Geometric Aspects of Ambrosetti-Prodi operators with Lipschitz nonlinearities
aa r X i v : . [ m a t h . A P ] J un Geometric aspects of Ambrosetti-Prodi operatorswith Lipschitz nonlinearities
Carlos Tomei and Andr´e ZaccurDepartamento de Matem´atica, PUC-Rio ∗ Dedicated to Bernhard, with affection and admiration
Abstract
Let the function u satisfy Dirichlet boundary conditions on a boundeddomain Ω. What happens to the critical set of the Ambrosetti-Prodi operator F ( u ) = − ∆ u − f ( u ) if the nonlinearity is only a Lipschitz map? It turns outthat many properties which hold in the smooth case are preserved, despiteof the fact that F is not even differentiable at some points. In particular, aglobal Lyapunov-Schmidt decomposition of great convenience for numericalsolution of F ( u ) = g is still available. Keywords:
Semilinear elliptic equations, Ambrosetti-Prodi theorem.
MSC-class:
A familiar set of hypotheses for the celebrated Ambrosetti-Prodi theorem is thefollowing. Let Ω ⊂ R n be an open, bounded, connected domain with smoothboundary ∂ Ω and denote by 0 < λ < λ ≤ . . . the eigenvalues of the free DirichletLaplacian − ∆ on Ω. Let f : R → R be a smooth, strictly convex function, withasymptotically linear derivative so thatRan f ′ = ( a, b ) , a < λ < b < λ . Under such hypotheses, the theorem states that the equation F ( u ) = − ∆ u − f ( u ) = g, u | ∂ Ω = 0 (1)for, say, g ∈ C ,α (Ω), has (exactly) zero, one or two solutions in C ,α (Ω). C and F ( C ) In the original arguments ([1], [11]), a fundamental role is played by the criticalset C of F : X = C ,αD (Ω) → Y = C ,α (Ω). Here C ,αD (Ω) is the subspace offunctions of C ,α (Ω) satisfying Dirichlet boundary conditions. The proof follows afew steps, which follow from the inverse function theory and the characterization offold points.1. The critical set C ⊂ X is a hypersurface; every critical point is a fold. ∗ R. Mq. S. Vicente 225, Rio de Janeiro 22451-900, Brazil. email: [email protected]. F is proper, its restriction to C is injective and F − ( F ( C )) = C .3. The spaces X − C and Y − F ( C ) have two components. Each component of X − C is taken injectively to the same component of Y − F ( C ). Berger and Podolak came up with a different approach [2], which is easier to phrasefor F : X → Y between Sobolev spaces, X = H (Ω) and Y = H − (Ω) ≃ H (Ω).Their main result is the construction of a global Lyapunov-Schmidt decompositionfor F . Let ϕ be the (positively normalized) eigenfunction associated to the groundstate λ , set V X = V Y = h ϕ i and consider the orthogonal decompositions X = V X ⊕ W X and Y = V Y ⊕ W Y into vertical and horizontal subspaces. With a different vocabulary, their proofessentially goes through the verification of the following properties, by making useof spectral estimates on the Jacobians DF ( u ) : X → Y .1. Horizontal affine subspaces of X are taken by F to sheets .2. The inverse under F of vertical affine subspaces of Y are fibers .3. Sheets are essentially flat , fibers are essentially steep .An affine horizontal subspace of X is a set of the form x + W X , for a fixed x ∈ X ;an affine vertical subspace of Y is of the form y + V Y , for y ∈ Y . Sheets are graphsof smooth functions from W Y to V Y and fibers are graphs of smooth functions from V X to W X . The third property states that the inclination of the tangent spaces tosheets, with respect to horizontal subspaces, and to fibers, with respect to verticalsubspaces, is uniformly bounded from above.In the words of [4], F is a flat map. In the Ambrosetti-Prodi case, verticalspaces and fibers are one dimensional. More generally, the dimension equals thenumber k of eigenvalues