Bounded imaginary powers of cone differential operators on higher order Mellin-Sobolev spaces and applications to the Cahn-Hilliard equation
aa r X i v : . [ m a t h . A P ] M a y BOUNDED IMAGINARY POWERS OF CONE DIFFERENTIAL OPERATORSON HIGHER ORDER MELLIN-SOBOLEV SPACESAND APPLICATIONS TO THE CAHN-HILLIARD EQUATION
NIKOLAOS ROIDOS AND ELMAR SCHROHE
Abstract.
Extending earlier results on the existence of bounded imaginary powers for cone differentialoperators on weighted L p -spaces H ,γp ( B ) over a manifold with conical singularities, we show how thesame assumptions also yield the existence of bounded imaginary powers on higher order Mellin-Sobolevspaces H s,γp ( B ) , s ≥ .As an application we consider the Cahn-Hilliard equation on a manifold with (possibly warped)conical singularities. Relying on our work for the case of straight cones, we first establish R -sectoriality(and thus maximal regularity) for the linearized equation and then deduce the existence of a short timesolution with the help of a theorem by Clément and Li. We also obtain the short time asymptotics ofthe solution near the conical point. Introduction
In this article we show the existence of bounded imaginary powers for a class of elliptic differentialoperators on higher order Mellin-Sobolev spaces over manifolds with conical singularities and use this toestablish the existence of maximally regular solutions to the Cahn-Hilliard equation.We model the underlying manifold with conical singularities by a smooth manifold B with boundary, ofdimension n +1 , n ≥ , endowed with a conically degenerate Riemannian metric. In a collar neighborhoodof the boundary, we choose coordinates ( x, y ) with ≤ x < and y ∈ ∂ B . The metric then is assumedto be of the form g = dx + x h ( x ) , (1.1)where x h ( x ) is a smooth family of Riemannian metrics on the cross-section, non-degenerate up to x = 0 . The cone is straight, if this family is constant near x = 0 ; otherwise it is warped.We call a differential operator A of order µ on the interior B ◦ of B a cone differential operator orconically degenerate, if, in local coordinates near the boundary, it can be written in the form A = x − µ µ X j =0 a j ( x, y, D y )( − x∂ x ) j (1.2)with x a j ( x, y, D y ) smooth families, up to x = 0 , of differential operators of order µ − j on the cross-section ∂ B . It is degenerate elliptic or shortly B -elliptic, if the principal pseudodifferential symbol σ µψ ( A ) is invertible on T ∗ B ◦ and, moreover, in local coordinates ( x, y ) near the boundary and correspondingcovariables ( ξ, η ) , the rescaled principal symbol x µ σ µψ ( A )( x, y, ξ/x, η ) = µ X j =0 σ µ − jψ a j ( x, y, η )( − iξ ) j (1.3) Date : June 28, 2018.2000
Mathematics Subject Classification. is invertible up to x = 0 .The Laplace-Beltrami operator with respect to the metric g above is an elliptic cone differential op-erator. In fact, a short computation shows that, in local coordinates near the boundary, it is of theform ∆ = x − (cid:0) ( x∂ x ) − ( n − H ( x ))( − x∂ x ) + ∆ h ( x ) (cid:1) , (1.4)where ∆ h ( x ) is the Laplace-Beltrami operator on the cross-section with respect to the metric h ( x ) and H ( x ) = x∂ x (det h ( x )) / (2 det h ( x )) .Conically degenerate operators act in a natural way on scales of weighted Mellin-Sobolev spaces H s,γp ( B ) , s, γ ∈ R , < p < ∞ , cf. Section 2.2. For s ∈ N , H s,γp ( B ) is the space of all functions u in H sloc ( B ◦ ) such that, near the boundary, x n +12 − γ ( x∂ x ) j ∂ αy u ( x, y ) ∈ L p (cid:16) [0 , × ∂ B , p det h ( x ) dxx dy (cid:17) , j + | α | ≤ s. (1.5)In particular, H ,γp ( B ) simply is a weighted L p -space. The factor p det h ( x ) could be omitted in thedefinition of the spaces; it will be important only for questions of symmetry of the (warped) Laplacian.Sometimes it is necessary to consider a cone differential operator A as an unbounded operator in afixed space H s,γp ( B ) . If A is B -elliptic, it turns out that, under a mild additional assumption, the domainof the minimal extension, i.e. the closure of A considered as an operator on C ∞ c ( B ◦ ) , is D ( A min ,s ) = H s + µ,γ + µp ( B ) , while the domain of the maximal extension is D ( A max ,s ) = { u ∈ H s,γp ( B ) : Au ∈ H s,γp ( B ) } = D ( A min ,s ) ⊕ E , where E is a finite-dimensional space consisting of linear combinations of functions of the form x − ρ log k x c ( y ) with ρ ∈ C , k ∈ N and a smooth function c on the cross-section. It can be chosen independent of s .This result has a long history, see e.g. [2], [18], [21], [26]; the present version is due to Gil, Krainer andMendoza [13].A densely defined unbounded operator A on a Banach space X is said to have bounded imaginarypowers with angle φ ≥ , provided that (i) its resolvent exists in a closed sector Λ θ of angle θ around thenegative real axis and decays there like λ − as λ → ∞ , and (ii) the purely imaginary powers A it , t ∈ R ,satisfy the estimate k A it k L ( X ) ≤ M e φ | t | for a suitable constant M .It was shown in [7], that B -elliptic cone differential operators have bounded imaginary powers on H ,γp ( B ) , provided their resolvent exists in a sector Λ θ as above and has a certain structure. For detailssee Theorem 2 in connection with Remark 6 in the cited article.In the present paper, we show that the same assumptions on the structure of the resolvent alsoguarantee the existence of bounded imaginary powers on higher order Mellin-Sobolev spaces H s,γp ( B ) , s ≥ .This comes as a small surprise. Of course, the boundedness of imaginary powers of elliptic pseudodif-ferential operators on L p ( M ) , where M is a closed manifold, carries over to all Sobolev spaces H sp ( M ) , s ∈ R . But already boundary value problems exhibit a more complicated behavior. Seeley [24] provedthe existence of bounded imaginary powers for certain elliptic boundary value problems in L p . A directcomputation, cf. Nesensohn [19], however, shows that the resolvent to the Dirichlet-Laplacian ∆ Dir , con-sidered as an unbounded operator in the Sobolev space H p (Ω) over a bounded domain Ω with domain D (∆ Dir ) = { u ∈ H p (Ω) : u | ∂ Ω = 0 } , decays only as | λ | − / − / (2 p ) as | λ | → ∞ , so that already the most OUNDED IMAGINARY POWERS OF CONE DIFFERENTIAL OPERATORS 3 basic condition for the boundedness of imaginary powers is violated. See Denk and Dreher [8] for a carefulanalysis of the resolvent decay for boundary value problems.For cone differential operators, Gil, Krainer and Mendoza studied the resolvent decay on higher orderMellin-Sobolev spaces. In [13, Theorem 6.36], they showed that, under certain conditions implyingexistence and O ( | λ | − ) -decay of the resolvent on H ,γ ( B ) for λ in a sector of the complex plane, theresolvent also exists on H s,γ ( B ) . Moreover, it is O ( | λ | M ( s ) ) as | λ | → ∞ in the sector, with a function M ( s ) of at most polynomial growth.It was therefore not to be expected that cone differential operators satisfying the assumptions in [7]would have bounded imaginary powers on H s,γp ( B ) , s ≥ . In fact, our proof reduces the higher ordercase to the case s = 0 with the help of commutator techniques.We use the result on bounded imaginary powers in order to improve earlier work on the Cahn-Hilliardequation on manifolds with conical singularities.The Cahn-Hilliard equation ∂ t u ( t ) + ∆ u ( t ) + ∆ (cid:0) u ( t ) − u ( t ) (cid:1) = 0 , t ∈ (0 , T ); (1.6) u (0) = u , (1.7)is a phase-field or diffuse interface equation which models phase separation of a binary mixture; u denotesthe concentration difference of the components. The sets where u = ± correspond to domains of purephases. Global solvability had been established before, cf. Elliott and Zheng Songmu [11] or Caffarelliand Muler [3]. We are interested in the properties of the solutions caused by the conical singularities.In [20] we studied the case of a straight conical singularity. Our analysis was based on understandingthe Laplacian ∆ associated with the cone metric. We fixed a suitable extension ∆ of ∆ in H ,γp ( B ) andchose the domain of the bilaplacian correspondingly as D (∆ ) = { u ∈ D (∆) : ∆ u ∈ D (∆) } . We could then show the existence of a maximally regular short time solution to the Cahn-Hilliard equation.An important ingredient was the existence of bounded imaginary powers for ∆ established in [7].We extend this in two directions: • For the case of straight conical singularities we consider extensions of ∆ in the higher orderMellin-Sobolev spaces H s,γp ( B ) , s ≥ , and establish higher regularity of the solutions using theabove mentioned result on bounded imaginary powers. • We treat the case, where the manifold has warped conical singularities with the help of R -sectoriality and suitable perturbation results.Our strategy is to prove first maximal regularity for the linearized equation. For manifolds with straightconical singularities, we obtain it from the existence of bounded imaginary powers for the Laplacian bya combination of the above results with [21, Theorem 5.7], cf. Theorem 3.3, below.For manifolds with warped cones, the existence of bounded imaginary powers for the Laplacian is notclear. We instead infer maximal regularity from R -sectoriality and perturbation results by Kunstmannand Weis [17]. Maximal regularity together with a theorem by Clément and Li then implies the existenceof a short time solution to the non-linear equation. We obtain the following result: Theorem.
