Discrete Fourier multipliers and cylindrical boundary value problems
aa r X i v : . [ m a t h . A P ] J un DISCRETE FOURIER MULTIPLIERS AND CYLINDRICALBOUNDARY VALUE PROBLEMS
R. DENK, T. NAU
Abstract.
We consider operator-valued boundary value problems in (0 , π ) n with periodic or, more generally, ν -periodic boundary conditions. Using theconcept of discrete vector-valued Fourier multipliers, we give equivalent con-ditions for the unique solvability of the boundary value problem. As an ap-plication, we study vector-valued parabolic initial boundary value problemsin cylindrical domains (0 , π ) n × V with ν -periodic boundary conditions inthe cylindrical directions. We show that under suitable assumptions on thecoefficients, we obtain maximal L q -regularity for such problems. Introduction
In this paper we first study boundary value problems with operator-valued coeffi-cients of the form P ( D ) u + Q ( D ) Au = f in (0 , π ) n , (1.1) D β u (cid:12)(cid:12) x j =2 π − e πν j D β u (cid:12)(cid:12) x j =0 = 0 ( j = 1 , . . . , n, | β | < m ) . (1.2)Here P ( D ) is a partial differential operator of order m acting on u = u ( x ) with x ∈ (0 , π ) n , Q ( D ) a partial differential operator of order m ≤ m , A is a closedlinear operator acting in a Banach space X , and ν , . . . , ν n ∈ C are given numbers.We refer to the boundary conditions as ν -periodic. Note that for ν j = 0 we haveperiodic boundary conditions in direction j , whereas for ν j = i we have antiperi-odic boundary conditions in this direction. In general, we have different boundaryconditions (i.e., different ν j ) in different directions.As a motivation for studying problem (1.1)-(1.2), we want to mention two classes ofproblems: First, the boundary value problem (1.1)-(1.2) includes equations of theform(1.3) u t ( t ) + Au ( t ) = f ( t ) ( t ∈ (0 , π ))and(1.4) u tt ( t ) − aAu t ( t ) − αAu ( t ) = f ( t ) ( t ∈ (0 , π ))with periodic or ν -periodic boundary conditions. Equations of the form (1.3) and(1.4) were considered in [AB02] and [KL06], respectively. These equations fit intoour context by taking n = 1, P ( D ) = ∂ t and Q ( D ) = 1 for (1.3) and P ( D ) = ∂ t , Q ( D ) = − a∂ t − α for (1.4). Mathematics Subject Classification.
Key words and phrases.
Discrete Fourier multipliers, maximal regularity, bounded cylindricaldomains.
As a second motivation for studying (1.1)-(1.2), we consider a boundary valueproblem of cylindrical type where the domain is of the form Ω = (0 , π ) n × V with V ⊂ R n V being a sufficiently smooth domain with compact boundary. The operatoris assumed to split in the sense that(1.5) A ( x, D ) = P ( x , D ) + Q ( x , D ) A V ( x , D )where the differential operators P ( x , D ) and Q ( x , D ) act on x ∈ (0 , π ) n onlyand the differential operator A V ( x , D ) acts on x ∈ V only. The boundary con-ditions are assumed to be ν -periodic in x -direction, whereas in V the operator A V ( x , D ) of order 2 m V may be supplemented with general boundary conditions B ( x , D ) , . . . , B m V ( x , D ). The simplest example of such an operator is theLaplacian in a finite cylinder (0 , π ) n × V with ν -periodic boundary conditions inthe cylindrical directions and Dirichlet boundary conditions on (0 , π ) n × ∂V .Our first main result (Theorem 3.6) gives, under appropriate assumptions on P , Q ,and A , equivalent conditions for the unique solvability of (1.1)-(1.2) in L p -Sobolevspaces. This results generalizes results from [AB02] and [KL06] on equations (1.3)and (1.4), respectively.In particular in connection with operators of the form (1.5) in cylindrical domains,one is also interested in parabolic theory. Therefore, in Section 5 we study problemsof the form(1.6) u t + A ( x, D ) u = f ( t ∈ [0 , T ] , x ∈ (0 , π ) n × V ) ,B j ( x, D ) u = 0 ( t ∈ [0 , T ] , x ∈ (0 , π ) n × ∂V, j = 1 , . . . , m V ) , ( D β u ) | x j =2 π − e πν j ( D β u ) | x j =0 = 0 ( j = 1 , . . . , n ; | β | < m ) ,u (0 , x ) = u ( x ) ( x ∈ (0 , π ) n × V ) . Here A ( x, D ) is of the form (1.5). If ( A V , B , . . . , B m V ) is a parabolic boundaryvalue problem in the sense of parameter-ellipticity (see [DHP03, Section 8]), weobtain, under suitable assumptions on P and Q , maximal L q -regularity for (1.6)(see Theorems 4.3 and 4.7 below). The proof of maximal regularity is based on the R -boundedness of the resolvent related to (1.6).Apart from its own interest, the consideration of ν -periodic boundary conditionsalso allows us to address boundary conditions of mixed type. As the simplest ex-ample, when a = 0 we can analyze equation (1.4) with Dirichlet-Neumann typeboundary conditions u (0) = 0 , u t ( π ) = 0 . The connection to periodic and antiperiodic boundary conditions is given by suit-able extensions of the solution. This was also considered in [AB02] where – startingfrom periodic boundary conditions – the pure Dirichlet and the pure Neumann casecould be treated.The main tool to address problems (1.1)-(1.2) and (1.6) is the theory of discretevector-valued Fourier multipliers. Taking the Fourier series in the cylindrical direc-tions, we are faced with the question under which conditions an operator-valuedFourier series defines a bounded operator in L p . This question was answered byArendt and Bu in [AB02] for the one-dimensional case n = 1, where a discreteoperator-valued Fourier multiplier result for UMD spaces and applications to pe-riodic Cauchy problems of first and second order in Lebesgue- and H¨older-spaces OURIER MULTIPLIERS AND CYLINDRICAL BOUNDARY VALUE PROBLEMS 3 can be found. For general n , the main result on vector-valued Fourier multipliers