of − ∆ D in the set f ′ ( R ) (we suppose non-resonance, i.e.,the asymptotic values of Ran f ′ are not eigenvalues) and similar results still hold. There is an interesting bonus obtained from considering this global Lyapunov-Schmidt decomposition. For many nonlinearities f , the related nonlinear map F : X → Y, F ( u ) = − ∆ u − f ( u )is not proper, and part of the technology related to degree theory simply breaksdown. The Ambrosetti-Prodi hypothesis yield properness of F , but it is not reallyessential to most of what we want to do. The first section of the paper is dedicatedto explicit examples of Lipschitz nonlinearities for which properness does not occur— there are functions g ∈ Y for which F − ( g ) contains a full half-line of functionsin X . From such examples, we may obtain smooth nonlinearities with the sameproperty, but we give no details.The results in the paper indicate that in most directions, properness holds.What we mean by this is something similar to the fact that even when a map F does not have an invertible Jacobian DF ( u ) at a point u , it may still have asubspace L ( u ) on which the restriction of DF ( u ) acts injectively — this is howbifurcation equations come up: they concentrate in a possibly small subspace S ( u )transversal to L ( u ) the difficulties which are not resolved by the linearization at u . What we shall see is that the domain X of F splits into special (nonlinear)2urfaces of finite dimension k , on which properness may break down, which have fortangent spaces the subspaces S ( u ) when u is critical, but on transversal directionsto these surfaces, the horizontal affine subspaces, properness is always present. Aswe shall see, counting or computing solutions of the differential equation F ( u ) = g boils down to a finite dimensional problem, simplifying both (abstract) analysis andnumerics. F ( u ) = g Smiley and Chun ([12],[13]) showed that an analogous global Lyapunov-Schmidtdecomposition exists for appropriate non-autonomous nonlinearities f ( x, u ( x )) andemphasized its relevance for numerical analysis: one might solve F ( u ) = g byrestricting F to the fiber α g containing F − ( g ). In [4], this project is accomplished:given a right hand side g , one first obtains numerically a point in α g , which inthe Ambrosetti-Prodi case is a curve, and then proceeds to search for solutionsby moving along it. The algorithm is sufficiently robust to handle more flexiblenonlinearities: it does not require convexity of f or properness of F , and the rangeof f ′ may include other (finite) sets of eigenvalues of − ∆ D . Now, what happens when f is not smooth anymore, but, say, Lipschitz? In par-ticular, this is the scenario considered in the proof of the so called one dimen-sional Lazer-McKenna conjecture ([9], [10]) by Costa, Figueiredo and Srikanth in[3]. We state the result, for the reader’s convenience. Let X = H D ([0 , π ]) be theSobolev space of functions satisfying Dirichlet boundary conditions with square in-tegrable second derivatives. Recall that u
7→ − u ′′ acting on X to Y = L ([0 , π ]) haseigenvalues λ k = k , k = 1 , , . . . with corresponding eigenfunctions sin( kx ). Take f : R → R , a strictly convex smooth function f with asymptotic values a and b forits derivative f ′ satisfying a < , λ k = k < b < ( k + 1) = λ k +1 . Then the equation F ( u ) = − u ′′ − f ( u ) = − t sin x , u (0) = u ( π ) = 0, has exactly 2 k solutions for t > f given by ˜ f ′ ( x ) = a or b , de-pending if x < x >
