General parabolic mixed order systems in Lp and applications

(1)

General parabolic mixed order systems in L _p and applications

Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften

(Dr. rer. nat.)

vorgelegt von

Mario Kaip

am Fachbereich Mathematik und Statistik an der Mathematisch-Naturwissenschaftlichen Sektion

der Universität Konstanz

Tag der mündlichen Prüfung: 8. Februar 2012 Referenten: Prof. Dr. Robert Denk

(Universität Konstanz) Prof. Dr. Jürgen Saal

(Technische Universität Darmstadt)

Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-186447

(2)

(3)

Dedicated to Christina and

my parents

Cornelia & Michael

(4)

(5)

Acknowledgment

I would like to express my gratitude to several people who helped me during the years in which this thesis arose. First and foremost, I want to thank Christina Metzdorf for her great patience and care during those past years. This thesis would have been unthinkable without her cheerful nature and her support.

In the same breath I would like to thank my parents Cornelia Konrad-Kaip and Michael Konrad. All this time, they supported me by never losing faith in me. I would also like to thank them for their great solidarity in good times as well as in hard times.

Furthermore, I feel a deep gratitude towards my advisors Prof. Dr. Robert Denk and Prof. Dr.

Jürgen Saal for not only being my mentors but also my friends. I would particularly like to thank them for many stimulating discussions and for bringing me into contact with actual research topics. I have also deeply enjoyed the collaboration with Prof. Dr. Robert Denk while organizing with him many of his lectures and weekly exercises. Furthermore, I would like to show my gratitude to Prof. Dr. Wolfgang Watzlawek who deeply influenced my basic approach to mathematics.

Moreover, my thanks goes to the staff of the faculty of mathematics, especially Gerda Baumann, Rainer Janßen, and Gisela Cassola. Their uncomplicated and straightforward help was not only very pleasant but also extremely helpful throughout the past years.

Last but not least, I would like to thank my friends Dr. Thilo Moseler, Tobias Nau, Johannes Schnur, Tim Seger, Dr. Olaf Weinmann and the other members of the PDE research group in F5 for the great atmosphere and also for the numerous discussions. During the final stage of this thesis Martin Saal and Alexander Schöwe helped tremendously when taking over my task of preparing the weekly exercises, a help that is very much appreciated. I would also like to thank Johannes Schnur and Tim Seger for using their precious time to read this thesis. Besides, my special thanks goes to Johannes for his constant encouragement and his ’Gute Laune Versicherung’.

Konstanz, December 12, 2001 Mario Kaip

v

(6)

(7)

List of Figures

1.1 Spectrum,S_θ, and admissable curveΓ_ϕ . . . 9

1.2 Spectrum,Σ_δ, and admissable curveΓ_ϕ∪Γ_−ϕ . . . 9

2.1 Path of integration Γ(ε, R). . . 31

3.1 Regular Newton polygonN . . . 53

3.2 Newton polygon which is not regular in time . . . 54

3.3 Newton polygon which is not regular in space . . . 54

3.4 Illustration of the partition . . . 64

5.1 Illustration of the set of tuples(p, q)satisfying (5.4) . . . 104

7.1 Newton polygon of the α-β-system . . . 141

7.2 Newton polygon of the Cahn-Hilliard-Gurtin problem . . . 144

7.3 Newton polygon ofdetL . . . 147

7.4 Newton polygon for the spin-coating process . . . 150

7.5 Newton polygon ofP,B= 0. . . 156

7.6 Newton polygon ofP,B 6= 0. . . 156

7.7 Newton polygon for the two-phase Stefan problem . . . 161

ix

(10)

(11)

Introduction

The aim of this thesis is to develop a theory for the treatment of linear parabolic partial differential equations. Here we consider problems on the whole space as well as boundary value problems with dynamic boundary conditions. In particular, we establish results about mixed order systems on L_p, which help us to give elegant and short proofs of the well-posedness of linear parabolic problems. For the treatment of non-linear partial differential equations a thorough understanding of the associated linearized problem is indispensable. Here we want to mention the works of D. Bothe, R. Denk, J. Escher, M. Geissert, B. Grec, M. Hieber, J. Prüss, E.V. Radkevich, J. Saal, O. Sawada, G. Simonett, Y. Shibata, and S. Shimizu, cf. [EPS03], [BP07], [GR07], [PSS07], [SS07], [PS09], [PS10], [DGH⁺11], [SS11b]. There the authors handle non-linear problems inLp by a linearization process. To enhance the clarity we want to give a short overview of this approach, which mainly depends on the understanding of the associated linearized problem. This also helps to understand where our results come into play in this process. The standard process can be roughly described as follows:

• If the non-linear problem has a free boundary/interface, we first have to apply the Hanzawa transform. Here we assume that the free boundary/interface is locally given as the graph of an additional time-dependent unknown function. With the aid of this transform we can describe the problem on a time-independent, fixed domain (cf. [Han81], [EPS03], [PSS07], [DGH⁺11] for instance).