iscontained in [BK04]. A shorter proof of this result by means of induction based onthe result for n = 1 in [AB02] is given in [Bu06]. As pointed out by the authors in[AB02] and [BK04], the results can as well be deduced from [ˇSW07, Theorems 3.7,3.8].A generalization of the results in [AB02] to periodic first order integro-differentialequations in Lebesgue-, Besov- and H¨older-spaces is given in [KL04]. Here theconcept of 1-regularity in the context of sequences is introduced (see Remark 2.11below).In [KL06] one finds a comprehensive treatment of periodic second order differentialequations of type (1.4) in Lebesgue- and H¨older-spaces. In particular, the specialcase of a Cauchy problem of second order, i.e. α = 0 , a = 1, where A is the generatorof a strongly continuous cosine function is investigated. In [KLP09] more generalequations are treated in the mentioned spaces as well as in Triebel-Lizorkin-spaces.Moreover, applications to nonlinear equations are presented.Maximal regularity of second order initial value problems of the type u tt ( t ) + Bu t ( t ) + Au ( t ) = f ( t ) ( t ∈ [0 , T )) ,u (0) = u t (0) = 0is treated in [CS05] and [CS08]. In particular, p -independence of maximal regularityfor this type of second order problems is shown. The same equation involving dy-namic boundary conditions is studied in [XL04]. The non-autonomous second orderproblem, involving t -dependent operators B ( t ) and A ( t ), is treated in [BCS08]. Wealso refer to [XL98] for the treatment of higher order Cauchy problems.In [AR09] various properties as e.g. Fredholmness of the operator ∂ t − A ( · ) asso-ciated to the non-autonomous periodic first order Cauchy-problem in L p -contextare investigated. Results on this operator based on Floquet theory are obtained inthe PhD-thesis [Gau01]. We remark that in Floquet theory ν -periodic (instead ofperiodic) boundary conditions appear in a natural way.For the treatment of boundary value problems in (0 ,
1) with operator-valued coef-ficients subject to numerous types of homogeneous and inhomogeneous boundaryconditions, we refer to [FLM + , n . In [FSY09] however, applications to boundary value problems inthe cylindrical space domain (0 , × V can be found.The usage of operator-valued multipliers to treat cylindrical in space boundary valueproblems was first carried out in [Gui04] and [Gui05] in a Besov-space setting. Inthese papers the author constructs semiclassical fundamental solutions for a class ofelliptic operators on infinite cylindrical domains R n × V . This proves to be a strongtool for the treatment of related elliptic and parabolic ([Gui04] and [Gui05]), aswell as of hyperbolic ([Gui05]) problems. Operators in cylindrical domains with asimilar splitting property as in the present paper were, in the case of an infinitecylinder, also considered in [NS]. R. DENK, T. NAU Discrete Fourier multipliers and R -boundedness In the following, let X and Y be Banach spaces, 1 < p < ∞ , n ∈ N , and Q n :=(0 , π ) n . By L ( X, Y ) we denote the space of all bounded linear operators from X to Y , and we set L ( X ) := L ( X, X ). By L p ( Q n , X ) we denote the standard Bochnerspace of X -valued L p -functions defined on Q n . For f ∈ L p ( Q n , X ) and k ∈ Z n the k -th Fourier coefficient of f is given by(2.1) ˆ f ( k ) := 1(2 π ) n Z Q n e − i k · x f ( x ) dx . By Fejer’s Theorem we see that f ( x ) = 0 almost everywhere if ˆ f ( k ) = 0 for all k ∈ Z n as well as f ( x ) = ˆ f ( ) almost everywhere if ˆ f ( k ) = 0 for all k ∈ Z n \{ } . Moreover for f, g ∈ L p ( Q n , X ) and a closed operator A in X it holds that f ( x ) ∈ D ( A ) and Af ( x ) = g ( x ) almost everywhere if and only if ˆ f ( k ) ∈ D ( A ) and A ˆ f ( k ) = ˆ g ( k ) for all k ∈ Z n . We will frequently make use of these observationswithout further comments. Definition . A function M : Z n → L ( X, Y ) is called a (discrete) L p -multiplier iffor each f ∈ L p ( Q n , X ) there exists a g ∈ L p ( Q n , Y ) such thatˆ g ( k ) = M ( k ) ˆ f ( k ) ( k ∈ Z n ) . In this case there exists a unique operator T M ∈ L ( L p ( Q n , X ) , L p ( Q n , Y )) associ-ated to M such that(2.2) ( T M f )ˆ( k ) = M ( k ) ˆ f ( k ) ( k ∈ Z n ) . The property of being a Fourier multiplier is closely related to the concept of R -boundedness. Here we give only the definition and some properties which will beused later on; as references for R -boundedness we mention [KW04] and [DHP03]. Definition . A family
T ⊂ L ( X, Y ) is called R -bounded if there exist a C > p ∈ [1 , ∞ ) such that for all N ∈ N , T j ∈ T , x j ∈ X and all independentsymmetric {− , } -valued random variables ε j on a probability space (Ω , A , P ) for j = 1 , ..., N , we have that(2.3) (cid:13)(cid:13)(cid:13) N X j =1 ε j T j x j (cid:13)(cid:13)(cid:13) L p (Ω ,Y ) ≤ C p (cid:13)(cid:13)(cid:13) N X j =1 ε j x j (cid:13)(cid:13)(cid:13) L p (Ω ,X ) . The smallest C p > R p -bound of T and denotedby R p ( T ).By Kahane’s inequality, (2.3) holds for all p ∈ [1 , ∞ ) if it holds for one p ∈ [1 , ∞ ).Therefore, we will drop the p -dependence of R p ( T ) in the notation and write R ( T ). Lemma 2.3. a) Let Z be a third Banach space and let T , S ⊂ L ( X, Y ) as well as U ⊂ L ( Y, Z ) be R -bounded. Then T + S , T ∪ S and UT are R -bounded as well andwe have R ( T + S ) , R ( T ∪ S ) ≤ R ( S ) + R ( T ) , R ( UT ) ≤ R ( U ) R ( T ) . Furthermore, if T denotes the closure of T with respect to the strong operatortopology, then we have R ( T ) = R ( T ) . OURIER MULTIPLIERS AND CYLINDRICAL BOUNDARY VALUE PROBLEMS 5 b) Contraction principle of Kahane:
Let p ∈ [1 , ∞ ) . Then for all N ∈ N , x j ∈ X, ε j as above, and for all a j , b j ∈ C with | a j | ≤ | b j | for j = 1 , . . . , N we have (2.4) (cid:13)(cid:13)(cid:13) N X j =1 a j ε j x j (cid:13)(cid:13)(cid:13) L p (Ω ,X ) ≤ (cid:13)(cid:13)(cid:13) N X j =1 b j ε j x j (cid:13)(cid:13)(cid:13) L p (Ω ,X ) . For M : Z n → L ( X, Y ) and 1 ≤ j ≤ n we inductively define the differences (discretederivatives)∆ ℓj M ( k ) := ∆ ℓ − j M ( k ) − ∆ ℓ − j M ( k − e j ) ( ℓ ∈ N , k ∈ Z n ) , where e j denotes the j -th unit vector in R n and where we have set ∆ j M ( k ) := M ( k ) ( k ∈ Z n ). As ∆ γ i i and ∆ γ j j commute for 1 ≤ i, j ≤ n , for a multi-index γ ∈ N n the expression ∆ γ M ( k ) := (cid:0) ∆ γ · · · ∆ γ n n M (cid:1) ( k ) ( k ∈ Z n )is well-defined. Given α, β, γ ∈ N n , we will write α ≤ γ ≤ β if α j ≤ γ j ≤ β j for all1 ≤ j ≤ n . We also set | α | := α + · · · + α n , := (0 , . . . , := (1 , . . . , X is called a UMD space or a Banach space of class HT if there exists a q ∈ (1 , ∞ ) (equivalently: if for all q ∈ (1 , ∞ )) the Hilberttransform defines a bounded operator in L q ( R , X ). A Banach space X is said tohave property ( α ) if there exists a C > N ∈ N , α ij ∈ C with | α ij | ≤
1, all x ij ∈ X , and all independent symmetric { +1 , − } -valued randomvariables ε (1) i on a probability space (Ω , A , P ) and ε (2) j on a probability space(Ω , A , P ) for i, j = 1 , . . . , N we have (cid:13)(cid:13)(cid:13) N X i,j =1 α ij ε (1) i ε (2) j x ij (cid:13)(cid:13)(cid:13) L (Ω × Ω ,X ) ≤ C (cid:13)(cid:13)(cid:13) N X i,j =1 ε (1) i ε (2) j x ij (cid:13)(cid:13)(cid:13) L (Ω × Ω ,X ) . The following result from Bu and Kim characterizes discrete Fourier multipliers by R -boundedness. Theorem 2.4 ([BK04]) . a) Let X, Y be UMD spaces and let
T ⊂ L ( X, Y ) be R -bounded. If M : Z n → L ( X, Y ) satisfies (cid:8) | k | | γ | ∆ γ M ( k ) : k ∈ Z n , ≤ γ ≤ (cid:9) ⊂ T , (2.5) then M defines a Fourier multiplier.b) If X, Y additionally enjoy property ( α ) , then (cid:8) k γ ∆ γ M ( k ) : k ∈ Z n , ≤ γ ≤ (cid:9) ⊂ T (2.6) is sufficient. In this case the set { T M : M satisfies condition (2.6) } ⊂ L (cid:0) L p ( Q n , X ) , L p ( Q n , Y ) (cid:1) is R -bounded again.Remark . In [BK04], Theorem 2.4 is stated with discrete derivatives ˜∆ definedin such a way that ∆ γ M ( k + γ ) = ˜∆ γ M ( k ). However, as for fixed γ ∈ { , } n thereexist c, C > c | k − γ | ≤ | k | ≤ C | k − γ | for k ∈ Z n \ { , } n , Lemma2.3 shows our formulation to be equivalent to the one in [BK04]. Throughout thisarticle, we will make use of this estimate frequently without any further comment. R. DENK, T. NAU
The following lemma states some properties for discrete derivatives, where ( S k ) k ∈ Z n and ( T k ) k ∈ Z n denote arbitrary commuting sequences in L ( X ). For α ∈ N n \ { } ,let Z α := n W = ( ω , . . . , ω r ); 1 ≤ r ≤ | α | , ≤ ω j ≤ α, ω j = , r X j =1 ω j = α o denote the set of all additive decompositions of α into r = r W multi-indices andset Z := {∅} and r ∅ := 0. For W ∈ Z α we set ω ∗ j := P rl = j +1 ω l . In the following, c α,β and c W will denote integer constants depending on α, β and W , respectively. Lemma 2.6. a) Leibniz rule:
For α ∈ N n we have ∆ α ( ST ) k = X ≤ β ≤ α c α,β (∆ α − β S ) k − β (∆ β T ) k ( k ∈ Z n ) . b) Let ( S − ) k := ( S k ) − exist for all k ∈ Z n . Then, for α ∈ N n we have ∆ α ( S − ) k = X W∈Z α c W ( S − ) k − α r W Y j =1 (∆ ω j S ) k − ω ∗ j ( S − ) k − ω ∗ j ( k ∈ Z n ) . Proof.
We will show both assertions by induction on | α | , the case | α | = 0 beingobvious.a) By definition, we have∆ e j ( ST ) k = ( ST ) k − ( ST ) k − e j = S k − e j (∆ e j T ) k + (∆ e j S ) k T k , and for α ′ := α − e j where α j = 0 we obtain∆ α ( ST ) k = ∆ e j X β ≤ α ′ c α ′ β (∆ α ′ − β S ) k − β (∆ β T ) k = X β ≤ α ′ c α ′ β (cid:16) (∆ α ′ − β S ) k − ( β + e j ) (∆ β + e j T ) k + (∆ α ′ + e j − β S ) k − β (∆ β T ) k (cid:17) = X β ≤ α c αβ (∆ α − β S ) k − β (∆ β T ) k . b) For | α | ≥
1, we apply a) to SS − and get0 = ∆ α ( SS − ) k = S k − α (∆ α S − ) k + X β<α c αβ (∆ α − β S ) k − β (∆ β S − ) k . Hence(∆ α S − ) k = − ( S − ) k − α X β<α c αβ (∆ α − β S ) k − β (∆ β S − ) k = − X β<α X W∈Z β c W ( S − ) k − α (∆ α − β S ) k − β ( S − ) k − β r W Y j =1 (∆ ω j S ) k − ω ∗ j ( S − ) k − ω ∗ j = X W∈Z α c W ( S − ) k − α (∆ ω S ) k − ω ∗ ( S − ) k − ω ∗ r W Y j =2 (∆ ω j S ) k − ω ∗ j ( S − ) k − ω ∗ j . (cid:3) OURIER MULTIPLIERS AND CYLINDRICAL BOUNDARY VALUE PROBLEMS 7
Definition . Consider a polynomial P : R n → C ; ξ P ( ξ ) and let P denoteits principal part.a) P is called elliptic if P ( ξ ) = 0 for ξ ∈ R n \ { } .b) Let φ ∈ (0 , π ) and let Σ φ := { λ ∈ C \ { } : | arg( λ ) | < φ } be the open sectorwith angle φ . Then P is called parameter-elliptic in Σ π − φ if λ + P ( ξ ) = 0 for( λ, ξ ) ∈ Σ π − φ × R n \ { (0 , ) } . In this case, ϕ P := inf { φ ∈ (0 , π ) : P is parameter-elliptic in Σ π − φ } is called the angle of parameter-ellipticity of P . Remark . a) By quasi-homogeneity of ( λ, ξ ) λ + P ( ξ ), we easily see that P is parameter-elliptic in Σ π − φ if and only if for all polynomials N with deg N ≤ deg P there exist C > G ⊂ R n such that the estimate | ξ | m | N ( ξ ) | ≤ C | λ + P ( ξ ) | holds for all λ ∈ Σ π − φ , all 0 ≤ m ≤ deg P − deg N andall ξ ∈ R n \ G .b) In the same way, P is elliptic if and only if the assertion in a) is valid for λ = 0.c) By induction, one can see that for | α | ≤ deg P the discrete polynomial ∆ α P ( k )defines a polynomial of degree not greater than deg P − | α | . If P is elliptic, thisimplies | k | | α | | ∆ α P ( k ) | ≤ C | P ( k ) | ( k ∈ Z n \ G ) with a finite set G ⊂ Z n . Proposition 2.9.