0. The related operator ˜ F ( u ) = − u ′′ − ˜ f ( u ) now requiressome care: it stops being differentiable everywhere and the usual differentiable nor-mal forms at regular points and folds break down.In this paper, we show that when f is merely Lipschitz, for appropriate condi-tions on the boundary of Ω, the operator F is still flat, in the sense that the globalLyapunov-Schmidt decomposition still holds, and sheets and fibers are still availableas graphs of Lipschitz functions. A word of caution: piecewise linear nonlinearitiesmay yield continua of points on which the map F takes a unique value. Such setsnecessarily lie in a single fiber. More, the numerical analysis of the PDE F ( u ) = g presented in [4] is still valid, after minor modifications.We take this material to be an intermediate step towards a more geometricdescription of the operators of Hamilton-Jacobi-Bellman type, as studied by Felmer,Quaas and Sirakov in [7].The authors gratefully acknowledge support from CAPES, CNPq and FAPERJ.3 Some cautionary examples
For a box Ω ⊂ R n , we consider again the differential equation F ( u ) = − ∆ u − f ( u ) = g, u | ∂ Ω = 0 (2)with a Lipschitz nonlinearity f ( x ). Once f is piecewise linear, there may be whole(straight line) segments on the domain restricted to which F is actually constant.This happens already for the one-dimensional case. Let ϕ be the (positive) groundstate associated to eigenvalue λ and take a < λ < b . Suppose that Ran ϕ =[0 , M ] and define f ( x ) to be continuous, with derivatives equal to a , λ and b in theintervals [ −∞ , , [0 , M ] and [ M, ∞ ]: clearly, F ( tϕ ) = 0, for t ∈ [0 , f with derivatives taking two values, n = 1 In a similar vein, we now provide examples of segments on which F is constant forthe nonlinearity f ( x ) = ax or bx , for x < x >
0, with the property that, forspecial values of a and b , there are right hand sides g ( g ≡ F − ( g ) contains a (straight) half-line of solutions. In particular,the map F : X = H (Ω) ∩ H (Ω) → L (Ω) is not proper and the equation hassolutions which are not isolated.Figure 1: A half-line of solutionsWe begin with the one dimensional case, n = 1. Set Ω = I = [0 , π ]. Split I into k closed intervals I i , i = 1 , .., k joined at their ends, of two different lengths, I odd i = β , I even i = α . In the figure, k = 4. The smallest eigenvalues for theoperator u
7→ − u ′′ with Dirichlet conditions in an interval of sizes β and α arerespectively λ β = (cid:0) πβ (cid:1) and λ α = (cid:0) πα (cid:1) , with positive (normalized) eigenfunctions ϕ β and ϕ α . We set b = λ β and a = λ α andconstruct a solution ψ by juxtaposing multiples of ϕ β and ϕ α as shown in the figure.On I , one may take pϕ β , for arbitrary p >
0. On I , the (negative) multiple of ϕ α isdetermined by matching the first derivative — recall that ψ ∈ X = H ( I ) ∪ H ( I ),so ψ ′ is absolutely continuous. The procedure extends to the remaining intervalsin a unique fashion. We have to make sure that the total length of the intervals I i equals π . Thus, for example, in the simplest case k = 2, we must have β + α = π ⇐⇒ √ b + 1 √ a = 1 . For any value a ∈ (1 , b , which turns out to be in (4 , ∞ ) whichsolves this equation. Said differently: any interval [ a, b ] containing λ = 4 for which a, b are not eigenvalues of the free problem admits a half-line of solutions of theequation above. For different numbers of intervals, one shows half-lines of solutionsfor any a ∈ ( λ k , λ k +1 ) and appropriate b ∈ ( λ k +1 , ∞ ).4las, the only situation for which this argument does not provide a half-line ofsolutions F − (0) is a < λ , the Ambrosetti-Prodi case. There are strong evidencesthat in this case there are no continua of solutions F − ( g ), but we have no realproof.Clearly, one may replace g ≡ g defined piecewise on intervals I i asfunctions in the range of u
