Fibers and global geometry of functions
aa r X i v : . [ m a t h . A P ] J un Fibers and global geometry of functions
Marta Calanchi, Carlos Tomei and Andr´e Zaccur
Dedicated to Djairo, an example to follow in many directions
Abstract
Since the seminal work of Ambrosetti and Prodi, the study of global foldswas enriched by geometric concepts and extensions accomodating new exam-ples. We present the advantages of considering fibers, a construction dating toBerger and Podolak’s view of the original theorem. A description of folds interms of properties of fibers gives new perspective to the usual hypotheses inthe subject. The text is intended as a guide, outlining arguments and statingresults which will be detailed elsewhere.
Keywords:
Dolph-Hammerstein theorem, Semilinear elliptic equations, Ambrosetti-Prodi theorem, folds.
MSC-class:
When we teach the first courses in calculus and complex or real analysis, a greatemphasis is given to geometric issues: we plot graphs, enumerate conformal map-pings among special regions, identify homeomorphisms. Alas, this is far from beingenough: mappings become too complicated soon. Still, the geometric approach,especially combined with numerical arguments, is very fruitful in some nonlinearcontexts.It is rather surprising that some infinite dimensional maps can be studied in asimilar fashion — one may even think about their graphs! The examples which areamenable to such approach are very few, and they elicit the same sense of wonderthat (the equally rare) completely integrable systems do: one is left with a feelingof deep understanding. This text is dedicated to some such examples.The interested reader could hardly do better than going through the reviewpapers by Church and Timourian ([12], [13]), which cover extremely well the ma-terial up to the mid nineties. Their approach is strongly influenced by the originalAmbrosetti-Prodi view of the problem, which we describe in Section 2.2. In a nut-shell, the global geometry of a proper function F is studied through certain proper-ties of its critical set C together with its image F ( C ), along with the stratificationof C in terms of singularities.This much less ambitious text is mainly an enumeration of techniques and ofsome recent developments, some of which have not been published. We mostly takethe Berger-Podolak route ([4]) which has been extended by Podolak in [27] and, webelieve, still allows for improvement. Instead of the critical set, we concentrate onthe restriction of F to appropriate low dimensional manifolds (one dimensional, inthe Ambrosetti-Prodi case), the so called fibers.Essentially, fibers are appropriate in the presence of finite spectral interaction ,which roughly states that the function F : X → Y splits into a sum of linear andnonlinear terms, F = L − N and N deforms L substantially only along a few eigen-vectors spanning a subspace V ⊂ X . The domain splits into orthogonal subspaces,1 = H ⊕ V and the hypotheses on the nonlinearities are naturally anisotropic. Dif-ferent requests on H and V yield a global Lyapunov-Schmidt decomposition of F :on affine subspaces obtained by translating H , F is a homeomorphism and compli-cations due to the nonlinear term manifest on fibers, which are graphs of functionsfrom V to H .Fibers are also convenient for the verification of properness of F . In particular,one may search for folds in nonlinear maps defined on functions with unboundeddomains, which are natural in physical situations. Fibers also provide the conceptualstarting point for algorithms that solve a class of partial differential equations, anidea originally suggested by Smiley ([31], [32]) and later implemented for finitespectral interaction of the Dirichlet Laplacian on rectangles in [8].An abstract setup in the spirit of the characterization of folds as in [12], or likethe one we present in Section 4, provide a better understanding of the role of thehypotheses in the fundamental example of Ambrosetti and Prodi. Elliptic theoryseems to be less relevant than one might think, it is just that it provides a contextin which the required hypotheses are satisfied.In Section 2, we present the seminal examples — the Dolph-Hammerstein home-omorphisms and the Ambrosetti-Prodi fold — in a manner appropriate for our ar-guments. Fibers and sheets are defined and constructed in Section 3. A globalchange of coordinates in Section 4 gives rise to adapted coordinates, in which thedescription of critical points is especially simple. A characterization of the criticalpoints strictly in terms of spectral properties of the Jacobian DF is given. Also,the three natural steps to identify global folds become easy to identify. Furtherstudy of how to implement each step is the content of Sections 5, 6 and 7. The lastsection is dedicated to some examples.The text is written as a guide: we try to convey the merits of a set of techniques,without providing details. Complete proofs will be presented elsewhere ([ ? ], [ ? ]).Alas, we stop at folds. There are scattered results in which local or globalcusps were identified: again, the excellent survey [13] covers the material up tothe mid nineties. So far, the description of cusps seems rather ad hoc. There arecharacterizations ([13]), but they are hard to verify and new ideas are needed. Onthe other hand, checking that maps are not global folds is rather simple, a matter ofshowing for example that some points in the image have more than two preimages.A numerical example is exhibited in Section 5.3. Among the simplest continuous maps between Hilbert spaces are homeomorphisms,in particular linear isomorphisms. A second class of examples are folds.