Alternatively, in some cases one can formulate the problem in Lagrangian coordinates. Then the domain of the transformed problem is also fixed, see for example the works of V.A. Solonnikov, Y. Shibata, and S. Shimizu, cf. [Sol84], [Sol03a], [Sol03b],[SS07], [SS11b], for instance. However, one disadvantage of this formulation is that it is not obvious how to recover the regularity of the free boundary or rather the height function.

• Then one can determine an associated linearization by replacing non-linear terms by their Fréchet derivative at an equilibrium.

• Localization and regularization techniques (freezing the coefficients) of the associated linearized problem then yield linear boundary value problems inRⁿ andRⁿ+ orR˙ⁿwith constant coefficients.

These kinds of problems are called model problems. One major challenge is the handling of the inhomogeneous and involved boundary conditions for those model problems arising in this context.

• In the next step the focus lies on the well-posedness of the associated model problem and the determination of the sharp regularities of the solution.

• Based on the well-posedness of the model problem and the regularity of the associated solution one can solve the non-linear problem by the contraction mapping principle.

With this approach it is possible to establish conditions for the local in time solvability of a non-linear problem. Our results come into play in the treatment of model problems inRⁿ andRⁿ+ orR˙ⁿ. Here we consider the well-posedness in anL_p- orL_p-L_q-setting. This is convenient because for sufficiently large pand q embeddings into the Hölder spacesC^α and BUC, algebra properties, and multiplier results are obtained. These properties are essential for handling the non-linear terms and therefore it is worthwhile to decouple the integrability condition for the time and space variables. Thus, a better understanding of the linear problem implies better results for the associated non-linear problem. In particular, the focus is on the reduction of regularity of the initial values.

1

(12)

Maximal Lp-regularity results of parabolic model problems have been established under various conditions, but often for homogeneous boundary conditions or rather restrictive conditions for the boundary operators. Here we want to state only a few of these results which are highly related to our approach.

Parabolic problems with boundary dynamics of relaxation type are discussed in a work of R. Denk, J. Prüss, and R. Zacher, cf. [DPZ08]. The authors establish a general L_p-theory of a special class of problems which contain equations in the interior and on the boundary of order one in time. Problems with both free boundary and surface diffusion are not contained in this class. Due to the pressure term and the necessary reductions, the linearizations of the two-phase Navier-Stokes equations can also not be handled by the results in [DPZ08]. Our results cover model problems with general boundary dynamics as well as the linearization of two-phase Navier-Stokes equations, cf. Section 7.7.

The problems considered in [DV08] exhibit inhomogeneous dynamical boundary conditions as well as additional unknown functions on the boundary. The authors use Newton polygon techniques and N-parabolicity (see the works of R. Denk, S. Gindikin, B. Grec, R. Mennicken, E.V. Radkevich, and L.R. Volevich, cf. [GV92], [DMV98], [Vol01], [GR07]) to handle the inherent inhomogeneities of the boundary operators. On basis of the analysis of the associated mixed order system on the boundary, the so-called Lopatinskii matrix, the authors derive well-posedness in the Hilbert spaceL2. The spaces there are defined by weight functions. With this approach it is not possible to obtain spaces with mixed scales.

The mixing of scales is actually not necessary in [DV08] due to the congruence of Bessel-potential spaces and Besov spaces (i.e.H₂^r=B₂^r). Here we generalize their approach to the general L_p-setting, which is more involved due to the appearance of Besov spaces on the boundary.

In [DHP07] the authors present anL_p-L_q-theory for inhomogeneous boundary conditions, where the boundary operators do not contain time-derivatives and the equation in the interior is of order one in time. A major difficulty of the Lp-Lq-theory developed there is the necessity of vector-valued Triebel- Lizorkin spaces. In Chapter 5 we present our main result for mixed order systems on Triebel-Lizorkin spaces, which is convenient for the analysis of free boundary value problems in anLp-Lq-setting.