7→ − u ′′ − f ( u ) (Dirichlet conditions on I i ) acting onpositive functions restricted to I i . Thus, for k intervals, the set of such g is avector subspace of L ( I ) of codimension k . This construction ascertains that g isin the range of F (now considered in the full interval I ), so that g = F ( u ). Bylinearity on each interval I i , adding an homogeneous solution ψ ∈ F − (0) gives riseto u + ψ ∈ F − ( g ).These ideas also suffice to prove that there is no nontrivial function in [0 , π ]which is taken to 0 if a < λ < b . n We now consider the case n = 2, the general situation being similar. We now haveΩ = I × I (rectangles would work also): just separate variables and proceed. Let ψ ( y ) as before, solving − ψ yy − f ( ψ ) = 0, where f is constructed from appropriate a and b , and let ϕ ( x ) ≥ I , so that − ϕ xx = ϕ ( x ). The product ˜ ψ ( x, y ) = ϕ ( x ) ψ ( y ) satisfies − ˜ ψ xx − ˜ ψ yy − f ( ˜ ψ ) = ˜ ψ + ϕ ( x )( − ψ yy − f ( ψ )) = ˜ ψ so that ˜ ψ and its positive multiples solve − u xx − u yy − ˜ f ( u ) = 0 , u | ∂ Ω = 0 , for ˜ f ( x ) = f ( x ) + x . Set Y = L (Ω) with inner product h u, v i = R Ω uv and norm k u k . Also let X = H (Ω) ∩ H (Ω), with inner product h u, v i = h− ∆ u , − ∆ v i and norm k u k .We always consider sets Ω for which − ∆ : X ⊂ Y → Y is a self-adjoint isomor-phism — we call such domains Ω appropriate . Also the same operator shouldhave C ∞ (Ω) as a core, i.e. it is essentially self-adjoint in this domain. Fromthe spectral theorem, there is an orthonormal basis of (Dirichlet) eigenfunctions ϕ i ∈ X, k ϕ i k = 1, satisfying − ∆ ϕ i = λ i ϕ i . Eigenfunctions associated to differenteigenvalues are orthogonal with respect to both inner products. Concretely, onemight take Ω to be a convex set or require ∂ Ω to be C , ([12],[6]). Notice that,from standard results in spectral theory, operators T : X ⊂ Y → Y, T u = − ∆ u − qu, for bounded real potentials q , are still self-adjoint with an orthonormal basis ofeigenfunctions.We assume that the nonlinearity f : R → R is Lipschitz, and f ′ takes values inan interval [ a, b ] with the property that the bounds a and b are not eigenvalues λ i .For this part of the paper, we make no assumptions about convexity for f . Noticethat a and b do not have to be the asymptotic values of f ′ , a degree of freedomwhich is convenient for numerical analysis.For starters, F : X → Y given by F ( u ) = − ∆ u − f ( u ) is a well defined map —it suffices to check that f ( u ) ∈ Y . This follows from the easy lemma below.5 emma 1 Say f : R → R is Lipschitz. Then the map ˆ f : Y = L (Ω) → Y givenby ˆ f ( u ) = f ◦ u is also well defined and Lipschitz with the same constant. Proof:
Take first u, v continuous functions. Since f is M -Lipschitz, it is absolutelycontinuous, so that | f ( u ( x )) | = | f (0) + u ( x ) Z f ′ ( tu ( x )) dt | ≤ | f (0) | + M | u ( x ) | , x ∈ Ω , and, since Ω is bounded, we have f ( u ) ∈ Y . Similarly, applying the fundamentaltheorem of calculus to the function ϕ ( t ) = f ( tu ( x ) + (1 − t ) v ( x )), one obtains k f ( u ( x )) − f ( v ( x )) k ≤ Z | f ′ ( tu ( x ) + (1 − t ) v ( x )) | dt k u ( x ) − v ( x ) k . Now take Cauchy sequences of continuous functions converging to arbitrary func-tions in Y : the estimates above extend to the required L estimates. (cid:4) F v : W X → W X is an isomorphism We now describe an orthogonal decomposition of X and Y . Take Ω ⊂ R n to be abounded appropriate domain and let Λ f = { λ i } i ∈ I be the set of eigenvalues λ i in( a, b ). The vertical subspaces V X = V Y equal the invariant subspace associated toΛ f and V X ⊂ X, V Y ⊂ Y . The horizontal subspaces are W