linear map E → E ′ : v
7→ K (0 , , v ) = J ′∗ , ( v ) where J ′∗ , ( v )[ η ] = Z Ω A ′ ∇ v ∇ η − λ ′ Z Ω vη and this map vanishes on E . Let us split E as E ⊕ E ⊥ where E ⊥ is the L -orthogonal complement of E in E . Set K ( ε, γ, v , v ) := K ( ε, γ, v + v ) | E ⊥ where v ∈ E and v ∈ E ⊥ . Clearly ∂ v K (0 , , u ,
0) = J ′∗ , : E ⊥ → ( E ⊥ ) ′ isinvertible. Thus the Implicit Function Theorem implies that, possibly taking ¯ ε , γ and ρ smaller, there exists a function ω : [0 , ¯ ε ] × [0 , γ ] × ¯ B ρ ( u ) → E ⊥ such that ∀ ( ε, γ, v , v ) ∈ [0 , ¯ ε ] × [0 , γ ] × ¯ B ρ ( u ) × ¯ B ρ , K ( ε, γ, v , v ) = 0 ⇔ v = ω ( ε, γ, v )where ¯ B ρ ( u ) (resp. ¯ B ρ ) denote the closed ball of radius ρ centered at u in E (resp. centered at 0 in E ⊥ ). Thus, solving K ( ε, γ, v + v ) = 0 for ( ε, γ, v , v ) closeto (0 , , u ,
0) amounts to find the roots of the following map K : [0 , ¯ ε ] × [0 , γ ] × ¯ B ρ ( u ) → E ′ : ( ε, γ, v )
7→ K (cid:0) ε, γ, v + ω ( ε, γ, v ) (cid:1) | E . Assumption (6.10) says that J ′ ε, ( v ) = 0 whenever v ∈ E . As a consequence, ω ( ε, , v ) = 0, H ( ε, , v ) = 0, so K ( ε, , v ) = 0 for ε > ε = 0by continuity) and, finally, ω ( ε, , v ) = 0 and K ( ε, , v ) = 0. This enables us todefine the continuous map L ( ε, γ, v ) := ( K ( ε, γ, v ) /γ if γ > ,∂ γ K ( ε, , v ) if γ = 0 . Note that ∂ γ K (0 , , v ) = J ′∗ ( v ) | E . In particular, the assumption that u is a non-degenerate critical point of J ∗ | E implies that, possibly taking ¯ ε , γ and ρ smaller, there exists a map ˆ σ : [0 , ¯ ε ] × [0 , γ ] → E such that ∀ ( ε, γ, v ) ∈ [0 , ¯ ε ] × [0 , γ ] × ¯ B ρ ( u ) , L ( ε, γ, v ) = 0 ⇔ v = ˆ σ ( ε, γ ) . Extending the map ˜ σ for γ close to 0 by˜ σ : [0 , ¯ ε ] × [0 , γ ] → E : ( ε, γ ) ˆ σ ( ε, γ ) + ω (cid:0) ε, γ, ˆ σ ( ε, γ ) (cid:1) one has that (possibly taking ρ smaller in order to account for the different way ofmeasuring the distance to u ): ∀ ( ε, γ, v ) ∈ [0 , ¯ ε ] × ]0 , γ ] × ¯ B E ρ ( u ) , K ( ε, γ, v ) = 0 ⇔ v = ˜ σ ( ε, γ )where B E ρ ( u ) is the ball of center u and radius ρ in E . The map σ ( ε, γ ) :=˜ σ ( ε, γ ) + ω ( ε, γ, ˜ σ ( ε, γ )) satisfies the claims of the theorem. (cid:3) Symmetry breaking on rectangles.