In the discussion of free boundary value problems dynamical boundary conditions with inherent inhomogeneity occur (i.e. the problems are not parabolic in the sense of Petrovskii, cf. [DV08, Section 2]).

To our knowledge a generalLp-theory, respectively Lp-Lq-theory, for the associated mixed order system on the boundary has not been established for such problems so far. The established results of this thesis cover theLp-theory, respectivelyLp-Lq-theory, of such dynamic boundary conditions with inherent inhomogeneity.

We want to motivate the appearance of mixed order systems with a short well-known example, compare [EPS03], [GR07], [DSS08]. The model problem on Rⁿ+ for the Stefan problem with Gibbs-Thomson correction reads as follows











∂tu−∆u = 0 inR+×Rⁿ+, u+ ∆⁰h = g1 onR+×Rⁿ⁻¹,

∂th−∂nu = g2 onR+×Rⁿ⁻¹, u(t= 0) = 0 inRⁿ+,

h(t= 0) = 0 inRⁿ⁻¹.

(1)

The usual approach in the analysis of such problems is the reduction to the boundary. After employing formal Laplace and Fourier transform in t and x⁰ = (x₁, . . . , x_n), respectively, we obtain an ordinary differential equation foru(λ, ξˆ ⁰,·)andh(λ, ξˆ ⁰,·)for fixed(λ, ξ⁰), which is given by







ω(λ, ξ⁰)²u(λ, ξˆ ⁰, y)−∂_n²u(λ, ξˆ ⁰, y) = 0, x_n>0, ˆ

u(λ, ξ⁰,0)− |ξ⁰|²ˆh(λ, ξ⁰) = gˆ1(λ, ξ⁰), λˆh(λ, ξ⁰)−∂nu(λ, ξˆ ⁰) = gˆ2(λ, ξ⁰),

(2)

whereω(λ, ξ⁰) :=p

λ+|ξ⁰|². A stable solution of this can be obtained by ˆ

u(λ, ξ⁰, xn) = ˆΦ(λ, ξ⁰) exp(−ω(λ, ξ⁰)xn), xn >0

with an unknown functionΦ. The boundary conditions in (2) then yield the conditional equationˆ 1 −|ξ⁰|²

ω(λ, ξ⁰) λ

Φˆ ˆh

= ˆg1

ˆ g2

(3)

(13)

INTRODUCTION 3 for Φ and h. In order to find spaces which turn (1) into a well-posed problem we can consider the mapping properties of the associated mixed order system on the boundary in (3). In Theorem 4.12 and Theorem 5.40 we present our main results onN-parabolic mixed order systems, which can be applied to (3). Using these theorems it is possible to treat mixed order systems in anL_por even in anL_p-L_q-setting.

Altogether, we present a framework which enables us to handle a whole class of model problems like (1) by a unified approach. Therefore we can give proofs of well-posedness which are shorter, easier to handle, and more direct as in the existing literature.

We also want to emphasize that the system in (1) can also be treated by semigroup theory on a product space. This approach, however, yields regularity classes which are not suitable for the study of the corresponding non-linear problem (cf. [EPS03, p. 6]). Hence a treatment by mixed order systems is preferable.

In particular, we are able to treat free boundary value problems in an Lp-Lq-setting with a precise specification of all regularities on the boundary. The appropriate spaces on the boundary for a maximal regularity result are given in terms of vector-valued Triebel-Lizorkin spaces. See the works of R. Denk, M.Z. Berkolaiko, M. Hieber, J. Prüss, and P. Weidemaier in [Ber85], [Ber87a], [Ber87b], [Wei02], [Wei05], and [DHP07] on this topic. In Section 7.8 one can find the discussion of the two-phase Stefan problem with Gibbs-Thomson correction











∂_tu−∆u = 0 inR+× ˙ Rⁿ, JuK = 0 onR+×Rⁿ⁻¹, u+ ∆⁰h = g₁ onR+×Rⁿ⁻¹,

∂th−J∂nuK = g2 onR⁺×Rⁿ⁻¹, u(t= 0) = 0 in ˙

Rⁿ, h(t= 0) = 0 inRⁿ⁻¹

in anLp-Lq-setting. To the knowledge of the author this problem has not been discussed so far in the literature forp6=q.