Dolph and Hammerstein ([15], [16]) obtained a simple condition under which non-linear perturbation of linear isomorphisms are still homeomorphisms. A version oftheir results is the following.Start with a real Hilbert space Y and a self-adjoint operator L : X ⊂ Y → Y for a dense subspace X of Y . Let σ ( L ) be the spectrum of L . Theorem 1
Let [ − c, c ] ∩ σ ( L ) = ∅ and suppose N : Y → Y is a Lipschitz map withLipschitz constant n < c . Equip X with the graph topology, k x k X = k x k Y + k Lx k Y .Then the map F = L − N : X → Y is a Lipschitz homeomorphism. F ( x ) = y , search for a fixed point of C y : Y → Y, C y ( z ) = N ( L − ( z )) + y which is a contraction because the operator L − : Y → Y has norm less than 1 /c by standard spectral theory and then the map N ◦ L − is Lipschitz with constantless than n/c < y . Clearly, F is Lipschitz. To show the same for F − , keep track of the Banach iteration.Notice that the statement allows for differential operators between Sobolevspaces. Very little is required from the spectrum of L . Clearly, for symmetricbounded operators one should take X = Y . What about more complicated functions? Ambrosetti and Prodi ([1]) obtained anexquisite example. After refinements by Micheletti and Manes ([24]), Berger andPodolak ([4]) and Berger and Church ([5]), the result may be stated as follows. LetΩ ⊂ R n be a connected, open, bounded set with smooth boundary (for nonsmoothboundaries, see [34]). Let H (Ω) and H (Ω) be the usual Sobolev spaces and set X = H (Ω) ∩ H (Ω) and Y = H (Ω) = L (Ω). The eigenvalues of the DirichletLaplacian − ∆ : X ⊂ Y → Y are σ ( − ∆) = { < λ < λ ≤ . . . → ∞} . Denote by φ the ( L -normalized, positive) eigenvector associated to λ andsplit X = H X ⊕ V X , Y = H Y ⊕ V Y in horizontal and vertical orthogonal subspaces,where V X = V Y = h φ i , the one dimensional (real) vector space spanned by φ . Theorem 2
Let F : X → Y be F = L − N , where L = − ∆ , N ( u ) = f ( u ) , for asmooth, strictly convex function f : R → R satisfying Ran f ′ = ( a, b ) , a < λ < b < λ . Then there are global homeomorphisms ζ : X → H Y ⊕ R and ξ : Y → H Y ⊕ R for which ˜ F ( z, t ) = ξ ◦ F ◦ ζ − ( z, t ) = ( z, − t ) . Said differently, the following diagram commutes. X F −→ Y ζ ↓ ↓ ξ H Y ⊕ R ( z, − t ) −→ H Y ⊕ R Functions which admit such dramatic simplification are called global folds . Thevertical arrows in the diagram above are (global) changes of variables and sometimeswill be C maps, but we will not emphasize this point.The original approach by Ambrosetti and Prodi is very geometric ([1]). In anutshell, they show that F is a proper map whose critical set C (in the standardsense of differential geometry, the set of points u ∈ X for which the derivative DF ( u )is not invertible) is topologically a hyperplane, together with its image F ( C ). Theythen show that F is proper, its restriction to C is injective and F − ( F ( C )) = C .Finally, they prove that both connected components of X − C are taken injectivelyto the same component of Y − F ( C ). Their final result is a counting theorem: thenumber of preimages under F can only be 0, 1 or 2.Berger and Podolak ([4]), on the other hand, construct a global Lyapunov-Schmidt decomposition for F . For V X = V Y = h φ i , consider affine horizontal resp. vertical) subspaces of X (resp. Y ), i.e., sets of the form H X + tφ , for afixed t ∈ R (resp. y + V Y , for y ∈ H Y ). Let P : Y → H Y be the orthogonalprojection. The map P F t : H X → H Y , P F t ( w ) = P F ( w + tφ ), is a bi-Lipschitzhomeomorphism, as we shall see below. Thus, the inverse under F of vertical lines y + V Y , for y ∈ H Y are curves α y : R ∼ V X ⊂ X → H X , which we call fibers . Fibersstratify the domain X . Thus, to show that F is a global fold, it suffices to verifythat each restriction F : α y → V Y ∼ R , essentially a map from R to R , is a fold.After such a remarkable example, one is tempted to push forward. This is notthat simple: if the (generic) nonlinearity f is not convex, there are points in Y withfour preimages ([ ? ]), so the associated map F : X → Y cannot be a global fold (fora numerical example, see Section 5.3). Fibers come up in [4] and [32] for C maps associated to second order differentialoperators and in [25] in the context of first order periodic ordinary differentialequations. Due to the lack of self-adjointness, the construction in [25] is of a verydifferent nature. We follow [27] and [34], which handle Lipschitz maps, allowing theuse of piecewise linear functions in the Ambrosetti-Prodi scenario, namely f givenby f ′ ( x ) = a or b , depending if x < x > X and Y be Hilbert spaces, X densely included in Y . Let L : X ⊂ Y → Y be a self-adjoint operator with a simple, isolated, eigenvalue λ p , with eigenvector φ p ∈ X with k φ p k Y = 1. Notice that λ p may be located anywhere in the spectrum σ ( L ) of L . As before, consider horizontal and vertical orthogonal subspaces, X = H X ⊕ V X , Y = H Y ⊕ V Y , for V X = V Y = h φ p i and the projection P : Y → H Y . Let P F t : H X → H Y be the projection on H Y ofthe restriction of F to the affine subspace H X + tφ p , P F t ( w ) = P F ( w + tφ p ). Inthe same fashion, the nonlinearity N : Y → Y gives rise to maps P N t : H Y → H Y ,which we require to be Lipschitz with constant n independent of t ∈ R so that[ − n, n ] ∩ σ ( L ) = { λ p } . ( H )The standard Ambrosetti-Prodi map fits these hypotheses. In this case, X ⊂ Y are Sobolev spaces and the derivative f ′ : R → R is bounded by a and b . Set γ = ( a + b ) / , L = − ∆ − γ, N ( u ) = f ( u ) − γu and λ p = λ , the smallest eigenvalue of − ∆. Then the Lipschitz constant n of themaps P N t satisfies n < γ − a = b − γ < λ − γ , so that λ − γ ≤ n . Theorem 3
Let F : X → Y satisfy ( H ) above. Then for each t ∈ R , the map P F t is a bi-Lipschitz homeomorphism, and a C k diffeomorphism if F is C k . TheLipschitz constants for P F t and ( P F t ) − are independent of t . Proof:
The proof follows Theorem 1 once the potentially nasty eigenvalue λ p isruled out. Let c be the absolute value of the point in σ ( L ) \{ λ p } closest to 0, so that0 ≤ n < c . The operator L : X → Y restricts to L : H X → H Y , which is invertibleself-adjoint, and again L − : H Y → H Y with k L − k ≤ /c . The solutions w ∈ H X of P F t ( w ) = g ∈ H Y solve P Lu − P N ( u ) = Lw − P N t ( w ) = g for u = w + tφ p .The solutions w correspond to the fixed points of C g : H Y → H Y , where C g ( z ) = P N t ( L − z ) + g, for Lw = z ∈ H Y . The map C g is a contraction with constant bounded by n/c < t ).Now follow the proof of Theorem 1. (cid:4) N along the vertical direction is irrelevant for the construction of fibers.The same construction applies when the interval [ − n, n ] defined by the Lipschitzconstant n of P N t : H X → H Y interacts with an isolated subset I of σ ( L ) — moreprecisely, I = [ − n, n ] ∩ σ ( L ) and there is an open neighborhood U of I ⊂ R forwhich I = U ∩ σ ( L ). In this case P is the orthogonal projection on I , which takesinto account possible multiplicities. In the special situation when I consists of afinite number of eigenvalues (accounting multiplicity), we refer to finite spectralinteraction between L and N .We concentrate on the case when I = { λ p } consists of a simple eigenvalue.A more careful inspection of the constants in the Banach iteration in the proofabove yields the following result ([8], [34]). The image under F of horizontal affinesubspaces of X are sheets . The inverse under F of vertical lines of Y are fibers . Proposition 1 If F is C , sheets are graphs of C maps from H Y to h φ p i andfibers are graphs of C maps from h φ p i to H X . Sheets are essentially flat, fibers areessentially steep. We define what we mean by essential flatness and steepness. Let ν ( y ) be thenormal at a point y ∈ Y of (the tangent space of) a sheet, and τ ( u ) be the tangentvector at u ∈ X of a fiber. Then there is a constant ǫ ∈ (0 , π/
2) such that φ p makesan angle less than ǫ (or greater than π − ǫ , due to orientation) with both vectors. Suppose L and N interact at a simple eigenvalue λ p . Write F ( u ) = P F ( u ) + h F ( u ) , φ p i φ p = P F ( u ) + h ( u ) φ p where the map h : X → R is called the height function . In the diagram below,invertible maps are bi-Lipschitz ([34]) or C k diffeomorphisms, depending if P F t isLipschitz or C k . The smoothness of h and h a = h ◦ Φ follow accordingly. X = H X ⊕ V X F −→ Y = H Y ⊕ V Y Φ − =( P F t ,Id ) ց ր F a = F ◦ Φ=(
Id,h a ) Y The map F has been put in adapted coordinates by the change of variables Φ: F a : Y → Y , ( z, t ) ( z, h a ( z, t )) . Notice that fibers of F are taken to vertical lines in the domain of F a = F ◦ Φ.Explicitly, the vertical lines { ( z , t ) : t ∈ R } parameterized by z ∈ H Y correspondto fibers u ( z , t ) = ( P F t ) − ( z ) + tφ p = w ( z , t ) + tφ p . Thus F a is just a rank onenonlinear perturbation: F a ( z, t ) = ( z, h a ( z, t )) ∼ z + h a ( z + tφ p ) φ p . In a very strict sense, this is also true of F . In order to make F similar to anAmbrosetti-Prodi map, define G = F a ◦ ( − ∆) : X → Y : u − ∆ z + tφ F a z + tφ + ( h a ( z + tφ ) − t ) φ = − ∆ u + ψ ( u ) φ , for some nonlinear functional ψ . We generalize slightly.5 roposition 2 Let N be a C map. Say L and N interact at a simple eigenvalue λ p and L is invertible. Then, after a C change of variables, the C function F = L − N : X → Y becomes G : X → Y , G = L + ψ ( u ) φ p , for some ψ : X → R . For Ambrosetti-Prodi operators F ( u ) = − ∆ u − f ( u ), the nonlinear perturbationis given by a Nemitskii map u f ( u ). It is not surprising that once we enlargethe set of nonlinearities new global folds arise. For a map F given in adaptedcoordinates by F a ( z, t ) = ( z, h a ( z, t )), appropriate choices of the adapted heightfunction h a yields all sorts of behavior.The critical set of F : X → Y is compatible with fibers as follows ([4], [ ? ]). Proposition 3