Let ∆ be the closed extension of the cone Laplacian in H s,γp ( B ) with domain D (∆) = H s +2 ,γ +2 p ( B ) ⊕ C . Then, for small T , the Cahn-Hilliard equation with initial value u has a uniquesolution u in the space W q (0 , T ; H s,γp ( B )) ∩ L q (0 , T ; D (∆ )) . NIKOLAOS ROIDOS AND ELMAR SCHROHE
Here we choose p ≥ dim B , q > and γ slightly larger than dim B / − ; the possible choices are limitedby spectral data for the boundary Laplacian ∆ h (0) . The initial value u necessarily is an element of thereal interpolation space X q = ( D (∆ ) , H s,γp ( B )) /q,q , and the solution belongs to C ([0 , T ]; X q ) .As the domain of ∆ (under a mild additional assumption) equals H s +4 ,γ +4 p ( B ) ⊕ F , where F is afinite-dimensional space of singular functions of the form x q log k x c ( y ) , the theorem gives us informationon the asymptotics of the solution close to the conic point. The asymptotics can be determined ratherexplicitly. For straight cones, they only depend on spectral data of ∆ h (0) ; see Section 4.2 for details. Onwarped cones; they additionally depend on the metric h and its derivatives in x = 0 .In principle, maximal regularity and the theorem of Clément and Li imply solvability even for quasi-linear equations, while the Cahn-Hilliard equation is only semilinear. Still, this equation already exhibitsvery clearly the difficulties that arise as a consequence of the combination of singular analysis with non-linear theory while many computations still can be performed explicitly, so that it seems a good example.Our results are complemented by a recent article by Vertman [28]. He studied the Cahn-Hilliard(and more general) equations on manifolds with edges, showed the existence of short time solutions andobtained results on their short time asymptotics. He works in the L -setting with the analysis based onthe Friedrichs extension of the Laplacian and a microlocal heat kernel construction. In [28, Definition2.2(i)] a spectral condition on ∆ h (0) is imposed which, in the language of this article, corresponds to theassumption that ε > in (4.5). Moreover, warping of the cone is admitted, but condition (ii) of thementioned definition requires the difference to the straight metric to be O ( x ) .This article is structured as follows: In Section 2, we recall basic notions such as bounded imaginarypowers, R -sectoriality and maximal regularity. We prove the existence of bounded imaginary powers inSection 3. Section 4 contains the analysis of the Cahn-Hilliard equation in higher order Mellin-Sobolevspaces for the case of a straight cone metric. The case of warped cones is treated in Section 5.2. Preliminary Results on Parabolic Problems and Mellin Sobolev Spaces
Maximal L p -regularity and parabolic problems. We start with the notion of sectoriality whichguarantees the existence of solutions for the linearized problem. For the rest of the section let X be aBanach space. Definition 2.1.
For θ ∈ [0 , π [ denote by P ( θ ) the class of all closed densely defined linear operators A in X such that S θ = { z ∈ C | | arg z | ≤ θ } ∪ { } ⊂ ρ ( − A ) and (1 + | z | ) k ( A + z ) − k ≤ K θ , z ∈ S θ , for some K θ ≥ . The elements in P ( θ ) are called sectorial operators of angle θ . The Dunford integral allows us to define the complex powers of a sectorial operator for negative realpart. The definition then extends to give arbitrary complex powers, cf. Amann [1, III.4.6.5].
Definition 2.2.
Let A ∈ P ( θ ) , θ ∈ [0 , π [ . We say that A has bounded imaginary powers if there existssome ε > and K ≥ such that A it ∈ L ( X ) and k A it k ≤ K for all t ∈ [ − ε, ε ] . In this case, there exists a φ ≥ such that k A it k ≤ M e φ | t | for all t ∈ R with some M ≥ , and we write A ∈ BIP ( φ ) . We continue with the notion of the R -sectoriality, that is slightly stronger than the standard sectorialityand will guarantee maximal regularity for the linearized problem. OUNDED IMAGINARY POWERS OF CONE DIFFERENTIAL OPERATORS 5
Definition 2.3.
Let θ ∈ [0 , π [ . An operator A ∈ P ( θ ) is called R -sectorial of angle θ if for any choice of λ , ..., λ n ∈ S θ , x , ..., x n ∈ X , and n ∈ N , we have (cid:13)(cid:13) n X k =1 ǫ k λ k ( A + λ k ) − x k (cid:13)(cid:13) L (0 , X ) ≤ C (cid:13)(cid:13) n X k =1 ǫ k x k (cid:13)(cid:13) L (0 , X ) , for some constant C ≥ , called the R -bound, and the sequence { ǫ k } ∞ k =1 of the Rademacher functions. Let A be a closed densely defined linear operator A : D ( A ) = X → X . It is well known that − A generates a bounded analytic semigroup if and only if c + A ∈ P ( θ ) for some c ∈ C and some θ > π/ .For such A , consider the Cauchy problem n u ′ ( t ) + Au ( t ) = g ( t ) , t ∈ (0 , T ) u (0) = u (2.1)in the X -valued L q -space L q (0 , T ; X ) , where < q < ∞ , T > . We say that A has maximal L q -regularity , if for some q (and hence by a result of Dore [9] for all) we have that, given any data g ∈ L q (0 , T ; X ) and u in the real interpolation space X q = ( X , X ) q ,q , the unique solution of (2.1) belongsto L q (0 , T ; X ) ∩ W ,q (0 , T ; X ) ∩ C ([0 , T ]; X q ) and depends continuously on g and u . If the space X is UMD (unconditionality of martingale differences property) then the following result holds. Theorem 2.4. (Weis, [29, Theorem 4.2])
In a UMD Banach space any R -sectorial operator of anglegreater than π has maximal L q -regularity. Remark 2.5.
In a UMD space, an operator A ∈ BIP ( φ ) with φ < π/ is R -sectorial with angle greaterthan π/ by [5, Theorem 4] and hence has maximal L q -regularity. This also is a classical result by Doreand Venni [10].Next, we consider quasilinear problems of the form n ∂ t u ( t ) + A ( u ( t )) u ( t ) = f ( t, u ( t )) + g ( t ) , t ∈ (0 , T ); u (0) = u (2.2)in L q (0 , T ; X ) , such that D ( A ( u ( t ))) = X , < q < ∞ and T is finite. The main tool we use forproving the existence of solutions of the above problems and regularity results is the following theoremthat is based on a Banach fixed point argument. Theorem 2.6. (Clément and Li, [4, Theorem 2.1])
Assume that there exists an open neighborhood U of u in X q such that A ( u ) : X → X has maximal L q -regularity and that (H1) A ∈ C − ( U, L ( X , X )) , (H2) f ∈ C − , − ([0 , T ] × U, X ) , (H3) g ∈ L q (0 , T ; X ) .Then there exists a T > and a unique u ∈ L q (0 , T ; X ) ∩ W q (0 , T ; X ) ∩ C ([0 , T ]; X q ) solving theequation (2.2) on (0 , T ) . Mellin Sobolev Spaces.
By a cut-off function (near ∂ B ) we mean a smooth non-negative function ω on B with ω ≡ near ∂ B and ω ≡ outside the collar neighborhood of the boundary.There are various ways of extending the definition of the Mellin-Sobolev spaces H s,γp ( B ) given in theintroduction for s ∈ N to arbitrary s ∈ R . A simple way, cf. [6], is given via the map S γ : C ∞ c ( R n +1+ ) → C ∞ c ( R n +1 ) , v ( t, y ) e ( γ − n +12 ) t v ( e − t , y ) . Let κ j : U j ⊆ ∂ B → R n , j = 1 , . . . , N, be a covering of ∂ B by coordinate charts and { ϕ j } a subordinatepartition of unity. NIKOLAOS ROIDOS AND ELMAR SCHROHE
Definition 2.7. H s,γp ( B ) , s, γ ∈ R , < p < ∞ , is the space of all distributions on B ◦ such that (2.3) k u k H s,γp ( B ) = N X j =1 kS γ (1 ⊗ κ j ) ∗ ( ωϕ j u ) k H sp ( R n ) + k (1 − ω ) u ) k H sp ( B ) is defined and finite. Here, ω is a (fixed) cut-off function and ∗ refers to the push-forward of distributions.Up to equivalence of norms, this construction is independent of the choice of ω and the κ j . Clearly, allthe spaces H s,γp ( B ) are UMD spaces. Corollary 2.8.