Results on mixed order systems can also be found in papers of R. Denk, M. Dreher, M. Faierman, R. Mennicken, J. Saal, J. Seiler, L.R. Volevich, cf. [DMV98], [DV02a], [DV02b], [DSS08], [DSS09], [DF10], [DD11], and [DS11] for example. The approach in [DSS09] and [DS11] is based on pseudo-differential operator theory. The dependance of the covariablesλandzin [DS11] is more general as in [DSS09]. Both references deal with spaces defined by weight functions. Therefore they cannot handle ground spaces with different scales in time and space. Moreover, the approach via pseudo-differential operator methods has the restrictive assumption that the symbol has to be smooth in zero. Another approach in the sense of parameter-ellipticity can be found in [DV02b], [DF10], and [DD11]. Especially, in [DMV98, Theorem 3.14]

one can find an equivalent characterization ofN-parameter-elliptic matrices by the so-called Kozhevnikov condition (cf. [Koz96]). R. Denk, J. Saal, and J. Seiler use an interpretation in [DSS08] of the mixed order system by a jointH^∞-calculus of the time-derivative and the Laplacian. This approach is highly related to Lp-Fourier multipliers, and in contrast to the pseudo-differential approach it is possible to handle symbols which are non-smooth in zero. We also treat mixed order systems with a bounded joint H^∞-calculus by generalizing the results of [DSS08] to a bounded jointH^∞-calculus of the time-derivative

∂tand∇= (∂1, . . . , ∂n). In many applications this generalization becomes necessary due to the fact that the systems can include symbols which are not rotation invariant in space (see Chapter 7). The bounded H^∞-calculus of

∇+:= (∂t, ∂1, . . . , ∂n),

based on an approach by G. Dore and A. Venni, is the main result of Chapter 2 and is stated in Theorem 2.47. For the treatment ofLp-Lqproblems we deduce the boundedH^∞-calculus of∇+on Bessel- valued Triebel-Lizorkin spaces in Theorem 5.31. For a holomorphic functionf the operator f(∇+)and its domainD(f(∇+))are given abstractly. To determine the mapping properties or a characterization of the domain in terms of standard spaces, we introduce the concept of upper order functions in Chapter 3.

Using this we can formulate in Theorem 4.6, Theorem 4.7, and Corollary 4.9 mapping properties of symbols possessing upper order functions. In Chapter 5 we present the analog versions for operators on Triebel-Lizorkin spaces. These results concerning upper order functions are essential for the whole work.

In Theorem 4.12 and Theorem 5.40, in [DSS08] and [DV08] respectively, it turns out that N-parabolicity, i.e.N-parabolicity of the determinant, is crucial to ensure invertibility of a mixed order system.

(14)

Therefore it is essential to provide an easy-to-handle characterization for a large class of N-parabolic symbols. In the book [GV92] of S. Gindikin and L.R. Volevich one can find such a characterization by so-called ’non-vanishing principal parts’ for polynomials in multiple variables but this is not sufficient for free boundary problems. Note that the determinant of the system in (3) is not polynomial due to the square root in the definition ofω. The authors of [DSS08] give a similar characterization for polynomial- shaped symbols P

m∈I(√

λ+z²)^m¹λ^m²z^m³ (I ⊆N³0,(λ, z) ∈C²). For applications as in Section 7.7, a larger class of admissible symbols is needed. As a main result of Chapter 3 we present in Theorem 3.58 a characterization of a more general class of, not necessarily regular,N-parabolic symbols similar to the characterizations in [DSS08], [DV08], and [GV92]. The treatable symbols are of the form

X

`∈I

τ_`(λ, z)ϕ_`(λ)ψ_`(z)

for functionsτ`, ϕ`, andψ`which fulfill appropriate homogeneity assumptions.

In the treatment of mixed order systems resulting from a reduction-to-the-boundary process, spaces of mixed scales like0Bp^1−1/(2p)(R+, Lp(Rⁿ))∩Lp(R+, Bp^2−1/p(Rⁿ))appear. This mixing of Bessel-potential and Besov scale is one of the major difficulties. These spaces occur as canonical trace spaces on the boundary. Further we need a compatibility embedding of the used scales in the sense of

0B_p^s⁰^+s(R+, H_p^r⁰(Rⁿ))∩0H_p^s⁰(R+, B_p^r⁰^+r(Rⁿ))