Recall that on rectangles, the nodalline of a l.e.n.s. is given by the small median. Moreover, Theorem 2.1 asserts thaton squares the nodal line is a diagonal. One can thus guess that for rectangles thatare almost square a symmetry breaking will occur.Consider the rectangle R ε = ]0 , × ]0 , ε [ and the problem ( P p ) on R ε ( − ∆ u = λ ( ε ) | u | p − u in R ε ,u = 0 on ∂R ε . ( P R ε )The change of variables v ( x, y ) = u ( x, (1 + ε ) y ) leads to the equivalent problem onthe square Ω = ]0 , : ( − v xx − ε ) v yy = λ ( ε ) | v | p − v in Ω ,v = 0 on ∂ Ω . ( P p,ε )Observe that u is a l.e.n.s. of ( P R ε ) if and only if v is a l.e.n.s. on Ω.Theorem 6 of [4] claims, based on unverified numerical computations, that for p > verified computations, validates their reasoning. The following theorem expands this resultand proves Conjecture 6.5 in [4]. Theorem 6.4. (1) There exists γ > and ¯ ε > such that, for any ( ε, p ) in the triangle defined by ε ∈ ]0 , ¯ ε ] and < p γ ε , every l.e.n.s. to ( P p,ε ) is symmetric with respect to the longest median and antisymmetricwith respect to the shortest one. COMPUTER ASSISTED PROOF OF THE SYMMETRIES OF L.E.N.S. 23 (2) There exists γ > γ such that, for all γ > γ , there exists ε γ > such that,for any ε ∈ ]0 , ε γ ] , l.e.n.s. to ( P p,ε ) with p = 2 + γε have no diagonal normedian (anti-)symmetries. This Theorem is illustrated in Figure 7.
Proof. 1.
First note that, for any ε > p ∈ ]2 , ∗ ], solutions to ( P p,ε ) correspondto critical points of J ε,γ with A ε = (cid:18) ε ) − (cid:19) , λ ( ε ) = π (cid:16) ε ) (cid:17) , and γ = p − ε . The associated reduced functional J ∗ ,γ is defined by (6.4) with A ′ = ( ∂ ε A ε ) | ε =0 = (cid:18) − (cid:19) and λ ′ = ∂ ε λ (0) = − π . It is easily verified that E is the one-dimensional subspace given by E = span { ϕ } and that (6.10) holds. Let us now give some properties of J ∗ ,γ when γ > J ∗ ,γ on N ∗ ,γ . Equivalently, we want to show that, for all γ ∈ ]0 , ¯ γ ], g γ ( θ ) := π J ∗ ,γ ( u θ ),where u θ is defined by (3.2), has a unique minimum up to symmetries in [0 , π ]; thisminimum occurs at θ = 0 and g ′′ γ (0) > g γ ( θ ) := π − S ∗ ,γ ( u θ ) = π − (cid:0) S ∗ , ( u θ ) + γλ S ∗ ( u θ ) (cid:1) = sin θ − γg ( θ )where S ∗ is given in (3.4) and g by (3.4). Given the symmetries of g γ , one only hasto examine θ ∈ [0 , π/ , ¯ γ ] by a family of small intervals Γ i and, on eachof them, we perform similar computations to those of section 5. This is illustratedby Fig. 8. Suppose now that there is a ¯ γ such that, for all γ ∈ ]0 , ¯ γ ], the minimum of J ∗ ,γ on N ∗ ,γ is achieved solely (up to symmetries) at a multiple of ϕ and thatthe corresponding critical point u γ is non-degenerate. Note that u γ := t γ ϕ where t γ > u γ ∈ N ∗ ,γ . It is given by t γ = exp (cid:18) R A ′ ∇ ϕ · ∇ ϕ − λ ′ R ϕ − γλ R u log | u | γλ R u (cid:19) = exp (cid:18) − R u log | u | R u (cid:19) where the second equality results from ϕ ∈ E . Obviously, γ t γ is continuousand, when γ = 0, u = t ϕ is a non-degenerate critical point of J ∗ on E (see (2.5)and remark 6.3). Moreover, u γ is symmetric with respect to the longest medianand antisymmetric with respect to the shortest one.Let ¯ ε and ρ be given by theorem 6.2 for this map γ u γ . If u is a solutionto ( P p,ε ) with ε ∈ ]0 , ¯ ε ], 0 < ε γε , and k u − u γ k ρ , then its even (resp.odd) symmetrization v w.r.t. the longest (resp. shortest) median is still a solutionto ( P p,ε ) and satisfies k v − u γ k ρ . In view of the local uniqueness