Let s > ( n + 1) /p . Then k uv k H s,γp ( B ) ≤ k u k H s,γp ( B ) k v k H s, ( n +1) / p ( B ) . In particular, H s,γp ( B ) is a Banach algebra whenever s > ( n + 1) /p and γ ≥ ( n + 1) / .Proof . Since s > ( n + 1) /p , H sp ( R n ) is a Banach algebra. We can therefore assume u and v to besupported close to the boundary. Then k e ( γ − n +12 ) t u ( e − t , y ) v ( e − t , y ) k H sp ( R n +1 ) ≤ k e ( γ n +12 ) t u ( e − t , y ) k H sp ( R n +1 ) k v ( e − t , y ) k H sp ( R n +1 ) ∼ k u k H s,γp ( B ) k v k H s, ( n +1) / p ( B ) , where ∼ denotes equivalence of norms. ✷ Corollary 2.9.
Let ≤ p < ∞ and s > ( n + 1) /p . Then a function u in H s,γp ( B ) is continuous on B ◦ ,and, near ∂ B , we have | u ( x, y ) | ≤ cx γ − ( n +1) / k u k H s,γp ( B ) for a constant c > . This is [20, Corollary 2.5]. For convenience, we recall the easy proof. The Sobolev embedding theoremimplies continuity as H s,γp ( B ) ֒ → H sp,loc ( B ◦ ) . Near the boundary, we deduce from (2.3) and the tracetheorem that for each t ∈ R , e ( γ − ( n +1) / t k u ( e − t , · ) k B s − /pp,p ( ∂ B ) ≤ c k u k H s,γp ( B ) . For x = e − t we obtain the assertion from the fact that the Besov space B s − /pp,p ( ∂ B ) embeds into theSobolev space H s − /p − εp ( ∂ B ) for every ε > and the Sobolev embedding theorem. ✷ Corollary 2.10.
Let < p < ∞ , s ≥ , ε > . Then the operator M m of multiplication by a function m in H s +( n +1) /p + ε, ( n +1) / p ( B ) defines a continuous map M m : H s,γp ( B ) → H s,γp ( B ) . Proof . Let v ∈ H s,γp ( B ) and denote by ∼ equivalence of norms. We can assume that v is supported ina single coordinate neighborhood and work in local coordinates. Then k mv k H s,γp ( B ) ∼ k e ( γ − ( n +1) / t m ( e − t , y ) v ( e − t , y ) k H sp ( R n +1 ) ≤ c k m ( e − t , y ) k C s + ε ∗ ( R n +1 ) k e ( γ − ( n +1) / t v ( e − t , y ) k H sp ( R n +1 ) ≤ c k m ( e − t , y ) k H s +( n +1) /p + εp ( R n +1 ) k e ( γ − ( n +1) / t v ( e − t , y ) k H sp ( R n +1 ) ∼ k m k H s +( n +1) /p + ε, ( n +1) / p ( B ) k v k H s,γp ( B ) . Here, the first inequality is due to the fact that multiplication by functions in the Zygmund space C τ ∗ defines a bounded operator in H s for − τ < s < τ , cf. [27, Section 13, Theorem 9.1], and the second is aconsequence of the fact that H s +( n +1) /pp ( R n +1 ) ֒ → C s ∗ , cf. [27, Section 13, Proposition 8.5]. ✷ OUNDED IMAGINARY POWERS OF CONE DIFFERENTIAL OPERATORS 7 Bounded Imaginary Powers of Cone Differential Operators
Cone differential operators.
We consider a cone differential operator A : C ∞ c ( B ◦ , E ) → C ∞ c ( B ◦ , E ) (3.1)of the form (1.2), acting on sections of a vector bundle E over B . We may assume that E respects theproduct structure near the boundary ∂ B , i.e. is the pull-back of a vector bundle E | ∂ B over ∂ B underthe canonical projection [0 , × ∂ B → ∂ B . In order to keep the notation simple, we shall not indicate thebundles in the function spaces and write C ∞ c ( B ◦ ) , H s,γp ( B ) , etc.. We moreover assume A to be B -ellipticas explained around (1.3) in the introduction.The conormal symbol of a A is the operator polynomial σ M ( A ) : C → L ( H s + µp ( ∂ B ) , H sp ( ∂ B )) , defined by σ M ( A )( z ) = µ X j =0 a j ( x, y, D y ) z j . To simplify matters, we shall assume: σ M ( A ) is invertible on the line Re z = ( n + 1) / − γ − µ (3.2)(note that the invertibility is independent of s ∈ R and < p < ∞ ). This implies the existence of aparametrix B : H s,γp ( B ) → H s + µ,γ + µp ( B ) , such that BA − I is compact on H s + µ,γ + µp ( B ) and AB − I iscompact on H s,γp ( B ) .We shall now consider A as an unbounded operator in H s,γp ( B ) . Under assumption (3.2) the domainof the minimal extension A min ,s of A , i.e. the closure of A acting as in (3.1) in H s,γp ( B ) , is D ( A min ,s ) = H s + µ,γ + µp ( B ) , and the domain of the maximal extension A max ,s is D ( A max ,s ) = D ( A min ,s ) ⊕ E , (3.3)where E is a finite-dimensional space of functions of the form X j,k c jk ( y ) ω ( x ) x − q j log k x (3.4)with c jk ∈ C ∞ ( ∂ B ) , a cut-off function ω , n + 12 − γ − µ < Re q j < n + 12 − γ, and k ∈ N . (3.5)As a subset of H ∞ ,γp ( B ) , E is independent of s . For details see Section 3 in [13] (the statements there aremade for the case p = 2 , but they extend to < p < ∞ ) in connection with [21, Sections 2.2, 2.3].It follows that the domain of an arbitrary closed extension A s of A in H s,γp ( B ) is of the form D ( A s ) = D ( A min ,s ) ⊕ E with a subspace E of the above space E . Remark 3.1.
In case condition (3.2) does not hold, the minimal domain of A , considered as an unboundedoperator in H s,γp ( B ) with representation (1.2) is given by D ( A min ,s ) = n u ∈ \ ε> H s + µ,γ + µ − εp ( B ) : x − µ µ X j =0 a j (0 , y, D y )( − x∂ x ) j u ∈ H s,γp ( B ) o ; (3.6)the maximal domain still is as in (3.3). NIKOLAOS ROIDOS AND ELMAR SCHROHE
The results on H ,γp ( B ) . We denote by A a closed extension of A , considered as an unboundedoperator on H ,γ ( B ) . It is the main result of [7] that A has bounded imaginary powers, provided theresolvent has a certain structure which we will now recall:For given θ and δ , let Λ θ = { z ∈ C : | z | ≤ δ or | arg z | ≥ θ } . Assume that the resolvent to A exists on Λ θ and is of the following form: ( A − λ ) − = ω ( x µ op γM ( g )( λ ) + G ( λ )) ω + (1 − ω ) P ( λ )(1 − ω ) + G ∞ ( λ ) . (3.7)Here ω , ω , ω are cut-off functions with ω ω = ω , ω ω = ω , and(i) g ( x, z, λ ) = ˜ g ( x, z, x µ λ ) with ˜ g ∈ C ∞ ( R ≥ , M − µ,µ O ( ∂ B ; Λ θ )) (ii) P ( λ ) ∈ L − µ,µcl ( B ◦ ; Λ θ ) (iii) G ( λ ) ∈ R − µ,µG ( R × ∂ B ; Λ θ , γ ) (iv) G ∞ (Λ θ ) ∈ C −∞ G ( B ; Λ θ , γ ) .Let us give a short description of these operator classes; full details can be found in [7]. • M − µ,µ O ( ∂ B ; Λ θ ) is the class of holomorphic Mellin symbols of order − µ , depending on the pa-rameter λ ∈ Λ θ in such a way that if we write λ = σ µ , then σ plays the role of an additionalcovariable. • L − µ,µcl ( B ◦ ; Λ θ ) denotes the classical pseudodifferential operators of order − µ depending on theparameter λ ∈ Λ θ in the same sense as above.We need more notation to explain the other two classes. By C ∞ ,γ ( B ) denote the space of all smoothfunctions u on B ◦ such that, in local coordinates near the boundary sup
Theorem 3.2.
Let A be an extension of the cone differential operator A on H ,γp ( B ) with domain D ( A ) = H µ,γ + µp ( B ) ⊕ E , (3.11) with a subspace E of E , whose resolvent is of the form (3.7) above. Then A has bounded imaginarypowers on H ,γp ( B ) of angle θ . OUNDED IMAGINARY POWERS OF CONE DIFFERENTIAL OPERATORS 9
Bounded imaginary powers on higher order Mellin-Sobolev spaces.