Suppose the C map F : X → Y admits fibers. Then u is a criticalpoint of F if and only if it is a critical point of the height function h along its fiber,or equivalently of the adapted height function h a . Isolated local extrema have to alternate between maxima and minima. In par-ticular, given the appropriate behavior at infinity at each fiber and the fact that allcritical points are of the same type, we learn from a continuity argument that thefull critical set C is connected, with a single point on each fiber ([11]).The study of a function F : X → Y reduces to three steps:1. Stratify X into fibers.2. Verify the asymptotic behavior of F along fibers.3. Classify the critical points of the restriction of F along fibers.The following result is natural from this point of view ([ ? ]). Let F : X → Y satisfies ( H ) of Section 3, so that, by Proposition 1, X stratifies in one dimensionalfibers { u ( z, t ) : t ∈ R } , one for each z ∈ H Y . Proposition 4
Suppose that, on each fiber, lim t →±∞ h F ( u ( z, t )) , φ p i = lim t →±∞ h ( u ( z, t )) = −∞ . Suppose also that each critical point of h restricted to each fiber is an isolated localmaximum. Then F : X → Y is a global fold, in the sense that there are homeomor-phisms on domain and image that give rise to a diagram as in Theorem 2. To verify that such limits exist, one might check hypotheses ( V ± ) in Section6.1, but there are alternatives. Similarly, there are ways of obtaining fibers whichdo not fit the construction presented in Section 3 (this is the case for perturbationsof non-self-adjoint operators, Section 8.4). The upshot is that there is some loss informulating the three step recipe into a clear cut theorem.As trivial examples, h a ( z, t ) = − t is a global fold, whereas h a ( z, t ) = t − t has a critical set consisting of two connected components having only (local) folds(from Section 7.1). More complicated singularities require the dependence on z :not every fiber of F (equivalently, vertical line in the domain of F a ) has the samenumber of critical points close to a cusp, for example. The reader is invited to checkthat ( z, t ) ( z, t − h z, ˜ φ i t ) is a global cusp, for ˜ φ any fixed vector in H Y . Higherorder Morin singularities, considered in Section 7, are obtained in a similar fashion.From the Proposition 2, changes of variables on such maps yield nonlinear rank oneperturbations of the Laplacian which are globally diffeomorphic to the standardnormal forms of Morin singularities. 6e consider the standard Ambrosetti-Prodi scenario in the light of this strategy.For the function F ( u ) = − ∆ u − f ( u ) defined in Theorem 2, elliptic theory yields allsort of benefits — the smallest eigenvalue of the Jacobian DF ( u ) is always simple,the ground state may be taken to be a positive function in X .The hypotheses required for the construction of fibers in Theorem 3 do not implythe simplicity of the relevant eigenvalue: there are examples for which there is nonaturally defined C functional λ p : X → R because two eigenvalues collide. Onemight circumvent this difficulty by forcing the nonlinearity N to be smaller, but itturns out that this is not necessary. The hypotheses instead imply the simplicityof λ p in an open neighborhood of the critical set C of F , and this is all we need, aswe shall see in Section 8.The positivity of the ground state and the convexity of the nonlinearity f areused in a combined fashion in the Ambrosetti-Prodi theorem to prove that alongfibers the height function only has local maxima. Clearly, this is a property onlyof critical points. On the other hand, the nonlinearity N ( u ) = f ( u ) is so rigid thatthe standard hypothesis of convexity of f is essentially necessary, as shown in [ ? ].More general nonlinearities require a better understanding of the singularities.We now provide more technical details on each of the three steps. For starters, what if L is not self-adjoint, or X is not Hilbert? Suppose momentarily that X and Y are Banach spaces. Let L : X → Y be aFredholm operator of index zero with kernel generated by a vector φ X and let φ Y be a vector not in Ran L . Podolak ([27]) considered the following scenario, for whichshe obtained a lower bound on the number of preimages for a region of Y of vectorswith very negative component along φ Y . Split X = H X ⊕ V X where V X = h φ X i and H X is any complement. Also, split Y = H Y ⊕ V Y where H Y = Ran L and V Y = h φ Y i . In particular L : H X → H Y is an isomorphism. Also, define theassociated projection P : Y → H Y . Write u = w + tφ X , y = g + sφ Y for w ∈ H X .The equation F ( u ) = Lu − N ( u ) = y becomes L ( w + tφ X ) − N ( w + tφ X ) = Lw − N ( w + tφ X ) = g + sφ Y , and, as in Theorem 3, we are reduced to solving the map C g : H Y → H Y , C g ( z ) = P N t ( L − z ) + g , for Lw = z ∈ H Y . Her hypotheses imply that such maps are contractions.