Let A be a closed linear operator in a UMD space X . Considerpolynomials P, Q : Z n → C such that • P and Q are elliptic, • (cid:0) P ( k ) + Q ( k ) A (cid:1) − exists for all k ∈ Z n , • (cid:8) P ( k ) (cid:0) P ( k ) + Q ( k ) A (cid:1) − : k ∈ Z n (cid:9) is R -bounded.Then for every polynomial N with deg N ≤ deg P the map M : Z n → L ( X ) : k N ( k ) (cid:0) P ( k ) + Q ( k ) A (cid:1) − defines an L p -multiplier for < p < ∞ .Proof. Lemma 2.6 yields | k | | γ | ∆ γ M ( k ) = X β ≤ γ X W∈Z β c W | k | | γ − β | (∆ γ − β N )( k − β ) (cid:0) P ( k − β ) + Q ( k − β ) A (cid:1) − · r W Y j =1 | k | | ω j | (cid:0) ∆ ω j P ( k − ω ∗ j ) + ∆ ω j Q ( k − ω ∗ j ) A (cid:1)(cid:0) P ( k − ω ∗ j ) + Q ( k − ω ∗ j ) A (cid:1) − . By Remark 2.8, we know that deg(∆ γ − β N ) ≤ deg N −| γ − β | . This and the ellipticityof P imply | k | | γ − β | | ∆ γ − β N ( k ) | ≤ C | P ( k ) | for k ∈ Z n \ G with a finite set G ⊂ Z n .By Kahane’s contraction principle, we obtain the R -boundedness of n | k | | γ − β | ∆ γ − β N ( k − β ) (cid:0) P ( k − β ) + Q ( k − β ) A (cid:1) − : k ∈ Z n \ G o . Since Q ( k ) A (cid:0) P ( k ) + Q ( k ) A (cid:1) − = id X − P ( k ) (cid:0) P ( k ) + Q ( k ) A (cid:1) − , in the same way the R -boundedness of n | k | | ω j | ∆ ω j Q ( k − ω ∗ j ) A (cid:0) P ( k − ω ∗ j ) + Q ( k − ω ∗ j ) A (cid:1) − : k ∈ Z n \ G o R. DENK, T. NAU follows from the ellipticity of Q . Now the assertion follows from Lemma 2.3 andTheorem 2.4. (cid:3) Proposition 2.9 is closely related to the concept of 1-regularity of complex-valuedsequences, introduced in [KL04] for the one dimensional case n = 1. In fact, if Q ( k ) = 0 for all k ∈ Z n , we may write M ( k ) = N ( k ) Q ( k ) (cid:0) P ( k ) Q ( k ) + A (cid:1) − . Hence, for n = 1we enter the framework of [KLP09, Proposition 5.3], i.e. M ( k ) = a k ( b k − A ) − with( a k ) k ∈ Z , ( b k ) k ∈ Z ⊂ C . We will give a generalization of this concept to arbitrary n and briefly indicate the connection to the results above. Definition . We call a pair of sequences ( a k , b k ) k ∈ Z n ⊂ C ≤ γ ≤ there exist a finite set K ⊂ Z n and a constant C > | k γ | max {| (∆ γ a ) k | , | (∆ γ b ) k |} ≤ C | b k | ( k ∈ Z n \ K ) . We say the pair ( a k , b k ) k ∈ Z n is strictly 1-regular if | k γ | can be replaced by | k | | γ | in (2.7). A sequence ( a k ) k ∈ Z n is called (strictly) 1-regular if ( a k , a k ) k ∈ Z n has thisproperty. Remark . a) In the case n = 1, a sequence ( a k ) k ∈ Z ⊂ C \ { } is 1-regular in Z inthe sense of Definition 2.10 if and only if the sequence (cid:0) k ( a k +1 − a k ) a k (cid:1) k ∈ Z is bounded.Hence our definition extends the one from [KL04] for a sequence ( a k ) k ∈ Z .b) With γ = 0 the definition especially requests | a k | ≤ C | b k | for k ∈ Z n \ K .c) Strict 1-regularity implies 1-regularity. If n = 1 both concepts are equivalent.d) Subject to the assumptions of Proposition 2.9, let Q ( k ) = 0 for k ∈ Z n . Thenthe pair (cid:0) N ( k ) Q ( k ) , P ( k ) Q ( k ) (cid:1) k ∈ Z n is strictly 1-regular.e) Again from Lemma 2.6 we deduce the following variant of Proposition 2.9: Let b k ∈ ρ ( A ) for all k ∈ Z n , let R ( { b k ( b k − A ) − : k ∈ Z n \ G } ) < ∞ for somefinite subset G ⊂ Z n , and let ( a k , b k ) k ∈ Z n be strictly 1-regular. Then M ( k ) := a k ( b k − A ) − defines a Fourier multiplier.3. ν -periodic boundary value problems Definition . Let X be a Banach space, m ∈ N , n ∈ N and ν ∈ C n . We set D α := D α . . . D α n n with D j = − i ∂∂j and denote by W m,pν,per ( Q n , X ) the space of all u ∈ W m,p ( Q n , X ) such that for all j ∈ { , . . . , n } and all | α | < m it holds that( D α u ) | x j =2 π = e πν j ( D α u ) | x j =0 . For sake of convenience we set W m,pper ( Q n , X ) := W m,p ,per ( Q n , X ).We give some helpful characterizations of the space W m,pν,per ( Q n , X ) where we omitthe rather simple proof. Lemma 3.2.
The following assertions are equivalent: (i) u ∈ W m,pν,per ( Q n , X ) . (ii) u ∈ W m,p ( Q n , X ) and for all | α | ≤ m it holds that ( e − ν · D α u )ˆ( k ) = ( k − iν ) α ( e − ν · u )ˆ( k ) for all k ∈ Z n . OURIER MULTIPLIERS AND CYLINDRICAL BOUNDARY VALUE PROBLEMS 9 (iii)
There exists v ∈ W m,pper ( Q n , X ) such that u = e ν · v . The following lemma characterizes multipliers such that the associated operatorsmap L p ( Q n , X ) into W α,pper ( Q n , X ). The proof follows the one for the case n = 1 of[AB02, Lemma 2.2]. Lemma 3.3.
Let ≤ p < ∞ , m ∈ N and M : Z n → L ( X ) . Then the followingassertions are equivalent: (i) M is an L p -multiplier such that the associated operator T M ∈ L ( L p ( Q n , X )) maps L p ( Q n , X ) into W m,pper ( Q n , X ) . (ii) M α : Z n → L ( X ) , k k α M ( k ) is an L p -multiplier for all | α | = m . Let X be a UMD space and A be a closed linear operator in X . With n ∈ N and ν ∈ C n we consider the boundary value problem in Q n given by(3.1) A ( D ) u = f ( x ∈ Q n ) , ( D β u ) | x j =2 π − e πν j ( D β u ) | x j =0 = 0 ( j = 1 , . . . , n ; | β | < m ) . In view of the boundary conditions, we refer to the boundary value problem (3.1)as ν -periodic. Here A ( D ) := P ( D ) + Q ( D ) A := X | α |≤ m p α D α + X | α |≤ m q α D α A with m , m ∈ N , m ≤ m , and p α , q α ∈ C . In what follows, with m := m wefrequently write A ( D ) = P | α |≤ m ( p α D α + q α D α A ) where additional coefficients q α ,that is, where m < | α | ≤ m , are understood to be equal to zero. Besides that wedefine the complex polynomials P ( z ) := P | α |≤ m p α z α and Q ( z ) := P | α |≤ m q α z α for z ∈ C n . Definition . A solution of the boundary value problem (3.1) is understood asa function u ∈ W m ,pν,per ( Q n , X ) ∩ W m ,p ( Q n , D ( A )) such that A ( D ) u ( x ) = f ( x ) foralmost every x ∈ Q n . Remark . Since the trace operator with respect to one direction and tangentialderivation commute, the ν -periodic boundary conditions as imposed in (3.1) areequivalent to( D ℓj u ) | x j =2 π − e πν j ( D ℓj u ) | x j =0 = 0 ( j = 1 , . . . , n, ≤ ℓ < m ) . Theorem 3.6.