We next extend The-orem 3.2 to higher order Mellin-Sobolev spaces, keeping γ, p and the space E in (3.11) fixed. Theorem 3.3.
For s ≥ denote by A s the unbounded operator on H s,γp ( B ) with domain D ( A s ) = H s + µ,γ + µp ( B ) + E . Assume that the resolvent of the corresponding extension A with domain (3.11) exists on Λ θ and is ofthe form (3.7) . Then A s has bounded imaginary powers on H s,γp ( B ) of angle θ .Proof . Step
1. First let us check that the restriction of the resolvent of A to H s,γp ( B ) is the resolventof A s . The identities ( A − λ ) − ( A − λ ) = I and ( A − λ )( A − λ ) − = I hold on D ( A s ) and H s,γp ( B ) , respectively. Thus it suffices to show that ( A − λ ) − ( H s,γp ( B )) ⊆ D ( A s ) .So suppose u ∈ D ( A ) and ( A − λ ) u ∈ H s,γp ( B ) . We first claim that u ∈ D ( A max ,s ) . Indeed, by (3.11), u ∈ H µ,γp ( B ) . If s ≤ µ , we are done. Otherwise, we have u ∈ D ( A max ,µ ) ⊆ H µ,γp ( B ) . After finitely manysteps we reach the assertion. Hence u is in the intersection of D ( A max ,s ) and D ( A ) , which is D ( A s ) . Step
2. We show that for arbitrary s ≥ A s − λ ) − = O ( | λ | − ) , λ ∈ Λ θ . We recall that the result was proven for s = 0 in [7, Proposition 1] making only use of the representation(3.7) of the resolvent. Using interpolation, it suffices to treat the case where s is a positive integer andto show the corresponding estimates for the four types of operators appearing in the resolvent.Ad (i). We write M ( λ ) = x µ op γM ( g )( λ ) . By Γ β we denote the set { z ∈ C : Re z = β } . For u ∈ C ∞ c ( R > , C ∞ ( ∂ B )) we then have ( x∂ x ) (op γM ( g )( λ )) ( u )( x )= ( x∂ x ) Z Γ n +12 − γ Z ∞ ( x ′ /x ) z e g ( x, z, x µ λ ) u ( x ′ ) dx ′ x ′ dz = Z Γ n +12 − γ Z ∞ ( x ′ /x ) z (( x∂ x e g )( x, z, x µ λ ) + µx µ ( ∂ λ e g )( x, z, x µ λ )) u ( x ′ ) dx ′ x ′ dz + Z Γ n +12 − γ Z ∞ ( x ′ /x ) z e g ( x, z, x µ λ )( x ′ ∂ x ′ u )( x ′ ) dx ′ x ′ dz. Hence ( x∂ x )( ω M ( λ ) ω ) = ( ω M ( λ ) ω )( x∂ x ) + f M ( λ ) (3.12)with f M ( λ ) = µ ω M ( λ ) ω + ( x∂ x ω ) M ( λ ) ω + ω M ( λ )( x∂ x ω )+ ω x µ op γM (( x∂ x e g )( x, z, x µ λ ) + µx µ ( ∂ λ e g )( x, z, x µ λ )) ω . The operator family λ f M ( λ ) is uniformly bounded on H ,γp ( B ) , since it satisfies the same assumptions as ω λM ( λ ) ω above.Moreover, the well-known commutator identity [ D y j , op p ] = op( D y j p ) , valid for a pseudodifferentialsymbol p = p ( y, η ) on R n , implies that we have in local coordinates D y j M ( λ ) = M ( λ ) D y j + x µ op γM ( g ( x, z, x µ λ )) for a symbol g ∈ C ∞ ( R ≥ , M − µ,µ O ( ∂ B , Λ θ )) . Together with equation (3.12) this shows that ω λM ( λ ) ω is uniformly bounded on H ,γp ( B ) . Iteration then implies boundedness on H s,γp ( B ) for all s ∈ N . Ad (ii). Away from the boundary, the space H s,γp ( B ) coincides with the usual Sobolev space of order s .The uniform boundedness of the operator family λ (1 − ω ) P ( λ )(1 − ω ) then is an immediate consequenceof the well-known continuity result for pseudodifferential operators.Ad (iii). In order to show the uniform boundedness of ω λG ( λ ) ω on H s,γp ( B ) , it is sufficient to provethat ω λ ( x∂ x ) j D αy G ( λ ) ω is uniformly bounded in L ( H ,γp ( B )) whenever j + | α | ≤ s . In order to see this,we simply note that, as a consequence of (3.9), ( x∂ x ) j D αy G ( λ ) has the integral kernel ( x∂ x ) s D αy k ( λ, x, y, x ′ , y ′ ) = [ λ ] ( n +1) /µ (( x∂ x ) s D αy e k )( λ, [ λ ] /µ x, y, [ λ ] /µ x ′ , y ′ ) . As ( x∂ x ) j D αy e k ∈ S − cl (Λ θ ) b ⊗ π S γ + ε b ⊗ π S γ + ε , we obtain the required boundedness from [7, Proposition 1].Ad (iv). Elements of C −∞ G ( B ; Λ θ , γ ) clearly have operator norms on H s,γp ( B ) which are O ( | λ | − N ) forarbitrary N . Step
3. Finally we prove the required estimate for the purely imaginary powers of A s . For Re z < , A z is defined by the Dunford integral i π Z C λ z ( A s − λ ) − dλ where C is a contour around Λ θ .We have to show that, as < Re z < , the norm of the Dunford integral in L ( H s,γp ( B )) can beestimated by M e θ | Im z | , uniformly in Re z . To this end we estimate the norms of the four terms that ariseby replacing the resolvent in the integrand by the terms in (i)-(iv).Ad (i). It is shown in [7, Proposition 3] that g z ( x, y, ( n + 1) / − γ + iτ, η ) = ω ( x ) x µ Z C λ z ˜ g ( x, y, ( n + 1) / − γ + iτ, η, x µ λ ) dλ is a zero order Mellin symbol with respect to the line { Re z = ( n + 1) / − γ } (an element of M S ( R > × R n × Γ ( n +1) / − γ × R n ) in the notation of [7, Section10]). Moreover, the symbol estimates of e − θ | Im z | g z are proven to be uniform in − ≤ Re z < . This implies the uniform boundedness of the operator normalso on H s,γp ( B ) for each s .Ad (ii). This is immediate from the result for standard pseudodifferential operators.Ad (iii). Conjugation by x γ reduces the task to the case γ = 0 . By definition G ( λ ) then is an integraloperator with an integral kernel k ( λ, x, y, x ′ , y ′ ) of the form given in (3.9) and (3.10). According to theproof of [7, Proposition 2] this is sufficient to establish that the norms of the operators K z with kernels k z ( x, y, x ′ , y ′ ) = Z C λ z k ( λ, x, y, x ′ , y ′ ) dλ on H , p ( B ) are O ( e θ | Im z | ) , uniformly for − ≤ Re z < .Now it is easy to infer much better mapping properties: For s ∈ N , the norms of K z : H , p ( B ) →H s, p ( B ) can be estimated by the operator norms on H , p ( B ) of the operators with kernels ( x∂ x ) j D αy k z ( x, y, x ′ , y ′ ) = Z C λ z ( x∂ x ) j D αy k ( λ, x, y, x ′ , y ′ ) dλ. Now (3.9) and (3.10) imply that ( x∂ x ) j D αy k ( λ, x, y, x ′ , y ′ ) has the same structure as k , since S ε is invariantunder derivatives ( x∂ x ) j and D αy . This implies the uniform boundedness of the operator norms.Ad (iv). This follows immediately from the fact that λ G ∞ ( λ ) is rapidly decreasing with values inbounded operators on H s,γp ( B ) for each s ∈ R .Hence we obtain the assertion. ✷ OUNDED IMAGINARY POWERS OF CONE DIFFERENTIAL OPERATORS 11
The model cone operator.