The estimates arising from spectral theorem in the Hilbert context are easy to obtainand possibly more effective. Podolak’s hypotheses are harder to verify. There isa possibility: getting fibers in Hilbert spaces and transplanting them to Banachspaces. This happens for example when moving from the Ambrosetti-Prodi exampleas a map between Sobolev spaces ([4]) to a map between H¨older spaces ([1]). Theclassification of singularities is simpler with additional smoothness (Section 7).
Proposition 5
Let F = L − N : X → Y satisfy hypothesis ( H ) of Section 3.Consider the densely included Banach spaces A ⊂ X and B ⊂ Y allowing for the restriction F : A → B for which V X = V Y ⊂ A . Suppose that DF ( a ) : A → B is a Fredholm operator of index zero for each a ∈ A . Then fibers of F : X → Y either belong to A or do not intersect A . Said differently, if a point u ∈ X belongs to A then the whole fiber does.In the Ambrosetti-Prodi scenario, this proposition seems to be a consequence ofelliptic regularity, which may be used to prove it. Regularity of eigenfunctions isirrelevant: fibers are the orbits of the vector field of their tangent vectors, which areinverses of the vertical vector under DF ( u ), and necessarily lie in A ([ ? ]). Tangentvectors are indeed eigenfunctions φ p ( u ) of DF ( u ) at critical points u .The fact that sheets and fibers are uniformly flat and steep (Proposition 1) allowsone to modify vertical spaces ever slightly and still obtain space decompositions forwhich the Lyapunov-Schmidt decomposition, and hence the construction of fibersin Theorem 3, apply. In particular, transplants may be performed even when theeigenvector φ p originally used to define the vertical spaces V X = V Y do not haveregularity, i.e., do not belong to A ⊂ X . We only have to require that A is densein X , so that φ p can be well approximated by a new vertical direction. Finite spectral interaction is a very convenient context for numerics. Any questionrelated to solving F ( u ) = g for some fixed g ∈ Y reduces to a finite dimensionalproblem in situations of finite spectral interaction, irrespective of additional hy-potheses. If the interaction involves a simple eigenvalue λ p , one simply has to lookat the restriction of F to the (one dimensional) fiber associated to the affine verticalline through g .Smiley and Chun realized the implications of this fact for numerical analysis([31], [32]). An implementation for functions F ( u ) = − ∆ u − f ( u ) defined on rect-angles Ω ⊂ R was presented in [8]. In the forecoming sections, we will requiremore stringent hypotheses with the scope of obtaining very well behaved functions F — we will mostly be interested in global folds. Such additional restrictions mightimprove on computations, but so far this has not seen to lead to substantial im-provements on the available algorithms. −100 −80 −60 −40 −20 0 20 40 60 80 1005101520253035 x f ’ ( x ) −100 −80 −60 −40 −20 0 20 40 60 80 100−19−18−17−16−15−14−13−12−11−10−9 u1 F We present an example obtained from programs by Jos´e Cal Neto ([8]) andOtavio Kaminski. For Ω = [0 , × [0 , λ ∼ .
337 and λ ∼ . − u xx − u yy − f ( u ) = g , ( x, y ) ∈ Ω , u = 0 in ∂ Ω ,f ′ ( x ) = λ − λ π (cid:0) arctan( x
10 ) − x e − ( x/ (cid:1) + λ , f (0) ∼ . ( x, y ) = − (cid:0) x ( x − y ( y − (cid:1) −
35 sin( πx ) sin( πy . On the left, we show the graphs of f ′ , which interacts only with λ . On the right,the height function h associated to the fiber obtained by inverting the vertical linethrough g . The height value − . λ of the Laplacian with Dirichlet conditions). F on fibers and vertical lines We stick to one dimensional fibers and consider two issues.1. How does F behave at infinity along fibers?2. How do fibers look like at infinity ?The first question, to say the very least, is tantamount to characterizing theimage of F . The second is not relevant for the theoretical study of the globalgeometry of F , since a (global) coordinate system leading to a normal form (like( z, t ) ( z, − t )) is insensitive to the shape of fibers. On the other hand, fornumerical purposes, a uniform behavior at infinity of the fibers is informative. F along fibers The inverse of a vertical line z + V Y , z ∈ H Y is the fiber u ( z , t ) = w ( z , t ) + tφ p : F ( u ( z , t )) = z + h a ( z , t ) φ p . ( ∗ )9or a fixed z ∈ H Y , the C map t h a ( z , t ) is the adapted height function of thefiber associated to z . Clearly, h a ( z , t ) = h F ( u ( z , t )) , φ p i = h L ( w ( z , t ) + tφ p ) − N ( u ( z , t )) , φ p i so that h a ( z , t ) = λ p t − h N ( u ( z , t )) , φ p i . In order to havelim t →±∞ h F ( u ( z , t )) , φ p i = lim t →±∞ h a ( z , t ) = −∞ and some uniformity convenient to obtain properness as discussed in Section 6.4,we require an extra hypothesis:For each z ∈ X , there is a ball U ( z ) ⊂ X and ǫ, T > , c ± such that, for z ∈ U ( z ), h N ( u ( z, t )) , φ p i > ( λ p + ǫ ) t + c + , for t > T , ( V +) h N ( u ( z, t )) , φ p i > ( λ p − ǫ ) t + c − , for t < − T . ( V − )Notice that the asymptotic behavior on each fiber is the same. Again, parameterize fibers as u ( z, t ) = w ( z, t ) + tφ p . Under mild hypotheses, thevectors w ( z, t ) /t have a limit for t → ±∞ , which is independent of z . A version ofthis result was originally obtained by Podolak ([27]). Proposition 6