offered bypoint (3), one deduces that u possesses the same symmetries as u γ .To conclude, one must show that, possibly taking ¯ ε smaller, l.e.n.s. u to ( P p,ε ),with ( ε, p ) as in the statement, satisfy, up to symmetries, k u − u γ k ρ with γ = ( p − ε . If not, there exists sequences ε n → p n ∈ ]2 , γε n ], and ( u n )l.e.n.s. such that k u − u γ n k > ρ , with γ n := ( p n − /ε n ∈ ]0 , ¯ γ ], for u being any even or odd symmetry of u n w.r.t. medians. Passing if necessary to a subsequence, wecan assume γ n → γ ∈ [0 , ¯ γ ]. Clearly, (6.1) and (6.2) are satisfied for λ ,n = λ ( ε n ).Therefore, theorem 6.1 asserts that u n → u ∗ for some u ∗ = 0 which is a criticalpoint of J ∗ ,γ . If γ = 0, u ∗ minimizes J ∗ ,γ on N ∗ ,γ and so, up to symmetries, is u γ .This contradicts the fact that k u n − u γ n k > ρ for n large. If γ = 0, u ∗ ∈ E andis a non-trivial critical point of J ∗ on E . Since E is one dimensional, u ∗ = ± u which again contradicts k u − u k > ρ . (cid:3) p = + ¯ γ ε ¯ ε ε p ε =0 Figure 7.
Symmetry breaking. g g θ π π − − Figure 8. g γ for some γ ∈ [0 , + ∞ [. References
1. Amandine Aftalion and Filomena Pacella,
Qualitative properties of nodal solutions of semi-linear elliptic equations in radially symmetric domains , Comptes Rendus Mathematique (2004), no. 5, 339 – 344.2. Antonio Ambrosetti and Paul H Rabinowitz,
Dual variational methods in critical point theoryand applications , Journal of Functional Analysis (1973), no. 4, 349 – 381.3. Thomas Bartsch, Tobias Weth, and Michel Willem, Partial symmetry of least energy nodalsolutions to some variational problems , Journal d’Analyse Math´ematique (2005), no. 1,1–18.4. Denis Bonheure, Vincent Bouchez, Christopher Grumiau, and Jean Van Schaftingen, Asymp-totics and symmetries of least energy nodal solutions of Lane-Emden problems with slowgrowth , Communications in Contemporary Mathematics (2008), no. 04, 609–631.5. Alfonso Castro, Jorge Cossio, and John M. Neuberger, A sign-changing solution for a super-linear Dirichlet problem , Rocky Mountain J. Math. (1997), no. 4, 1041–1053. MR 16276546. Philip J. Davis and Philip Rabinowitz, Methods of numerical integration , second ed., Com-puter Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. MR 7606297. S. Galdino,
Interval integration revisited , Open Journal of Applied Sciences (2012), 108–111.8. B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximumprinciple , Comm. Math. Phys. (1979), no. 3, 209–243. MR 5448799. G¨unter Mayer, Interval analysis—and automatic result verification , De Gruyter Studies inMathematics, vol. 65, De Gruyter, Berlin, 2017. MR 372685410. Ramon E. Moore, R. Baker Kearfott, and Michael J. Cloud,
Introduction to interval analysis ,Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009. MR 248268211. Knut Petras,
Principles of verified numerical integration , J. Comput. Appl. Math. (2007),no. 2, 317–328. MR 226951412. Sergey A. Smolyak,
Quadrature and interpolation formulas for tensor products of certainclasses of functions , Soviet Math. Dokl. (1963), 240–243. COMPUTER ASSISTED PROOF OF THE SYMMETRIES OF L.E.N.S. 25 (A. Salort)
Departamento de Matem´atica, FCEyN - Universidad de Buenos Aires andIMAS - CONICETCiudad Universitaria, Pabell´on I (1428) Av. Cantilo s/n.Buenos Aires, Argentina.
Email address : [email protected] URL : http://mate.dm.uba.ar/~asalort (C. Troestler) D´epartment de Math´ematique, Universit´e de Mons, Place du Parc, 20,B-7000 Mons, Belgium
Email address : [email protected] URL ::