For completeness, we recall the definition of the model cone operator b A associated with A . In the notation (1.2), it is given by b A = x − µ µ X j =0 a j (0 , y, D y )( − x∂ x ) j . It naturally acts on the spaces K s,γp ( R > × ∂ B ) , s, γ ∈ R , < p < ∞ . In order to introduce these, denotefirst by H sp,cone ( R > × ∂ B ) the space of all tempered distributions u on R > × ∂ B which belong to theSobolev space H sp ( R > × ∂ B ) with respect to the cone metric g = dx + x h (0) on R > × ∂ B . Then let K s,γp ( R > × ∂ B ) = { u ∈ S ′ ( R > × ∂ B ) : ωu ∈ H s,γp ( B ) and (1 − ω ) u ∈ H sp,cone ( R > × ∂ B ) } . Here ω is an arbitrary cut-off function on [0 , × ∂ B .Similarly as for A acting in the spaces H s,γp ( B ) , one can study the closed extensions of b A in K s,γp ( R > × ∂ B ) . Under the assumption (3.2), the domain of minimal extension b A min ,s on K s,γp ( R > × ∂ B ) is givenby K s + µ,γ + µp ( R > × ∂ B ) and the maximal domain is D ( b A max ,s ) = K s + µ,γ + µp ( R > × ∂ B ) ⊕ b E , where b E is a set of singular functions as in (3.4). If the operators a j ( x, y, D y ) are independent of x closeto x = 0 , then b E coincides with the space E in (3.3). In the general case, there is a 1-1-correspondencebetween b E and E ; in particular both spaces have the same dimension. For more details see [13].4. Higher Regularity for the Cahn-Hilliard Equation on Manifolds with StraightConical Singularities
The straight cone Laplacian in H s,γp ( B ) . In this section we assume that in a neighborhood ofthe boundary, say on [0 , / × ∂ B , our Riemannian metric – which we now denote by g in order todistinguish it from the more general Riemannian metric g used above – is of the form g = dx + x h (0) with the metric h in (1.1), so that g models a manifold with a straight conical singularity.We write ∆ for the Laplace-Beltrami operator on B with respect to g . Our aim is to show thatsmoother data u in the Cahn-Hilliard equation produce smoother short time solutions.By λ > λ > λ > . . . we denote the different eigenvalues of ∆ h (0) . We recall that dim B = n + 1 and that the conormal symbol of ∆ is the operator-valued polynomial z σ M (∆ )( z ) = z − ( n − z + ∆ h (0) . (4.1)It is invertible as an operator H ( ∂ B ) → L ( ∂ B ) , provided z = q ± j , j = 0 , , , . . . , where q ± j = n − ± s(cid:18) n − (cid:19) − λ j . (4.2)The inverse is given by ( σ M (∆ )( z )) − = ∞ X j =0 z − q + j )( z − q − j ) π j (4.3)with the orthogonal projection π j onto the eigenspace E j of λ j in L ( ∂ B ) . As a pseudodifferentialoperator, it is then also invertible in L ( H s +2 p ( ∂ B ) , H sp ( ∂ B )) for any s ∈ R , < p < ∞ . With q ± j we associate the spaces E q ± j = { x − q ± j ω ( x ) e ( y ) : e ∈ E j } , except for the case, where j = 0 and dim B = 2 when, according to (4.3), we have a double pole in q ± = 0 and let E = { ω ( x ) e ( y ) + log x ω ( x ) e ( y ) : e , e ∈ E } . The asymptotics space in (3.3) is then E = M E ρ with ρ ranging over the q ± j in ] n +12 − γ − , n +12 − γ [ .For arbitrary s ≥ and < p < ∞ we consider ∆ as an unbounded operator in H s,γp ( B ) . We fix aweight γ with n − < γ < min (cid:26) n −
32 + ε, n + 12 (cid:27) , (4.4)where ε = − n −
12 + s(cid:18) n − (cid:19) − λ > . (4.5)This implies in particular that σ M (∆ )( z ) is invertible whenever Re z = ( n + 1) / − γ .By ∆ we denote the closed extension of ∆ on H s,γp ( B ) with domain D s (∆ ) = H s +2 ,γ +2 p ( B ) ⊕ C , (4.6)where C stands for the constant functions. In this way, the domain of ∆ (and hence that of ∆ ) willconsist of bounded functions only and will contain functions that do not vanish at the tip of the cone.For more information on the choice of γ see [20, Section 2.3 and Proposition 3.1]; it is a special case ofthat in [21, Theorem 5.7].We will now show that ∆ has spectrum in R ≤ . To this end we first note that ∆ is symmetric andbounded from above by zero on C ∞ c ( B ◦ ) with respect to the scalar product on H , ( B ) given by (1.5)(for h ≡ h (0) ) and thus has a Friedrichs extension with spectrum in R ≤ . Theorem 4.1.
Let ∆ be the closed extension (4.6) of ∆ in H s,γp ( B ) . Then σ (∆ ) ⊆ R ≤ .Proof. According to our choice of γ , the conormal symbol of λ − ∆ is invertible on the line Re z =( n + 1) / − γ . By [22, Corollaries 3.3 and 3.5], λ − ∆ has a parametrix in the cone calculus; moreover,it is a Fredholm operator in L ( H s +2 ,γ +2 p ( B ) , H s,γp ( B )) for < p < ∞ with index and kernel independentof s and p . The same applies to the formal adjoint, which is also B -elliptic, cf. [22, Theorem 2.10]. Thusthe invertibility of λ − ∆ : H s +2 ,γ +2 p ( B ) ⊕ C → H s,γp ( B ) is independent of s and p , and we may assume s = 0 , p = 2 .We also know from [21], specifically Theorem 4.3 in connection with Theorem 5.7, that λ − ∆ isinvertible with compact inverse for sufficiently large λ outside any sector containing R ≤ . Hence anypoint in the spectrum is an eigenvalue.Let ∆ ,F be the Friedrichs extension of ∆ . According to [21, Corollary 5.4], its domain is D (∆ ,F ) = ( D (∆ , min ) ⊕ L ρ E ρ ⊕ C , dim B = 2 D (∆ , min ) ⊕ L ρ E ρ , dim B > , (4.7) OUNDED IMAGINARY POWERS OF CONE DIFFERENTIAL OPERATORS 13 where ∆ , min / max refers to the minimal/maximal extension on H , ( B ) and the summation is over all ρ in ] − , in the first case and in ]( n − / , ( n − / in the second.We note that C is always contained in D (∆ ,F ) . Indeed, for n = 1 this is trivial. For n = 2 , C ⊆ E ,and for n ≥ , it is a subset of D (∆ , min ) . Moreover, D (∆ ,F ) contains D (∆ , max ) ∩ H , ( B ) .Let λ / ∈ R ≤ , u ∈ D (∆ ) and ( λ − ∆ ) u = 0 . Write u = u + µ with u ∈ H ,γ +2 p ( B ) and µ ∈ C . Then ∆ u − λu = λµ ∈ H , ( n +1) / − δ ( B ) for all δ > . Hence u belongs to the maximal domain of ∆ in H , ( n +1) / − δ ( B ) , thus to D (∆ , max ) ∩ H , ( B ) ⊆ D (∆ ,F ) in view of the fact that γ > ( n + 1) / ≥ .We conclude that u belongs to the domain of the Friedrichs extension and therefore is zero. Hence λ − ∆ : H ,γ +22 ( B ) ⊕ C → H ,γ ( B ) is invertible and so is λ − ∆ : H s +2 ,γ +2 p ( B ) ⊕ C → H s,γp ( B ) . (cid:3) The following theorem extends the statements in [20] to the case of higher s and arbitrary c > . Theorem 4.2.
Let c > , θ ∈ [0 , π [ , φ > and γ , ∆ be chosen as above. Then c − ∆ ∈ P ( θ ) ∩ BIP ( φ ) on H s,γp ( B ) for any s ≥ .Proof . It follows from [21, Theorem 5.7] in connection with [20, Remark 2.10] that the resolvent of c − ∆ on H ,γp ( B ) has the structure required in Theorem 3.3 for large c > . As the resolvent moreoverexists outside R ≤ by Theorem 4.1 and θ ∈ [0 , π [ is arbitrary, we conclude that c − ∆ ∈ P ( θ ) for all c > . Also, c − ∆ ∈ BIP ( φ ) by a shift of the integration contour. ✷ The associated domain of ∆ . We first recall the basic facts from [20, Section 3.2]. The conormalsymbol of ∆ is given by σ M (∆ )( z ) = σ M (∆ )( z + 2) σ M (∆ )( z ) . By (4.3), its inverse is ( σ M (∆ )( z )) − = ∞ X j =0 z − q + j )( z − q − j )( z + 2 − q + j )( z + 2 − q − j ) π j . (4.8)It is meromorphic with poles of order ≤ in the points z = q ± j and z = q ± j − .The domain of the maximal extension of ∆ on H s,γp ( B ) then is of the form D ((∆ ) min ,s ) ⊕ M ρ ˜ E ρ . Here D ((∆ ) min ,s ) ⊆ H ,γ +4 − εp ( B ) for all ε > , ρ runs over the poles of σ M (∆ ) between dim B / − − γ and dim B / − γ , and the spaces ˜ E ρ are of the form ˜ E ρ = { x − ρ log xω ( x ) e ( y ) + x − ρ ω ( x ) e ( y ) } with e and e in the corresponding eigenspace E j (the logarithmic terms will only appear for a doublepole in ρ ).By assumption D (∆ ) is a subset of D (∆ ) = H ,γ +2 p ( B ) ⊕ C . In view of the fact that a function u ( x, y ) = x − ρ log xω ( x ) e ( y ) with = e ∈ C ∞ ( ∂ B ) belongs to H ∞ ,γ +2 p ( B ) if and only if Re ρ < dim B / − γ − , we obtain: Lemma 4.3.