X = V ⊥ X ⊂ X and W Y = V ⊥ Y ⊂ Y where orthogonality takes into account the (different) inner products in X and Y . These induce orthogonal decompositions X = W X ⊕ V X , Y = W Y ⊕ V Y andcorresponding orthogonal projections P Y and Q Y from Y to W Y and V Y . Finally,we consider affine horizontal subspaces in X , which are sets of the form x + W X ,for a fixed x , and affine vertical subspaces in Y , of the form y + V Y .We need a label for this construction: a nonlinearity f induces an I -decomposition X = W X ⊕ V X , Y = W Y ⊕ V Y associated to bounds a and b . Theorem 1
Let Ω be an appropriate domain, f : R → R Lipschitz,
Ran f ′ ⊂ [ a, b ] ,where a and b are not eigenvalues λ i . and X = W X ⊕ V X , Y = W Y ⊕ V Y be the I -decomposition specified above. For v ∈ V X , let F v : W X → W Y be the horizontalprojection of the restriction of F to the affine subspace v + W X , F v ( w ) = P Y F ( w + v ) .Then F v is a bi-Lipschitz homeomorphism. The Lipschitz constants for F v and F − v are independent of v . Proof:
For γ = ( a + b ) /
2, set ˜ f ( x ) = f ( x ) − γx . Then T : W X → W Y given by u → − ∆ u − γu is well defined and invertible, with eigenvalues λ j − γ with j / ∈ I .Let λ m − γ be the eigenvalue of T of smallest absolute value: clearly, || T − || = | λ m − γ | − > | a − γ | = b − γ. For u = w + v, w ∈ W X , v ∈ V X , we have F v ( w ) = P Y [ − ∆( w + v ) − f ( w + v )] = T w − P Y ˜ f ( w + v ) . The composition F v ◦ T − : W Y → W Y is of the form I − K v , where K v ( w ) = P Y ˜ f ( T − w + v ). We show that K v : W Y → W Y is a contraction with constantuniformly bounded away from 1. For w, ˜ w ∈ W Y ⊂ L (Ω), k K v ( w ) − K v ( ˜ w ) k ≤ k ˜ f ( T − w + v ) − ˜ f ( T − ˜ w + v ) k ≤ ( b − γ ) k T − ( w − ˜ w ) k ≤ b − γ | λ m − γ | k w − ˜ w k = c k w − ˜ w k . b is not an eigenvalue λ j , the Lipschitz constant c is uniformly bounded awayfrom 1. From the Banach contraction theorem, F v : W Y → W Y is a homeomorphismand standard estimates using the familiar formula( I − K v ) − = I + K v + K v ◦ K v + K v ◦ K v ◦ K v + . . . show that ( F v ) − is Lipschitz, where the constant depends on c and not on v . (cid:4) In particular, if [ a, b ] does not contain eigenvalues λ i , the result above recoversthe Dolph-Hammerstein theorem for Lipschitz nonlinearities ([5], [8]). We are ready to extend to the Lipschitz context the global Lyapunov-Schmidt de-composition which is known for the smooth case, from the works of Berger andPodolak and Smiley. Consider the following diagram. X = W X ⊕ V X F −→ Y = W Y ⊕ V Y Φ − =( F v ,Id ) ց ր ˜ F = F ◦ Φ=(
Id,φ ) Y = W Y ⊕ V Y Thus the change of variables Φ yields ˜ F ( w, v ) = F ◦ Φ( w, v ) = ( w, φ ( w, v )),from which we will derive some convenient geometric properties. We first clarify atechnicality: Φ is indeed a global change of variables in the Lipschitz category. In W X ⊕ V X , we use the norm obtained by adding the norms in each coordinate.Figure 2: The change of variables Proposition 1
The map
Φ = (( F v ) − , Id ) : Y = W Y ⊕ V Y → X = W X ⊕ V X is abi-Lipschitz homeomorphism. Proof:
The invertibility of Φ follows from the previous theorem. We use someelementary facts. The identity map Id : ( V X , k . k ) → ( V Y , k . k ) between normedspaces of the same finite dimension is bi-Lipschitz. Also, since W X and V X areorthogonal (in L and H ), k w k + k v k ≤ k w ± v k for w ∈ W X and v ∈ V X . Toshow that Φ − is Lipschitz, take w + v, ˜ w + ˜ v ∈ X . For appropriate constants C , ˜ C , k Φ − ( w + v ) − Φ − ( ˜ w + ˜ v ) k = k F v ( w ) − F ˜ v ( ˜ w ) k + k v − ˜ v k ≤ k − ∆( w − ˜ w ) − P Y ( f ( w + v ) − f ( ˜ w + ˜ v )) k + C k v − ˜ v k k w − ˜ w k + k f ( w + v ) − f ( ˜ w + ˜ v ) k + C k v − ˜ v k ≤ ˜ C k w + v − ( ˜ w + ˜ v ) k , where the last inequality follows from Lemma 1.We obtain a Lipschitz estimate for Φ = (( F v ) − , Id ) : W Y ⊕ V Y → W X ⊕ V X .Take z + v, ˜ z + ˜ v ∈ Y = W Y ⊕ V Y . Then k Φ( z + v ) − Φ(˜ z + ˜ v ) k ≤ k F − z ( v ) − F − z ( v ) k + k F − z ( v ) − F − z (˜ v ) k + k v − ˜ v k Again from finite dimensionality of V X = V Y , there is an estimate of the form k v − ˜ v k ≤ C k v − ˜ v k . We also have k F − z ( v ) − F − z (˜ v ) k ≤ C k v − ˜ v k from theproof of the previous theorem. The first term is handled in a similar fashion. (cid:4) The picture should help putting pieces together. Here, dim V X = dim V Y = 1,as in the Ambrosetti-Prodi theorem: the convex span of Ran f ′ ( f is Lipschitz!)contains only the eigenvalue λ . The map F takes an affine horizontal subspace v + W X to a sheet, and the inverse of the vertical affine subspace g + V Y is a fiber,which crosses W X at w (0) and v + W X at v + w ( v ), in the notation of the proofabove. Clearly, sheet and fiber are graphs, as stated above. The change of variablesΦ preserves horizontal affine subspaces and ˜ F preserves affine vertical subspaces.We are ready to prove the fundamental geometric property of such F : X → Y :there are uniformly flat sheets and uniformly steep fibers. Proposition 2
Let F : X = W X ⊕ V X → Y = W Y ⊕ V Y with the hypothesis given inthe beginning of the section. The image of each horizontal affine space v + W X ⊂ X under F is the graph of a Lipschitz function σ v : W Y → V Y . Similarly, the inverseof each vertical affine subspace g + V Y ⊂ Y under F is the graph of a Lipschitzfunction α g : V X → W X . The Lipschitz constant can be taken to be the same, forall v ∈ V X , g ∈ W Y . Proof:
We prove the result for fibers F − ( g + V Y ): the statement for sheets F ( v + W X ) is easier. Clearly, ˜ F − ( g + V Y ) ⊂ g + V Y , which is taken by thechange of variables Φ to a set of the form α g = { ( F v ) − ( g ) + v, v ∈ V X } ⊂ X .From the theorem, for every v ∈ V X , there is a unique w ( v ) ∈ W X for which P Y F ( w ( v )+ v ) = g — thus F − ( g + V Y ) = { ( F v ) − ( g )+ v, v ∈ V X } . Said differently, w ( v ) + v ∈ α g : the set { ( w ( v ) , v ) , v ∈ V X } ⊂ X is the graph of a Lipschitz map.The uniformity (on g ) of the Lipschitz constant of the maps v w ( v ) is respon-sible for the uniform steepness of the fibers. (cid:4) In opposition to the arguments in [2], [13] and [4] for the smooth case, thegeometric statements follow without recourse to implicit function theorems. Noticealso that the uniform flatness of sheets and steepness of fibers are a counterpart to(differential) transversality between fibers and horizontal affine spaces in the domainand between sheets and vertical affine spaces in the counterdomain.The restriction of F to horizontal affine subspaces is injective but the restrictionto fibers α is not. In particular, in the standard Ambrosetti-Prodi case, F restrictedto each fiber is simply the map x ∈ R
7→ − x ∈ R , after global changes of variables.The theorem becomes evident from this fact, first proved in [2].Vertical lines may be taken by F to the horizontal plane, indicating yet anotherrelevant transversality property of the fibers. To see this, take Ω = [ − π/ , π/
2] and F ( u ) = − u ′′ − f ( u ), so that λ = 4 and the corresponding eigenvector ϕ is odd.Set a = 3 and b = 5, split X = W X ⊕ h ϕ i and take f ( x ) = e ( x ) + 4 x , where e ( x )is even (convexity is not necessary!) and Ran f ′ = ( a, b ). By symmetry, we have h F ( tϕ ) , ϕ i = Z Ω ( − t ∆ ϕ − e ( tϕ ) − t ϕ ) ϕ = 0 . Geometry and numerics