Let < p < ∞ , and assume P and Q to be elliptic. Then thefollowing assertions are equivalent: (i) For each f ∈ L p ( Q n , X ) there exists a unique solution of (3.1) . (ii) (cid:0) P ( k − iν ) + Q ( k − iν ) A (cid:1) − ∈ L ( X ) exists for k ∈ Z n and M α ( k ) := k α (cid:0) P ( k − iν ) + Q ( k − iν ) A (cid:1) − defines a Fourier multiplier for every | α | = m . (iii) (cid:0) P ( k − iν ) + Q ( k − iν ) A (cid:1) − ∈ L ( X ) exists for k ∈ Z n and for all | α | = m there exists a finite subset G ⊂ Z n such that the sets { M α ( k ); k ∈ Z n \ G } are R -bounded. Proof. (i) ⇒ (ii): Let f ∈ L p ( Q n , X ) be arbitrary and let u be a solution of (3.1)with right-hand side e ν · f . Then e − ν · A ( D ) u = f .To compute the Fourier coefficients, we first remark that (cid:0) e − ν · P ( D ) u (cid:1) ˆ( k ) = P ( k − iν )( e − ν · u )ˆ( k )by Lemma 3.2. Concerning e − ν · Q ( D ) Au , note that by definition of a solution wehave Au ∈ W m ,p ( Q n , X ). Due to the closedness of A , we obtain D α Au = AD α u for | α | ≤ m , and consequently Au ∈ W m ,pν,per ( Q n , X ). Now we can apply Lemma 3.2to see (cid:0) e − ν · Q ( D ) Au (cid:1) ˆ( k ) = Q ( k − iν )( e − ν · Au )ˆ( k ) = Q ( k − iν ) A ( e − ν · u )ˆ( k ) . Writing k ν := k − iν for short, we obtain(3.2) (cid:0) P ( k ν ) + Q ( k ν ) A (cid:1)(cid:0) e − ν · u (cid:1) ˆ( k ) = ˆ f ( k ) . For arbitrary y ∈ X and k ∈ Z n , the choice f := e i k · y shows (cid:0) P ( k ν ) + Q ( k ν ) A (cid:1) tobe surjective. Let z ∈ D ( A ) such that (cid:0) P ( k ν ) + Q ( k ν ) A (cid:1) z = 0. For fixed k ∈ Z n set v := e i k · z and u := e ν · v . Then P ( k ν ) (cid:0) e − ν · u (cid:1) ˆ( k ) + Q ( k ν ) A (cid:0) e − ν · u (cid:1) ˆ( k ) = 0 . As ( e − ν · u )ˆ( m ) = 0 for all m = k , this gives A ( D ) u = 0, hence v = u = 0 and z = 0.Altogether we have shown bijectivity of P ( k ν )+ Q ( k ν ) A for k ∈ Z n . The closednessof A yields (cid:0) P ( k ν ) + Q ( k ν ) A (cid:1) − ∈ L ( X ).For f ∈ L p ( Q n , X ) let u be a solution of (3.1) with right hand side e ν · f and v := e − ν · u . Then v ∈ W m ,pper ( Q n , X ), and (3.2) impliesˆ v ( k ) = (cid:0) P ( k ν ) + Q ( k ν ) A (cid:1) − ˆ f ( k ) . This shows M : Z n → L ( L p ( Q n , X )); k (cid:0) P ( k ν ) + Q ( k ν ) A (cid:1) − to be a Fourier multiplier such that T M maps L p ( Q n , X ) into W m ,pper ( Q n , X ). Dueto Lemma 3.3, we have that M α is a Fourier multiplier for all | α | = m .(ii) ⇒ (iii): This follows as in [AB02, Prop. 1.11].(iii) ⇒ (i): For k = it holds that P ( k ν ) (cid:0) P ( k ν ) + Q ( k ν ) A (cid:1) − = P ( k ν ) n P j =1 k m e j (cid:18) n X j =1 k m e j (cid:0) P ( k ν ) + Q ( k ν ) A (cid:1) − (cid:19) and as there exists C > | P ( k ν ) | ≤ C | P nj =1 k m e j | for k ∈ Z n \ G withsuitably chosen finite G ⊂ Z n , Lemma 2.3 shows that the set n P ( k ν ) (cid:0) P ( k ν ) + Q ( k ν ) A (cid:1) − : k ∈ Z n \ G o OURIER MULTIPLIERS AND CYLINDRICAL BOUNDARY VALUE PROBLEMS 11 is R -bounded as well. By Proposition 2.9 it follows that M α for | α | = m as wellas P ( · − iν ) M are Fourier multipliers. For arbitrary f ∈ L p ( Q n , X ) we thereforeget v := T M ( e − ν · f ) ∈ W m ,pper ( Q n , X ). As(3.3) Q ( k ν ) A (cid:0) P ( k ν ) + Q ( k ν ) A (cid:1) − = id X − P ( k ν ) (cid:0) P ( k ν ) + Q ( k ν ) A (cid:1) − ,Q ( · − iν ) AM is a Fourier multiplier, too. By ellipticity of Q and Lemma 2.3 again,the same holds for k α A (cid:0) P ( k ν ) + Q ( k ν ) A (cid:1) − , | α | ≤ m .Set u := e ν · v = e ν · T M e − ν · f . Then u solves (3.1) by construction, and Lemma 3.3yields u ∈ W m ,pν,per ( Q n , X ) and Au ∈ W m ,pν,per ( Q n , X ). Finally, uniqueness of u followsimmediately from the uniqueness of the representation as a Fourier series. (cid:3) Remark . We have seen in the proof that if one of the equivalent conditions inTheorem 3.6 is satisfied, we have Au ∈ W m ,pν,per ( Q n , X ). In particular, we get( D β Au ) | x j =2 π − e πν j ( D β Au ) | x j =0 = 0 ( j = 1 , . . . , n ; | β | < m )as additional boundary conditions in (3.1).Theorem 3.6 enables us to treat Dirichlet-Neumann type boundary conditions on˜ Q n := (0 , π ) n for symmetric operators, provided P and Q are of appropriate struc-ture. More precisely, we call a differential operator A ( D ) = P | α |≤ m ( p α D α + q α D α A )symmetric if for all | α | ≤ m either p α = q α = 0 or α ∈ N n . In particular, m is even. As examples, the operators A ( D t ) := D t + A and A ( D , D ) :=( D + D ) + ( D + D ) A are symmetric and satisfy the conditions on P and Q from Theorem 3.6.In each direction j ∈ { , . . . , n } , we will consider one of the following boundaryconditions:(i) D ℓj u | x j =0 = D ℓj u | x j = π = 0 ( ℓ = 0 , , . . . , m − D ℓj u | x j =0 = D ℓj u | x j = π = 0 ( ℓ = 1 , , . . . , m − D ℓj u | x j =0 = D ℓ +1 j u | x j = π = 0 ( ℓ = 0 , , . . . , m − D ℓ +1 j u | x j =0 = D ℓj u | x j = π = 0 ( ℓ = 0 , , . . . , m − u | x j =0 = 0 and D j u | x j = π = 0. Therefore, we refer to these boundary conditionsas conditions of Dirichlet-Neumann type. Note that the types may be different indifferent directions. Theorem 3.8.