The domain of ∆ as an unbounded operator on H s,γp ( B ) is given by D (∆ ,s ) = D (∆ , min ,s ) ⊕ M ρ ˜ E ρ ⊕ C with the summation now over dim B / − γ − < ρ < dim B / − γ − . Interpolation spaces.
We want to treat the Cahn-Hilliard equation in H s,γp ( B ) for s ≥ with thehelp of Clément and Li’s Theorem 2.6. To this end we have to come to a good understanding of the realinterpolation space X q = ( X , X ) /q,q for X = H s,γp ( B ) and X = D (∆ ,s ) . Let us suppose that < q < ∞ . For real η with / < η < − /q and c > we then have X q = ( X , X ) − q ,q ֒ → [ X , X ] η = [ H s,γp ( B ) , D (∆ ,s )] η = [ D (( c − ∆ ,s ) ) , D (( c − ∆ ,s ) )] η (4.9) = [ D (( c − ∆ ,s )) , D (( c − ∆ ,s ) )] η − by the reiteration theorem for interpolation, cf. [1, (I.2.8.4)], using the boundedness of the imaginarypowers. See Section 5.4 for an alternative way.Since c − ∆ ∈ BIP ( φ ) , φ > , we apply (I.2.9.8) in [1] and obtain [ D (( c − ∆ ,s ) ) , D (( c − ∆ ,s ) )] η ֒ → D (( c − ∆ ,s ) (1 − η )0+2 η ) = D (( c − ∆ ,s ) η ) . (4.10)In particular, as η > , we have X q ֒ → D (∆ ,s ) . (4.11)4.4. The linearized equation.
We write the Cahn-Hilliard equation in the form ∂ t u + A ( u ) u = F ( u ) , u (0) = u with A ( v ) u = ∆ u + ∆ u − v ∆ u and(4.12) F ( u ) = − u ( ∇ u, ∇ u ) g , (4.13)where ( ∇ u, ∇ v ) g = 1 x (cid:16) ( x∂ x u )( x∂ x v ) + X h ij (0)( ∂ y i u )( ∂ y j v ) (cid:17) . (4.14)Our next goal is to establish the existence of bounded imaginary powers for A ( u ) , u ∈ X q , q > . Tothis end we use the following lemma. A proof can be found in [20, Lemma 3.6]. Lemma 4.4.
Let X be a Banach space and T ∈ P ( θ ) in the sense of Definition . with θ ≥ π/ . Then T ∈ P (˜ θ ) for ˜ θ = 2 θ − π and ( T ) z = T z for z ∈ C . Proposition 4.5.
Let s ≥ , p > ( n +1) / , and q > . For every choice of u ∈ X q , φ > , and θ ∈ [0 , π [ ,the operator A ( u ) + c I , considered as an unbounded operator in H s,γp ( B ) with domain D (∆ ,s ) belongsto P ( θ ) ∩ BIP ( φ ) for all sufficiently large c > .Proof . This proposition was shown as [20, Proposition 3.7] for the base space X = H ,γp ( B ) and u ∈ L ∞ ( B ) . In order for the statement to extend to the present situation we need to prove thatmultiplication by a function in X q defines a bounded operator on H s,γp ( B ) . For q > , however, we knowfrom (4.11) that X q ֒ → D (∆ ,s ) = H s +2 ,γ +2 p ( B ) ⊕ C . Clearly, multiplication by constants furnishes abounded operator. By assumption s + 2 > s + ( n + 1) /p ; by (4.4) we have γ + 2 ≥ ( n + 1) / . HenceCorollary 2.10 shows that also the functions in H s +2 ,γ +2 p ( B ) define continuous multipliers. This completesthe argument. ✷ OUNDED IMAGINARY POWERS OF CONE DIFFERENTIAL OPERATORS 15
Short time solutions of the Cahn-Hilliard equation.Theorem 4.6.
Let p ≥ n + 1 , q > . Given any u ∈ X q , there exists a T > and a unique solution in u ∈ L q (0 , T ; D (∆ )) ∩ W q (0 , T ; H s,γp ( B )) ∩ C ([0 , T ] , X q ) solving Equation (1.6) on ]0 , T [ with initial condition (1.7) .Proof . We write the Cahn-Hilliard equation in the form ∂ t u + A ( u ) u = F ( u ) with A and F defined in(4.12) and (4.13), respectively, and apply Theorem 2.6. We have seen in Proposition 4.5 that A ( u ) + cI has BIP ( φ ) for arbitrary φ > and hence maximal regularity, provided c ≥ is large. It remains tocheck conditions (H1) and (H2); note that (H3) is not required.Let U be a bounded neighborhood of u in X q and ˜ x a function which equals x near ∂ B , is strictlypositive on B ◦ and is ≡ outside a neighborhood of ∂ B .Concerning (H1): Let u , u ∈ U . We have seen in the proof of Proposition 4.5 that multiplication byan element in D (∆ ,s ) defines a bounded operator on H s,γp ( B ) . Moreover, in view of our assumptions on p and γ , we see from Corollary 2.9 that D (∆ ,s ) is an algebra. Therefore k A ( u ) − A ( u ) k L ( X ,X ) = 3 k ( u − u )∆ k L ( X ,X ) ≤ c k ( u + u )( u − u ) I k L ( D (∆ ,s ) ,X ) ≤ c k u + u k D (∆ ,s ) k u − u k D (∆ ,s ) ≤ c ( k u k X q + k u k X q ) k u − u k X q ≤ c k u − u k X q for suitable constants c, c , c , and c , where the last inequality is a consequence of the boundedness of U . Concerning (H2): In view of (4.9) and (4.10) the operators x∂ x and ∂ y j , defined near the boundary,map X q to H s +1+ ε,γ +2 p ( B ) for some ε > . Hence we conclude from (4.14) that ∇ u ∈ H s +1+ ε,γ +1 p ( B ) for u ∈ X q . We let θ = ( n + 1) / − γ − . Then ˜ x θ ∇ u ∈ H s +1+ ε, ( n +1) / p ( B ) and ˜ x − θ u ∈ H s,γp ( B ) for u ∈ X q since γ > ( n − / .In the following estimate we use the facts that, by Corollary 2.8, H s +1+ ε, ( n +1) / p ( B ) is an algebraand, by Corollary 2.10, multiplication by an element in H s +1+ ε, ( n +1) / p ( B ) defines a bounded operator on H s,γp ( B ) . k F ( u ) − F ( u ) k X = k u ( ∇ u , ∇ u ) g − u ( ∇ u , ∇ u ) g k X (4.15) ≤ k u ( ∇ u , ∇ u ) g − u ( ∇ u , ∇ u ) g k X + 6 k u ( ∇ u , ∇ u ) g − u ( ∇ u , ∇ u ) g k X +6 k u ( ∇ u , ∇ u ) g − u ( ∇ u , ∇ u ) g k X ≤ k ( u − u )˜ x − θ k X k ˜ x θ ∇ u k H s +1+ ε, ( n +1) / p k ˜ x θ ∇ u k H s +1+ ε, ( n +1) / p +6 k u ˜ x − θ k X k ˜ x θ ∇ u k H s +1+ ε, ( n +1) / p k ˜ x θ ∇ ( u − u ) k H s +1+ ε, ( n +1) / p +6 k u ˜ x − θ k X k ˜ x θ ∇ u k H s +1+ ε, ( n +1) / p k ˜ x θ ∇ ( u − u ) k H s +1+ ε, ( n +1) / p ≤ c ( k u k X q + k u k X q k u k X q + k u k X q ) k u − u k X q for u , u in X q , with a suitable constant c . As U is bounded, the Lipschitz continuity of F follows. ✷ Higher Regularity on Warped Cones
Let now ∆ be as in (1.4), the Laplacian on B induced by the warped cone metric (1.1). In this case,the results in [21] are no longer applicable. We will instead infer maximal regularity of the operator A ( u ) in Equation (4.12) from the results for straight cone Laplacians and perturbation theory for R -sectorialoperators. In addition to g we therefore choose a metric g which coincides with g outside the collar neighborhood [0 , × ∂ B and is of the form g = dx + x h (0) on [0 , / × ∂ B . As above we fix γ according to (4.4) and denote by ∆ the Laplace-Beltrami operatorwith respect to g and by ∆ the extension in H s,γp ( B ) with domain D (∆ ) = H s +2 ,γ +2 p ( B ) ⊕ C . The choice of the extension of ∆ . It is clear that the constant functions also belong to themaximal domain of ∆ on H s,γp ( B ) . We can therefore study the extension ∆ of ∆ with the domain D (∆) = H s +2 ,γ +2 p ( B ) ⊕ C . (5.1)Next fix a cut-off function ω . For < ε ≤ let ω ε ( t ) = ω ( t/ε ) . This is a cutoff function with supportin [0 , ε ) × ∂ B . We define ∆ ε = ω ε ∆ + (1 − ω ε )∆ and consider the closed extension ∆ ε of ∆ ε with domain given by (5.1).Clearly, each of the operators ∆ , ∆ and ∆ ε