Suppose that F : X → Y , F = L − N satisfies hypothesis ( H ) ofSection 3. Suppose also that, for every u ∈ X , lim t → + ∞ P N ( tu ) t = N ∞ ( u ) ∈ Y. Then there exist w + , w − ∈ H X such that, for every fiber u ( z, t ) = w ( z, t ) + tφ p , lim t → + ∞ k w ( z, t ) t − w + k X = 0 , lim t →−∞ k w ( z, t ) t − w − k X = 0 which are respectively the unique solutions of the equations Lw − P N ∞ ( w + φ p ) = 0 , Lw + P N ∞ ( − w − φ p ) = 0 . It turns out that N ∞ = P N ∞ satisfies the same Lipschitz bound that the func-tions P N t in Theorem 3, which is why both equations are (uniquely) solvable.Fibers are asymptotically vertical if and only if lim | t |→∞ w ( z, t ) /t = 0, or equiv-alently, P N ∞ ( ± φ p ) = 0. Indeed, in this case, w = 0 is the unique solution ofboth equations. This is what happens in the Ambrosetti-Prodi scenario, where P N ∞ ( u ) = ( b − γ ) P u + − ( a − γ ) P u − (recall u = u + − u − ), since φ p = φ > F on fibers and on vertical lines One might wish to relate the heights of F along fibers and vertical lines, which areeasier to handle. In [27] Podolak presented a scenario in which this is possible. Westate a version of her result for the case t → + ∞ . Theorem 4
Let X ⊂ Y be Hilbert spaces with X dense in Y . Let L : X → Y be a self-adjoint operator with ∈ σ ( L ) , a simple, isolated eigenvalue, associatedto the normalized kernel vector φ p . Set H Y = h φ p i ⊥ . Take N : Y → Y and F = L − N : X → Y so that . k N ( u ) − N ( u ) k Y ≤ ǫ k u − u k Y , lim t → + ∞ N ( tu ) /t = N ∞ ( u ) h N ∞ ( φ p ) , φ p i = − lim t → + ∞ h F ( tφ p ) , φ p i /t > ǫ k (cid:0) L | H Y (cid:1) − k < / , ǫ k (cid:0) L | H Y (cid:1) − k < / h N ∞ ( φ p ) , φ p i .Then, for each fiber ( z , t ) in adapted coordinates, (cid:12)(cid:12) lim t → + ∞ h a ( z , t ) t − h N ∞ ( φ p ) , φ p i (cid:12)(cid:12) < h N ∞ ( φ p ) , φ p i . The number h N ∞ ( φ p ) , φ p i gives the asymptotic behaviour of the height of F along the vertical line through the origin. The theorem implies that F along theupper part of each fiber converges to the same infinity that F along { tφ p , t ≥ } .A context in which these hypotheses apply is the Ambrosetti-Prodi operatorwith a piecewise nonlinearity f ( u ) = ( λ p + c ) u + − ( λ p − c ) u − for a sufficiently smallnumber c >
0. However, for pairs ( λ p − c , λ p + c ) , p = 1 in the Fuˇcik spectrum ofthe (Dirichlet) negative second derivative, for which necessarily c = c (near λ p ),the condition involving ǫ does not hold and indeed the thesis is not true. F From a more theoretical point of view, fibers circumvent the fundamental issue ofdeciding if F is proper. For example ([25]), the map F : C ( S ) → C ( S ) , u u ′ + arctan( u )is a diffeomorphism from the domain to the open region between two parallel planes, (cid:8) y ∈ C ( S ) , − π < Z π y ( θ ) dθ < π (cid:9) . Indeed, fibers in this case are simply lines parallel to the vertical line of constantfunctions, and each is taken to such region.Perhaps, it would be more appropriate to think of fibers as a tool to showproperness ([ ? ]). As far as we know, for the Ambrosetti-Prodi map F : X → Y inunbounded domains, the properness has been proved only by making use of fibers(see Section 8). Proposition 7
The map F : X → Y satisfying hypotheses ( H ) of Section 3 and ( V ± ) above is proper if and only if the restriction of F to each fiber is proper. Points in the Fuˇcik spectrum of the (Dirichlet) second derivative give rise tomaps F which take the half-fiber { u (0 , t ) , t ≥ } to a single point 0 ([34]), whichshows that F is not proper, although the image of every vertical line has its verticalcomponent taken to infinity.A possible definition of a topological degree for F becomes innocuous — therelevant information is essentially the asymptotic behavior of F along each fiber. Generic singularities both of F and of each height function are very special — theyare Morin singularities. Morin classified generic singularities of functions from R n to R n whose derivative at the singularity has one dimensional kernel ([26]). Thisis sufficient for the study of critical points of height functions on one dimensionalfibers, by Proposition 3. In order to do the same for the critical points of thewhole function F : X → Y , we need an equivalent classification for singularities offunctions between