Let A ( D ) be symmetric, with P and Q being elliptic, and let theboundary conditions be of Dirichlet-Neumann type as explained above. Define ν ∈ C n by setting ν j := 0 in cases (i) and (ii) and ν j := i/ in cases (iii) and (iv). Iffor this ν one of the equivalent conditions of Theorem 3.6 is fulfilled, then for each f ∈ L p ( ˜ Q n , X ) there exists a unique solution u ∈ W m,p ( ˜ Q n , X ) of A ( D ) u = f satisfying the boundary conditions.Proof. Following an idea from [AB02], the solution is constructed by a suitable evenor odd extension of the right-hand side from (0 , π ) n to ( − π, π ) n . For simplicity ofnotation, let us consider the case n = 2 and boundary conditions of type (ii) in direction x and of type (iii) in direction x . By definition, this leads to ν = 0 and ν = i .Let f ∈ L p ( ˜ Q n , X ) be arbitrary. First considering the even extension of f to therectangle ( − π, π ) × (0 , π ) and afterwards its odd extension to ( − π, π ) × ( − π, π ),we end up with a function F which fulfills F ( x , x ) = F ( − x , x ) as well as F ( x , x ) = − F ( x , − x ) a.e. in ( − π, π ) .Now we can apply Theorem 3.6 with ν = ( ν , ν ) T as above. (Here and in thefollowing, the result of Theorem 3.6 has to be shifted from the interval (0 , π ) n tothe interval ( − π, π ) n .) This yields a unique solution U of(3.4) A ( D ) U = F in ( − π, π ) × ( − π, π ) ,D ℓ U | x = − π = D ℓ U | x = π ( ℓ = 0 , . . . , m − , − D ℓ U | x = − π = D ℓ U | x = π ( ℓ = 0 , . . . , m − . Symmetry of A ( D ) now shows that V ( x , x ) := U ( − x , x ) and V ( x , x ) := − U ( x , − x ) ( x ∈ ( − π, π ) ) are solutions of (3.4) as well. By uniqueness, V = U = V follows.Hence U x := U ( · , x ) ∈ W m,p (( − π, π ) , X ) ⊂ C m − (( − π, π ) , X ) for a.e. x ∈ ( − π, π ) is even. Together with symmetry of U x due to (3.4), this yields U ( ℓ ) x (0) = U ( ℓ ) x ( π ) = 0 ( ℓ = 1 , , . . . , m − . ) . Accordingly for a.e. x ∈ ( − π, π ) we have that U x is odd, and antisymmetry dueto (3.4) gives U ( ℓ ) x (0) = U ( ℓ +1) x ( π ) = 0 ( ℓ = 0 , , . . . , m − . Therefore, u := U | (0 ,π ) n solves A ( D ) u = f with boundary conditions (ii) for j = 1and (iii) for j = 2.For arbitrary n ∈ N and arbitrary boundary conditions of Dirichlet-Neumann type,the construction of the solution follows the same lines. Here we choose even exten-sions in the cases (ii) and (iv) and odd extensions in the cases (i) and (iii).On the other hand, let u be a solution of A ( D ) u = f satisfying boundary conditionsof Dirichlet-Neumann type. We extend u and f to U and F on ( − π, π ) n as describedabove. Then U ∈ W m,p (( − π, π ) n , X ), Q ( D ) AU ∈ L p (( − π, π ) n , X ) and due tosymmetry of A ( D ) we see that, apart from a shift, U solves (3.1) with right-handside F and ν defined as above. Thus, uniqueness of U yields uniqueness of u andthe proof is complete. (cid:3) Remark . In case n = 1 ellipticity of P does no longer force P to be of even order.Hence the same results can be achieved if A ( D ) is antisymmetric in the obvioussense, e.g. A ( D t ) := D t + D t + D t A .4. Maximal regularity of cylindrical boundary value problems with ν -periodic boundary conditions Let F be a UMD space and let Ω := Q n × V ⊂ R n + n V with V ⊂ R n V . For x ∈ Ω wewrite x = ( x , x ) ∈ Q n × V , whenever we want to refer to the cylindrical geometryof Ω. Accordingly, we write α = ( α , α ) ∈ N n × N n V for a multiindex α ∈ N n + n V and D α = D ( α ,α ) =: D α D α . OURIER MULTIPLIERS AND CYLINDRICAL BOUNDARY VALUE PROBLEMS 13
In the sequel we investigate the vector-valued parabolic initial boundary value prob-lem(4.1) u t + A ( x, D ) u = f ( t ∈ J, x ∈ Q n × V ) ,B j ( x, D ) u = 0 ( t ∈ J, x ∈ Q n × ∂V, j = 1 , . . . , m V ) , ( D β u ) | x j =2 π − e πν j ( D β u ) | x j =0 = 0 ( j = 1 , . . . , n ; | β | < m ) ,u (0 , x ) = u ( x ) ( x ∈ Q n × V ) . Here J := [0 , T ), 0 < T ≤ ∞ , denotes a time interval, and the differential operator A ( x, D ) has the form A ( x, D ) = P ( x , D ) + Q ( D ) A V ( x , D ):= X | α |≤ m p α ( x ) D α + X | α |≤ m q α D α A V ( x , D ) . The operator A V ( x , D ) is assumed to be of order 2 m V and is augmented withboundary conditions B j ( x, D ) = B j ( x , D ) ( j = 1 , . . . , m V )with operators B j ( x , D ) of order m j < m V acting on the boundary of V .This class of equations fits into the framework of Section 3 if we define the op-erator A = A V in Section 3 as the L p -realization of the boundary value problem(( A V ( x , D ) , B ( x , D ) , . . . , B m V ( x , D )). More precisely, for 1 < p < ∞ wedefine the operator A V in L p ( V, F ) by D ( A V ) := { u ∈ W m,p ( V, F ) : B j ( x , D ) u = 0 ( j = 1 , . . . , m V ) } ,A V u := A V ( x, D ) u := A V ( x , D ) u ( u ∈ D ( A V )) . Throughout this section, we will assume that the boundary value problem ( A V , B ,. . . , B m V ) satisfies standard smoothness and parabolicity assumptions as, e.g., givenin [DHP03, Theorem 8.2]. In particular, V is assumed to be a domain with compact C m V -boundary, and ( A V , B , . . . , B m V ) is assumed to be parameter-elliptic withangle ϕ A V ∈ [0 , π ). For the notion of parameter-ellipticity of a boundary valueproblem, we refer to [DHP03, Section 8.1].Recall that a sectorial operator A is called R -sectorial if there exists a θ ∈ (0 , π )such that(4.2) R (cid:0) { λ ( λ + A ) − : λ ∈ Σ π − θ } (cid:1) < ∞ . For an R -sectorial operator, φ R A := inf { θ ∈ (0 , π ) : (4.2) holds } is called the R -angleof A (see [DHP03, p. 42]). The R -sectoriality of an operator is closely related tomaximal regularity. Recall that a closed and densely defined operator in a Banachspace X has maximal L q -regularity if for each f ∈ L q ((0 , ∞ ) , X ) there exists aunique solution w : (0 , ∞ ) → D ( A ) of the Cauchy problem w t + Aw = f in (0 , ∞ ) ,w (0) = 0satisfying the estimate k w t k L q ((0 , ∞ ) ,X ) + k Aw k L q ((0 , ∞ ) ,X ) ≤ C k f k L q ((0 , ∞ ) ,X )4 R. DENK, T. NAU with a constant C independent of f . By a well-known result due to Weis [Wei01,Thm. 4.2], R -sectoriality in a UMD space with R -angle less than π is equivalent tomaximal L q -regularity for all 1 < q < ∞ . In [DHP03] it was shown that standardparameter-elliptic problems lead to R -sectorial operators: Proposition 4.1 ([DHP03, Theorem 8.2]) . Under the assumptions above, for each φ > ϕ A V there exists a δ V = δ V ( φ ) ≥ such that A V + δ V is R -sectorial with R -angle φ R A V + δ V ≤ φ . Moreover, (4.3) R ( { λ − | α | mV D α ( λ + A V + δ V ) − ; λ ∈ Σ π − φ , ≤ | α | ≤ m V } ) < ∞ . We will show that under suitable assumptions on P and Q , R -sectoriality of A V implies R -sectoriality of the operator related to the cylindrical problem (4.1). Forthis consider the resolvent problem corresponding to (4.1) which is given by(4.4) λu + A ( x, D ) u = f ( x ∈ Q n × V ) ,B j ( x, D ) u = 0 ( x ∈ Q n × ∂V, j = 1 , . . . , m V ) , ( D β u ) | x j =2 π − e πν j ( D β u ) | x j =0 = 0 ( j = 1 , . . . , n, | β | < m ) . For sake of readability, we assume that m = 2 m V . The L p (Ω , F )-realization of theboundary value problem (4.4) is defined as D ( A ) := { u ∈ W m ,p (Ω , F ) ∩ W m ,pν,per ( Q n , L p ( V, F )) : B j ( x, D ) u = 0 ( j = 1 , ..., m V ) , A V ( x, D ) u ∈ W m ,p ( Q n , L p ( V, F )) } , A u := A ( x, D ) u ( u ∈ D ( A )) . Remark . a) Since m ≤ m it holds that D ( A ) = W m ,p (Ω , F ) ∩ W m ,pν,per ( Q n , L p ( V, F )) ∩ W m ,p ( Q n , D ( A V )) . b) The following techniques apply as well to equations with mixed orders m =2 m V . Then, in the definition of D ( A ), the space W m ,p (Ω , F ) has to be replacedby { u ∈ L p (Ω , F ) : D α u ∈ L p (Ω , F ) for | α | m + | α | m V ≤ } .4.1. Constant coefficients.