induces (by pointwise action) a closed operator in L q (0 , T ; H s,γp ( B )) , < q < ∞ , with domain L q (0 , T ; H s +2 ,γ +2 p ( B ) ⊕ C ) .5.2. R-boundedness and maximal L p -regularity. It follows from (1.4) that ∆ = ∆ + ∂ x | h | x | h | ( x∂ x ) + 1 x (cid:0) ∆ h ( x ) − ∆ h (0) (cid:1) , (5.2)where | h | = det( h ij ( x )) . Hence ∆ − ∆ ε = B ε , where B ε = ω ε ( x ) (cid:16) ∂ x | h | x | h | ( x∂ x ) + 1 x (cid:0) ∆ h ( x ) − ∆ h (0) (cid:1)(cid:17) ∈ L ( D (∆) , H s,γp ( B )) . Lemma 5.1. B ε → in L ( D (∆) , H s,γp ( B )) as ε → + .Proof . The smoothness of h implies that B ε = xω ε ( x ) C , where C is a second order cone differ-ential operator. It maps the constants to zero; hence it suffices to show that B ε tends to zero in L ( H s +2 ,γ +2 p ( B ) , H s,γp ( B )) or, even simpler, that the norm of multiplication by xω ε ( x ) tends to zero in L ( H s,γp ( B )) . To see the latter, we write xω ε ( x ) = ε xε ω (cid:16) xε (cid:17) , note that ( x/ε ) ω ( x/ε ) is uniformly bounded with respect to ε in H k, ( n +1) / p ( B ) for each k ≥ , and inferfrom Corollary 2.10 that its norm as a multiplier is uniformly bounded. The factor ε then yields theassertion. ✷ The extensions ∆ , ∆ and ∆ ε all have the same model cone operator, namely b ∆ = 1 x (cid:0) ( x∂ x ) − ( n − − x∂ x ) + ∆ h (0) (cid:1) . We consider the extension b ∆ in K ,γp ( R × ∂ B ) with domain D ( b ∆) = K ,γ +2 p ( R × ∂ B ) ⊕ C . OUNDED IMAGINARY POWERS OF CONE DIFFERENTIAL OPERATORS 17
It was shown in [21, Theorem 5.7] that λ − b ∆ : K ,γ +2 p ( R × ∂ B ) ⊕ C −→ K ,γp ( R × ∂ B ) is invertible for λ / ∈ R − . According to [21, Theorem 4.2] the invertibility is independent of < p < ∞ ; itholds in particular for p = 2 . Moreover, it was pointed out shortly before Theorem 4.2 in [21] that theinverse is given as the sum of two principal edge symbols. They are parameter-dependent operators, andhence k ( λ − b ∆) − k L ( K ,γ ( R × ∂ B )) = O ( | λ | − ) . We now apply [13, Theorem 6.36] and conclude that for λ / ∈ R − , | λ | sufficiently large λ − ∆ : H s +2 ,γ +22 ( B ) ⊕ C → H s,γ ( B ) is invertible.(5.3)(Note that the authors explain in the beginning of Section 3 how to reduce the weight γ to the fixedchoice they use in their formulation of the theorem). Theorem 5.2.
For all < p < ∞ , s ≥ and λ / ∈ R − λ − ∆ : H s +2 ,γ +2 p ( B ) ⊕ C → H s,γp ( B ) (5.4) is invertible.Proof . We argue similarly as in the proof of Theorem 4.1. On the domain where (5.3) holds, λ − ∆ ,has compact resolvent. Hence every point in the spectrum necessarily is an eigenvalue. In order to seethat λ − ∆ is actually invertible for all λ / ∈ R ≤ , we first note that we may assume s = 0 and p = 2 . Wemoreover observe that ∆ is symmetric and bounded from above by zero on C ∞ c ( B ◦ ) with respect to the H , ( B ) -scalar product given by (1.5). It therefore has a Friedrichs extension ∆ F with spectrum in R ≤ .The domain of ∆ F has been determined in [12, Theorem 8.12]. For our purposes it suffices to knowthat D (∆ F ) contains the constants and, by [12, Lemma 8.1], the set D (∆ max ) ∩ H , ( B ) , where ∆ max isthe maximal extension of ∆ on H , ( B ) . With this information, the proof proceeds as in Theorem 4.1. ✷ Lemma 5.3.
Given c > there exist δ > and ε > such that k ( c − ∆ ε ) u k H s,γp ( B ) ≥ δ k u k D (∆) , < ε < ε , u ∈ D (∆) . (5.5) Proof . The invertibility of c − ∆ implies that there is a constant δ > such that k ( c − ∆) u k H s,γp ( B ) ≥ δ k u k D (∆) . Now the assertion follows from Lemma 5.1. ✷ Proposition 5.4.
Let φ > and c > . By possibly increasing c we can achieve that the BIP ( φ ) -boundsof c − ∆ ε can be estimated uniformly with respect to ε .Proof . We will show that, for sufficiently large λ ∈ Λ θ , we can choose all components in the repre-sentation (3.7) of the resolvent of c − ∆ ε to be uniformly bounded with respect to ε in the respectiveseminorms.Let ε be fixed and ε < ε . As already observed in the proof of Lemma 5.1, the difference ∆ ε − ∆ ε = B ε − B ε is of the form x ( ω ε ( x ) − ω ε ( x )) C with a second order cone differential operator C , independentof ε . The support of the difference therefore lies in ε < x < ε .Next we note that the conormal symbol of ∆ ε is independent of ε . Moreover, the symbol seminorms ofthe x -coefficient ( ω ε ( x ) − ω ε ( x )) C are uniformly bounded. (For this we measure x∂ x -derivatives insteadof the usual seminorms in C ∞ ( R ≥ , M , O ( ∂ B , Λ θ )) , which is sufficient for our purposes, cf. [7, Proposition3].) Hence the symbol inversion process produces a Mellin symbol g ε for the parametrix to c − ∆ ε − λ with uniformly bounded seminorms. This has an important consequence: As ∆ ε and ∆ ε coincide outside − ε < x < ε , the model coneoperators of both coincide, and the pseudodifferential parts agree for x > ε . Write, as in (3.7), ( c − ∆ ε − λ ) − = ω (op γM ( g ε )( λ ) + G ( λ )) ω + (1 − ω ) P (1 − ω ) + G ∞ ( λ ) with cut-off functions satisfying ω ω = ω and ω ω = ω . In view of the parametrix constructionprocess, see Section 3.2, in particular Eq. (3.10), in [21], we then obtain a parametrix to c − ∆ ε − λ bysimply replacing g ε with the corresponding Mellin symbol g ε as indicated above. Hence ( c − ∆ ε − λ ) ( ω (op γM ( g ε )( λ ) + G ( λ )) ω + (1 − ω ) P (1 − ω )) = I + G ∞ ,ε ( λ ) for a uniformly bounded family G ε, ∞ ∈ C −∞ G ( B , Λ θ , γ ) .As G ε, ∞ ( λ ) → , ( I + G ε, ∞ ( λ )) − exists as an operator in L ( H ,γ ( B )) for sufficiently large λ in Λ θ ,say | λ | ≥ R , where R is independent of ε . Writing ( I + G ε, ∞ ( λ )) − = I + H ε ( λ ) with H ε ( λ ) = G ε, ∞ ( λ ) − G ε, ∞ ( λ )( I + G ε, ∞ ( λ )) − G ε, ∞ ( λ ) and noting that ∂ λ ( I + G ε, ∞ ( λ )) − = − ( I + G ε, ∞ ( λ )) − ∂ λ G ε, ∞ ( λ )( I + G ε, ∞ ( λ )) − , we see that, given k, m ∈ N , the seminorms for λ k ∂ mλ H ε ( λ ) : H ,γ ( B ) → C ∞ ,γ + δ ( B ) (with suitably small δ > ) are uniformly bounded in ε . A similar consideration applies to the adjoints. Weinfer from [25, Corollary 4.3] that H ε has an integral kernel in S (Λ θ ∩{| λ | ≥ R } , C ∞ ,γ + δ ( B ) b ⊗ π C ∞ ,γ + δ ( B )) which satisfies the corresponding estimates uniformly in ε . As ( I + G ε, ∞ ( λ )) − = I + H ε ( λ ) and ( c − ∆ ε − λ ) − = ( ω (cid:0) x op γM ( g ε )( λ ) + G ( λ ) (cid:1) ω + (1 − ω ) P (1 − ω ))( I + G ε, ∞ ( λ )) − , the symbol seminorms for the components of the resolvent to c − ∆ ε with respect to the decompositionin (3.7) are independent of the choice of ε for λ ∈ Λ θ , | λ | ≥ R . Replacing c by c + R , the estimates willhold in all of Λ θ . The argument in [7] then shows that the BIP ( φ ) -bound is uniformly bounded in ε . ✷ Corollary 5.5.