infinite-dimensional spaces, which is very similar ([11], [25], [28])— this is how we proceed next. 11 .1 Morin theory in adapted coordinates The first step in Morin’s proof makes use of the implicit function theorem to writesuch a singularity at a point ( z , t ) in adapted coordinates, as in Section 4: F a : Y = H Y ⊕ V Y → Y = H Y ⊕ V Y , ( z, t ) ( z, h a ( z, t )) . Say F a is C k +1 . The point ( z , t ) is a Morin singularity of order k if and only if1. D t h a ( z , t ) = · · · = D kt h a ( z , t ) = 0, D k +1 t h a ( z , t ) = 0.2. The Jacobian D ( h a , D t h a , . . . , D k − t h a )( z , t ) has maximum rank.Then, in a neighborhood of ( z , t ) there is an additional change of variables whichconverts F a to the normal form(˜ z, x, t ) (˜ z, x, t k +1 + x t k − + · · · + x k − t ) . Here the coordinates (˜ z, x ) correspond to an appropriate splitting of Y = ˜ Y ⊕ R k − .Morin singularities of order 1, 2, 3 and 4 are called, respectively, folds, cusps,swallowtails and butterflies.Thus, the classification of critical points of F boils down to the study of afamily of one dimensional maps, the height functions restricted on fibers. Thefirst requirement is specific to each fiber (i.e., one checks it for every fixed z near z ), whereas the second relates nearby fibers, i.e., one has to change z . Folds arestructurally simpler than deeper singularities: the behavior along fibers near a foldpoint is always the same — essentially like t
7→ − t , whereas this is not the case forcusps, where close to t t one finds t t ± ǫt .There is something unsatisfying in the fact that the relevant properties of thecritical points of F requires knowledge of some version of the height function. Thisis circumvented by the next result ([ ? ]). Proposition 8
Suppose F : X → Y is C k +1 and admits one dimensional fibers.Then there is an open neighborhood U of the critical set C with the properties below.1. There is a unique C k map λ p : U → R for which λ p = 0 on C and is aneigenvalue of DF elsewhere.2. There is a strictly positive C k function p : U → R + such that λ p ( u ( z, t )) = p ( u ( z, t )) D t h ( u ( z, t )) , u ( z, t ) ∈ U .
A point u = u ( z , t ) is a Morin singularity of order k of F if and only if1. λ p ( u ) = · · · = D k − t λ p ( u ) = 0 , D kt λ p ( u ) = 0 ,2. The image of D ( λ p , . . . , D k − t λ p )( u ) together with D t λ p ( u ) span R n . There is an analogous characterization in adapted coordinates.
Consider a critical point u ∈ C ⊂ X and the fiber u ( z , t ) through it, u ( z , t ) = u .From Proposition 8, u is a (topological) fold of the height function h restricted tothe fiber if and only if u is a topologically simple root of λ p ( u ) along the fiber, i.e., λ p is strictly negative on one side of u and strictly positive on the other.12nce we reduce the issue to checking an eigenvalue along a fiber, derivatives areirrelevant : just study the quadratic form of the Jacobian. Clearly, this only handlestopological equivalence between the function and a fold.More explicitly, in standard Ambrosetti-Prodi contexts, λ ( u ) is the minimumvalue of the quadratic form h DF ( u ) v, v i . The derivative D t u ( z , t ) of the ( C )fiber is the eigenfunction φ ( u ) >
0, and it is easy to check that λ increases with t by the convexity of the nonlinearity f . This should be compared with differentia-bility arguments, which require some estimate on φ ( u ) (say, boundedness).The fact that all critical points are local maxima for height functions on fibers, asrequired in Proposition 4, suggest hypotheses to be checked only on the critical setof F . This is not the case in the original Ambrosetti-Prodi theorem: the statementof the theorem has the merit that it makes no reference to the critical set at all,an object which in principle is hard to identify. The convexity of the nonlinearityhandles the difficulty and, rather surprisingly, is essentially necessary ([ ? ]). Furtherexamples yielding local maximality are somewhat contrived. The geometric formulation F = L − N is not sufficient to accomodate situationsof the form F ( u ( x )) = − ∆ u ( x ) + f ( x, u ( x )), the so called non- autonomous case.Hammerstein ([16]) had already considered homeomorphisms of that form. A pos-sibility is requiring that X and Y are function spaces defined on a domain Ω, sothat the variable x makes sense. The formalism above carries over to this scenariowithout surprises.More precisely, as usual X and Y are Hilbert spaces, X dense in Y . The linearoperator L : X ⊂ Y → Y is self-adjoint with a simple eigenvalue λ p associated to anormalized eigenvector φ p . Let P : Y → H Y = h φ p i ⊥ be the orthogonal projection.From the nonlinear term N : Ω × Y → Y , define as before P N t : H Y → H Y , t ∈ R and require a Lipschitz estimate, k P N t ( x, w ) − P N t ( x, w ) k Y ≤ n k w − w k Y , for w , w ∈ H Y , so that [ − n, n ] ∩ σ ( L ) = { λ p } , which is the same hypothesis ( H ) in Section 3. Thisobtains fibers for F : X → Y as in Theorem 3, which satisfy the same properties asthose in the autonomous case, in particular, Proposition 1.The hypothesis which obtain appropriate asymptotic behavior of F along fibersare the obvious counterparts of ( V +) and ( V − ) in Section 6.1. For the classificationof critical points, we simply do not distinguish between the autonomous and non-autonomous case: the subject has become a geometric issue. R n As was surely known by Ambrosetti and Prodi (and [2] is an interesting exam-ple), the Laplacian with Dirichlet conditions might be replaced by more generalself-adjoint operators. The approach in this text is flexible enough to handle non-linear perturbations of Schr¨odinger operators on unbounded domains yielding globalfolds. In our knowledge there are no similar results in the literature. Tehrani ([33])obtained counting results for Schr¨odinger operators in R n in the spirit of thoseobtained by Podolak ([27]), indicated in Section 5.1 .We state the by now natural hypotheses. Here Y = L ( R n ).1. The free operator T = − ∆ + v ( x ) : X ⊂ Y → Y is self-adjoint, with simple,isolated, smallest eigenvalue λ and positive ground state φ .13. F : X ⊂ Y → Y, F ( u ) = T u − f ( u ) is a C map.3. The function f ∈ C ( R ) satisfies f (0) = 0, M ≥ f ′′ > f ′ ( R ) = ( a, b ) and a < λ < b < min { σ ( T ) \ { λ }} .4. The Jacobians DF ( u ) : X → Y are self-adjoint operators with eigenpair( λ ( u ) , φ ( u )) sharing the properties of ( λ , φ ). Theorem 5
Under these hypotheses, the map F : X → Y is a global fold. Such hypotheses are satisfied for v ( x ) = x /
2, the one dimensional quantumharmonic oscillator, as well as for the hydrogen atom in R , for which v ( x ) = − / | x | .Hypotheses on the potential of a Schr¨odinger operator in order to obtain suchproperties are commonly studied in mathematical physics. The interested readermight consider [7], [20], [29]. More about this in [ ? ]. We recall Mandhyan’s second example of a global fold ([22]), or better, a specialcase of the extension given by Church and Timourian ([12]).For Ω ⊂ R n a compact subset, let X = C (Ω) and define the compact operator K : X → X, K ( u )( x ) = Z Ω k ( x, y ) u ( y ) dy where the kernel k ∈ C (Ω × Ω) is symmetric and positive. Let µ > µ be thelargest eigenvalues of K . Now let f : R → R be a strictly convex C functionsatisfying 0 < lim x →−∞ f ′ ( x ) < /µ < lim x →∞ f ′ ( x ) < / | µ | . Theorem 6
Under these hypotheses for K and f , the map G : X → X, G ( u )( x ) = u ( x ) − Kf ( u ( y )) is a global fold. This is the kind of nonlinear map obtained if one started from the Ambrosetti-Prodi original operator F ( u ) = − ∆ u − f ( u ) and inverted the Laplacian. Actually,one could take another track: instead of inverting the linear part, one might considerthe inversion of the nonlinear map u f ( u ), since f ′ is bounded away from zero.For maps G ( u ) = Ku − f ( u ) obtained this way, we handle the case when K is ageneral compact symmetric operator K .More precisely, let Ω ⊂ R n , B = C (Ω) and Y = L (Ω). Let K : B → B and K : Y → Y be compact operators which preserve the cone of positive functions.Also, K : Y → Y has simple largest eigenvalue λ p = k K k and second largesteigenvalue λ s . Let f : R → R be a strictly convex C function, with f (0) = 0 if Ωis unbounded. Suppose λ s < a = lim t →−∞ f ′ ( t ) < λ p < b = lim t →∞ f ′ ( t ) . Theorem 7
The map F : B → B , F ( u ) = Ku − f ( u ) is a global fold. The reader should notice that F is Lipschitz but not differentiable as a mapfrom L (Ω) to itself. Still, the direct construction of fibers in C (Ω) is not a simplematter, because properness of F is not immediate. Transplanting fibers in thisexample is convenient, and was also used in Mandhyan’s context.14 .4 Folds as perturbations of non-self-adjoint operators McKean and Scovel ([23], [12]) studied the Riccati-like map on functions u ∈ L ([0 , u + ( D ) − f ( u ) ∈ L ([0 , , f ( x ) = x / , where ( D ) − is the inverse of the second derivative acting on W , ([0 , f , F : C ( S ) → C ( S ) , u u ′ + f ( u ) . McKean and Scovel ([23]) and Kappeler and Topalov ([18]) considered the samemap among Sobolev spaces, the celebrated
Miura map , used as a change of variablesbetween the Korteweg-deVries equation and its so called modified version.More recently, a perturbation of a non-self-adjoint elliptic operator (as in [3],but with Lipschitz boundary) has been shown to yield a global fold ([30]).
The first author thanks the Departamento de Matem´atica, PUC-Rio, for its warmhospitality. The second and third authors gratefully acknowledge support fromCAPES, CNPq and FAPERJ. We thank Jos´e Cal Neto and Otavio Kaminski forthe numerical examples.