We first assume P ( x , D ) = P ( D ) and Q ( x , D ) = Q ( D ) to have constant coefficients and set A := A ( x , D ) := P ( D ) + Q ( D ) (cid:0) A V + δ V (cid:1) . With A u := A u for u ∈ D ( A ) := D ( A ) we formally get ( λ + A ) − = e ν · T M λ e − ν · where T M λ denotes the associated operator to M λ ( k ) := (cid:0) λ + P ( k − iν ) + Q ( k − iν )( A V + δ V ) (cid:1) − . More generally, the Leibniz rule shows D α ( λ + A ) − = D α e ν · T M λ e − ν · = X β ≤ α g β ( ν ) e ν · T M βλ e − ν · , where g β is a polynomial depending on β . Here T M βλ denotes the associated operatorto M βλ ( k ) := k β D β (cid:0) λ + P ( k − iν ) + Q ( k − iν )( A V + δ V ) (cid:1) − where β = ( β , β ) T ≤ α . OURIER MULTIPLIERS AND CYLINDRICAL BOUNDARY VALUE PROBLEMS 15
Theorem 4.3.
Let < p < ∞ , let F be a UMD space enjoying property ( α ), letthe boundary valued problem ( A V , B ) fulfill the conditions of [DHP03, Theorem 8.2] with angle of parameter-ellipticity ϕ A V , and let ϕ > ϕ A V .For P and Q assume that (i) P is parameter-elliptic with angle ϕ P ∈ [0 , π ) , (ii) Q is elliptic, (iii) Q ( k − iν ) = 0 for all k ∈ Z n and there exists ϕ > ϕ P such that λ + P ( k − iν ) Q ( k − iν ) ∈ Σ π − ϕ holds true for all k ∈ Z n and all λ ∈ Σ π − ϕ .Then for each δ > the L p -realization A + δ of A + δ is R -sectorial with R -angle φ R A + δ ≤ ϕ . Moreover, it holds that (4.5) R ( { λ − | α | m D α ( λ + A + δ ) − : λ ∈ Σ π − φ , α ∈ N n + n v , ≤ | α | ≤ m } ) < ∞ . In particular, if ϕ < π then A + δ has maximal L q -regularity for every < q < ∞ ,i.e., the initial-boundary value problem (4.1) is well-posed in L q ( T, L p (Ω , F )) .Proof. Let α ∈ N n + n V , 0 ≤ | α | ≤ m = 2 m V , ≤ β ≤ α , ≤ γ ≤ , and φ > ϕ .For sake of convenience we drop the shift of A V , i.e. we assume δ V = 0. To prove(4.5), for arbitrary δ >
0, we apply Lemma 2.6 in order to calculate k γ ∆ γ M βλ + δ ( k ).In what follows we write k ν := k − iν for short again. As in the proof of Proposition2.9 it suffices to show that(4.6) { λ − | α | m k ω ∆ ω N ( k ) D β (cid:0) λ + δ + P ( k ν ) + Q ( k ν ) A V (cid:1) − : λ ∈ Σ π − φ , k ∈ Z n } with N ( k ) := k β and arbitrary ω ≤ γ ,(4.7) { k ω ∆ ω P ( k ν ) (cid:0) λ + δ + P ( k ν ) + Q ( k ν ) A V (cid:1) − : λ ∈ Σ π − φ , k ∈ Z n } , with < ω ≤ γ , and(4.8) { k ω ∆ ω Q ( k ν ) A V (cid:0) λ + δ + P ( k ν ) + Q ( k ν ) A V (cid:1) − : λ ∈ Σ π − φ , k ∈ Z n } with < ω ≤ γ are R -bounded. Due to our assumptions and Proposition 4.1, inparticular due to (4.3), for 0 ≤ | β | ≤ m = 2 m V the set (cid:26)(cid:18) λ + δ + P ( k ν ) Q ( k ν ) (cid:19) − | β | m D β (cid:18) λ + δ + P ( k ν ) Q ( k ν ) + A V (cid:19) − : λ ∈ Σ π − φ , k ∈ Z n (cid:27) is R -bounded. For β = 0 this yields the R -boundedness of(4.9) { (cid:0) λ + δ + P ( k ν ) (cid:1)(cid:0) λ + δ + P ( k ν ) + Q ( k ν ) A V (cid:1) − : λ ∈ Σ π − φ , k ∈ Z n } and with it the R -boundedness of(4.10) { Q ( k ν ) A V (cid:0) λ + δ + P ( k ν ) + Q ( k ν ) A V (cid:1) − : λ ∈ Σ π − φ , k ∈ Z n } . In particular, Q ( D ) A V ( λ + δ + A ) − f ∈ L p (Ω , F )) for f ∈ L p (Ω , F )).Since λ + P ( k ν ) = 0 for λ ∈ Σ π − φ by condition (iii), for each finite set G ⊂ Z n thereexists C > (cid:12)(cid:12) k ω ∆ ω P ( k ν ) (cid:12)(cid:12) ≤ C (cid:12)(cid:12) λ + δ + P ( k ν ) (cid:12)(cid:12) uniformly in λ ∈ Σ π − φ and k ∈ G . Together with parameter-ellipticity of P and Remark 2.8 this allows toapply the contraction principle of Kahane to prove (4.7). Similarly, ellipticity of Q proves (4.8) as well as D α A V ( λ + A ) − f ∈ L p (Ω , F )) for | α | ≤ m .To prove (4.6) first note that there exists C φ > | λ | ≤ C φ | λ + δ | holdsfor all λ ∈ Σ π − φ . As | β | ≤ | α | , with the same arguments as above, it suffices toshow the existence of a finite set G ⊂ Z n and C > | λ + δ | − | α | m (cid:12)(cid:12)(cid:12)(cid:12) k ω ∆ ω N ( k ) Q ( k ν ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:12)(cid:12)(cid:12)(cid:12) λ + δ + P ( k ν ) Q ( k ν ) (cid:12)(cid:12)(cid:12)(cid:12) − | β | m holds independently of λ ∈ Σ π − φ and k ∈ Z n \ G . Again by ellipticity of Q thereexists C > | k ν | m | Q ( k ν ) | ≤ C . Thus, it is sufficient to show | λ + δ | − | α | m | k ω ∆ ω N ( k ) || k ν | − m m | β | ≤ C (cid:12)(cid:12) λ + δ + P ( k ν ) (cid:12)(cid:12) − | β | m . The polynomial ∆ ω N has degree no larger than | β − ω | if ω ≤ β and we have∆ ω N ≡ k = , it is sufficient to consider (cid:12)(cid:12) M ( k ) (cid:12)(cid:12) | k | − m m | β | instead of | k ω ∆ ω N ( k ) || k ν | − m m | β | , where M ( k ) denotes a monomial of degree no larger than | β | . Hence, it remains to prove | λ + δ | − | α | m | M ( k ) || k | − m m | β | ≤ C (cid:12)(cid:12) λ + δ + P ( k ν ) (cid:12)(cid:12) − | β | m . Therefore, we end up with a left-hand side that is ( m , λ + δ, k ) of order no larger than m − | α | + | β | − m m | β | ≤ m − | β | . Thus,parameter-ellipticity of P yields existence of a finite set G ⊂ Z n such that | λ + δ | − | α | m | M ( k ) || k | − m m | β | ≤ C (cid:12)(cid:12) λ + δ + P ( k ν ) (cid:12)(cid:12) − | β | m for λ ∈ Σ π − φ and k ∈ Z n \ G .The last claim on maximal L q -regularity now follows from [Wei01, Thm. 4.2]. (cid:3) Remark . a) Since Q is elliptic, there exists a finite set G ⊂ Z n , such that Q ( k − iν ) = 0 for k ∈ Z n \ G . Instead of the stronger condition (iii), assume thatthere exists ϕ > ϕ P such that • λ + P ( k − iν ) = 0 for k ∈ G and λ ∈ Σ π − ϕ and • λ + P ( k − iν ) Q ( k − iν ) ∈ Σ π − ϕ for k ∈ Z n \ G and λ ∈ Σ π − ϕ .Set D ( ˜ A ) := W m ,pν,per ( Q n , L p ( V, F )) ∩ W m ,p ( Q n , D ( A V )). Then for each δ > L p -realization ˜ A + δ of A + δ is R -sectorial with R -angle φ R ˜ A + δ ≤ ϕ .b) Let P = P be given as homogeneous polynomial, let Q ≡ ν = i r with r ∈ R n ,and ϕ P + ϕ A V < π . Then for each ϕ > max { ϕ P , ϕ A V } condition (i) implies thatthere exists ϕ > ϕ A V such that condition (iii) holds true. Proof. a) First note that (cid:0) λ + δ + P ( k ν ) + Q ( k ν ) A V (cid:1) − still exists for all k ∈ Z n .In view of the domain of definition of ˜ A , equation (4.6) only has to be consideredwith β = and α = β . Moreover, the terms of (4.8) only appear in the formularfor k γ ∆ γ M βλ + δ ( k ) if k ∈ Z n \ G . Now the proof copies.b) Since ϕ P + ϕ A V < π , the claim follows readily due to homogeneity and parameter-ellipticity of P = P . Note that k ν = k + r ∈ R n and k ν = if and only if r = k = . (cid:3) OURIER MULTIPLIERS AND CYLINDRICAL BOUNDARY VALUE PROBLEMS 17
Remark . We have seen in the proof that A V u ∈ W m ,pν,per ( Q n , L p ( V, F )), i.e. thesolution u of (4.4) fulfills the further boundary condition( D β A V u ) | x j =2 π − e πν j ( D β A V u ) | x j =0 = 0 ( j = 1 , . . . , n ; | β | < m )(cf. Remark 3.7). Additionally, we have seen in the proof that(4.11) R ( { D α A V ( λ + A + δ ) − : λ ∈ Σ π − φ , ≤ | α | ≤ m } ) < ∞ . Remark . Consider again boundary value problems in (0 , π ) n × V with Dirichlet-Neumann type boundary conditions and a symmetric setting with respect to (0 , π ) n .As the extension and restriction operators defined above are bounded, Theorem3.8 immediately yields the related result for Dirichlet-Neumann type boundaryconditions. In particular, we obtain maximal regularity results also for boundaryconditions of mixed type (iii) and (iv).4.2. Non-constant coefficients of P . In this subsection, P ( x , D ) is allowed tohave non-constant coefficients, where we assume that(4.12) p α ∈ C per ( Q n ) for | α | = m ,p α ∈ L r η ( Q n ) for | α | = η < m , r η ≥ p, m − ηn − k > r η . Here C per ( Q n ) := { f ∈ C ([0 , π ] n ) : f | x j =0 = f | x j =2 π ( j = 1 , . . . , n ) } . However, inorder to apply perturbation results similar to [DHP03] or [NS], we assume Q ≡ A ( x, D ) := P ( x , D ) + A V ( x , D ) . Theorem 4.7.
Let < p < ∞ , let F be a UMD space enjoying property ( α ), let Ω := Q n × V , and let the boundary valued problem ( A V , B ) fulfill the conditions of [DHP03, Theorem 8.2] with angle of parameter-ellipticity ϕ A V .For P assume that • the coefficients satisfy (4.12) and • P is parameter-elliptic in Q n with angle ϕ P ∈ [0 , π − ϕ A V ) .Then for each ϕ > max { ϕ P , ϕ A V } there exists δ = δ ( ϕ ) ≥ such that the L p -realization A + δ of A + δ is R -sectorial with R -angle φ R A + δ ≤ ϕ . Moreover, wehave (4.13) R ( { λ − | α | m D α ( λ + A + δ ) − : λ ∈ Σ π − φ , α ∈ N n + n v , ≤ | α | ≤ m } ) < ∞ . In particular, if ϕ < π then A + δ has maximal L q -regularity for every < q < ∞ .Proof. In a first step, we consider P ( x, D ) to be a homogeneous differential operatorwith slightly varying coefficients. That is, we consider A va ( x, D ) := P ( D ) + R ( x , D ) + A V ( x , D ) , where P ( D ) := P | α | =2 m p α D α is assumed to have constant coefficients and R ( x , D ) := P | α | =2 m r α ( x ) D α fulfills P | α | =2 m k r α k ∞ ≤ η with η > R -sectorial operators(see [DHP03], [NS]) from Theorem 4.3 and Remark 4.4 b). In a second step, we choose a finite but sufficiently fine open covering of Q n . Inview of the periodicity of the top order coefficients, we may assume every open setof the covering, which intersects with R n \ Q n to be cut at the boundary of Q n andcontinued within Q n on the opposite side. By means of reflection, this enables us todefine local operators with slightly varying coefficients. With the help of a partitionof the unity and perturbation results for lower order terms subject to condition(4.12), just as in [NS], the claim follows. (cid:3) References [AB02] W. Arendt and S. Bu,
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