Fix c and ε as in Lemma . , with c possibly increased according to Proposition . . ByTheorem . , the operators c − ∆ ε belong to P ( θ ) ∩ BIP ( φ ) , for any θ ∈ [0 , π [ and φ > . Since H s,γp ( B ) is UMD, we deduce from [5, Theorem 4] that c − ∆ ε is R -sectorial with angle φ . Moreover, Proposition . shows that the BIP ( φ ) -bounds and hence - see the proof of [5, Theorem 4] - the R -bounds for c − ∆ ε are uniformly bounded in < ε < ε for sufficiently small ε . Theorem 5.6.
For any θ ∈ [0 , π [ and c > , c − ∆ is R -sectorial of angle θ . Hence c − ∆ generates ananalytic semigroup and has maximal L p -regularity.Proof . We first assume that c > is sufficiently large and infer from (5.5) and Lemma . that k B ε u k H s,γp ( B ) ≤ c ε k u k D (∆) ≤ c ε δ k ( c − ∆ ε ) u k H s,γp ( B ) , (5.6)where the constant c ε > can be made arbitrarily small by taking ε sufficiently close to zero. In particular,it can be made smaller than the inverse of the supremum of the R -bounds of the operators c − ∆ ε , cf.Corollary 5.5. From [17, Theorem 1], we infer that c − ∆ is R -sectorial of angle θ for large c > .On the other hand, since the resolvent exists outside R ≤ and θ ∈ [0 , π [ is arbitrary, we obtain the R -sectoriality for all c > . ✷ As in Section 4 we now study the linearized equation with g replaced by g and ∆ by ∆ . We useTheorem 5.6 to infer maximal regularity for the operator A ( v ) given by A ( v ) u = ∆ u + ∆ u − v ∆ u, OUNDED IMAGINARY POWERS OF CONE DIFFERENTIAL OPERATORS 19 cf. (4.12):
Proposition 5.7.
Let v ∈ H s +( n +1) /p + ε, ( n +1) / p ( B ) ⊕ C for some ε > . For every θ ∈ [0 , π [ the operator A ( v ) + c I is R -sectorial of angle θ , provided c > is sufficiently large. Hence A ( v ) has maximal L q -regularity, < q < ∞ .Proof. We write λ (cid:0) ( c − ∆) + λ (cid:1) − = ( i √ λ )( c − ∆ + i √ λ ) − ( − i √ λ )( c − ∆ − i √ λ ) − . (5.7)As arg( ± i √ λ ) = arg λ ± π , this implies that ( c − ∆) ∈ P ( θ ) for any θ ∈ [0 , π [ , whenever c > is largeenough. Similarly, ± i √ λ will lie in the domain of R -boundedness of z ( c − ∆ + z ) − . Thus, both families ( i √ λ )( c − ∆ + i √ λ ) − and ( − i √ λ )( c − ∆ − i √ λ ) − are R -bounded. By the definition of R -boundedness, we deduce from (5.7) that ( c − ∆) is R -sectorialwith angle θ .Next let µ > . The sectoriality of c − ∆ implies that the norm of the bounded operator µ ( c − ∆) − in L ( H s,γp ( B )) will become arbitrarily small, provided c > is chosen sufficiently large. Given any α > we will have k µx k H s,γp ( B ) < α k ( c − ∆) x k H s,γp ( B ) , x ∈ D (( c − ∆) ) , for sufficiently large c > . Thus, by Theorem 1 in [17], ( c − ∆) + µ is R -sectorial with angle θ for large c > .For any f ∈ H s +( n +1) /p + ε, ( n +1) / p ( B ) ⊕ C we have (with norms taken in L ( H s,γp ( B )) ) k f ∆ (cid:0) ( c − ∆) + µ (cid:1) − k = k f ∆( c − ∆ + i √ µ ) − ( c − ∆ − i √ µ ) − k≤ k M f kk ( c − ∆ + i √ µ − i √ µ − c )( c − ∆ + i √ µ ) − kk ( c − ∆ − i √ µ ) − k = k M f kk I − ( i √ µ + c )( c − ∆ + i √ µ ) − kk ( c − ∆ − i √ µ ) − k . Here, we have written M f for the operator of multiplication by f . By Corollary 2.10, it is bounded on H s,γp ( B ) .We can make the last term in the above inequality arbitrarily small by taking µ large. Hence, givenan arbitrary α > , we can take c > so large that k f ∆ x k < α k (cid:0) ( c − ∆) + µ (cid:1) x k , x ∈ D (( c − ∆) ) . Another application of [17, Theorem 1] then furnishes that ∆ + ( f − c )∆ + ( c + µ ) is R -sectorial with angle θ for c and µ large. Given v ∈ H s +( n +1) /p + ε, ( n +1) / p ( B ) ⊕ C , the assertion followsfor f = 2 c + 1 − v , noting that the latter space is an algebra by Corollary 2.8. (cid:3) The domain of ∆ . Let D (∆) be as in (5.1) and D (∆ ) = { u ∈ D (∆) : ∆ u ∈ D (∆) } . In view of (3.3), it is of the form D (∆ ) = D (∆ ,s ) ⊕ M F ρ ⊕ C with D (∆ ,s ) ⊆ H s +4 ,γ +4 − εp ( B ) for every ε > , and asymptotics spaces F ρ , which can be determinedexplicitly from the conormal symbol of ∆ as well as the metric h in (1.4) and its derivatives at x = 0 . Fordetails see e.g. [13, Section 6], [21, Section 2.3] or [26]. Here it suffices to say that ρ varies over the non-invertibility points of the conormal symbol of ∆ in ]( n +1) / − γ − , ( n +1) / − γ [ , and the elements in each F ρ are suitable linear combinations of functions of the form x − ρ + j log k xω ( x ) e ( y ) , where j ∈ { , , , } , k ∈ { , } and e ∈ C ∞ ( ∂ B ) . The space C has been listed separately, because, as D (∆ ) ⊆ D (∆) , we mayconclude that the F ρ are subsets of H s +2 ,γ +2 p ( B ) . In particular, D (∆ ) ⊆ H s +4 , ( n +1) / − εp ( B ) for every ε > .5.4. Short time solutions of the Cahn-Hilliard equation.
In order to determine the interpolationspace X q = ( X , X ) /q,q between X = H s,γp ( B ) and X = D (∆ ) we invoke the following result of Haase[14, Corollary 7.3]: Theorem 5.8.
Let A be a sectorial operator on the Banach space X , and let α, β, γ ∈ C with < Re γ < Re β ≤ Re α , σ ∈ (0 , , q ∈ [1 , ∞ ] ,and x ∈ X . Then x ∈ ( X, D ( A α )) θ,q ⇒ x ∈ D ( A γ ) and A γ x ∈ ( X, D ( A β − γ )) σ,q , where θ = (1 − σ ) Re γ Re α + σ Re β Re α . Moreover, ( X, D ( A α )) θ,q = ( D ( A γ ) , D ( A β )) σ,q . (5.8)For q > we apply the above theorem for A = ∆ on X with α = 2 , σ = 1 / and < γ < β ≤ sosmall that the resulting θ satisfies < θ ≤ − q . We conclude that X q ֒ → ( X , D (∆ )) θ,q ֒ → ( D (∆ γ ) , D (∆ β )) σ,q ֒ → D (∆ γ ) ֒ → D (∆) = H s +2 ,γ +2 p ( B ) ⊕ C . (5.9)We then obtain a complete analog of Theorem 4.6: Theorem 5.9.
Let s ≥ and ∆ as explained in Section . . Choose p ≥ n + 1 and q > . Given any u ∈ X q , there exists a T > and a unique solution in u ∈ L q (0 , T ; D (∆ )) ∩ W q (0 , T ; H s,γp ( B )) ∩ C ([0 , T ] , X q ) , solving the Cahn-Hilliard equation (1.6) on ]0 , T [ with initial condition u (0) = u .Proof. By Proposition 5.7, A ( u ) has maximal regularity. It remains to establish properties (H1) and(H2).As p ≥ n + 1 and γ + 2 > n +12 , D (∆) is an algebra by Corollary 2.9. Moreover, (5.9) implies that X q embeds into H s +( n +1) /p + ε, ( n +1) / p ( B ) ⊕ C for some ε > . By Corollary 2.10, multiplication by functionsin X q defines continuous operators on H s,γp ( B ) . With this information, (H1) and (H2) can be verified asin the proof of Theorem 4.6. (cid:3) Acknowledgment.
The authors thank G. Mendoza and J. Seiler for valuable discussions and the refereefor very helpful comments which led to an improvement in Theorem 5.9.
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