,→ ₀B_p^s⁰^+σs(R+, H_p^r⁰^+(1−σ)r(Rⁿ))∩₀H_p^s⁰^+σs(R+, B_p^r⁰^+(1−σ)r(Rⁿ)), σ∈(0,1) (4) and analog versions for Triebel-Lizorkin spaces. The importance of such embeddings can already be seen in a simple example. Definingω(λ, ξ) :=p

λ+|ξ|² we have ω⁻¹(∇+) ∈ LIsom

0B^s_p(R+, Lp(Rⁿ)), 0B^s+1/2_p (R+, Lp(Rⁿ))∩0B_p^s(R+, H_p¹(Rⁿ)) , ω⁻¹(∇+) ∈ LIsom

Lp(R+, B_p^2s(Rⁿ)), 0H_p^1/2(R+, B_p^2s(Rⁿ))∩Lp(R+, B^2s+1_p (Rⁿ)) . Hence we need embeddings as in (4) to obtain the desired result

ω⁻¹(∇+)∈LIsom

0B_p^s(R+, Lp(Rⁿ))∩Lp(R+, B_p^2s(Rⁿ)),0B_p^s+1/2(R+, Lp(Rⁿ))∩Lp(R+, B_p^2s+1(Rⁿ)) . Sections 4.3 and 5.4 are devoted to the study of compatibility embeddings. For the proofs in Section 5.4 we need the concept of anisotropic Triebel-Lizorkin spaces with mixed norms as well as the representation of these spaces as an intersection containing vector-valued Triebel-Lizorkin spaces. Our approach to Triebel-Lizorkin spaces is based on interpolation. In the scalar case there is the well-known result

F_pq^s(Rⁿ) =

H_p^s₀⁰(Rⁿ), B_p^s¹₁(Rⁿ)

θ, θ∈(0,1) (5)

with1/p= (1−θ)p0+θp1,1/q= (1−θ)/2 +θp1, ands= (1−θ)s0+θs1(cf. [Tri78, Section 2.4.2 (12)]).

To the knowledge of the author a vector-valued version of (5) has not been discussed so far. In Propo- sition 5.10 we present an analog representation for Bessel-valued spaces. This result paves the way to generalize the results of Chapter 2 and Chapter 4 to Triebel-Lizorkin spaces by interpolation arguments.

Note that there is a coupling betweenpandqin (5). This coupling also occurs in Chapter 5 and cannot be avoided in the approach via interpolation.

In Chapter 7 we present a selection of applications. This chapter is divided into two parts. In the first part we present applications of our main results on partial differential equations on the whole space Rⁿ. Here we directly formulate the equations as a mixed order system without a reduction to a first order system. In the second part we present applications of our main results to boundary value problems onRⁿ+, respectively ˙

Rⁿ. This situation is more involved because we have to handle ground spaces which are related to trace spaces. In both cases this approach gives very short and straight proofs of well- posedness. Our main application in the second part are the two-phase Navier-Stokes equations with

(15)

INTRODUCTION 5 Boussinesq-Scriven surface introduced in [BP10]:











ρ∂tu−µ∆u+∇π = 0 in R+×R˙ⁿ⁺¹, divu = 0 in R+×R˙ⁿ⁺¹,

−µs∆xv−λs∇xdivxv−Jµ∂yvK−Jµ∇xwK = gv onR+×Rⁿ,

−2Jµ∂ywK+JπK−σ∆h−Gh = gw onR+×Rⁿ, JuK = 0 onR+×Rⁿ,

∂th−w+hb0,∇ih = gh onR+×Rⁿ, u(t= 0) = 0 in R˙ⁿ⁺¹, h(t= 0) = 0 in Rⁿ.

(6)

In addition to the dynamical boundary condition, (6) also includes a boundary condition of order two.

This surface viscosity changes the behavior of the system in the highest order fundamentally. We show in Section 7.7 that the associated mixed order system on the boundary fits into our theory. The well- posedness inL_pof this problem has not been proved in the literature before. Especially, we give a proof which solves the problem either with or without surface viscosity, i.e. λ_s, µ_s = 0 or λ_s ≥ 0, µ_s > 0, simultaneously.

As a final remark we want to highlight some advantages of our approach by mixed order systems. The mapping properties of a mixed order system can be determined easily by an application of Theorem 4.12, Theorem 5.40, respectively the condensed versions in Corollary 4.22 and Corollary 5.43. The conditions therein can be verified easily in applications. In particular, this comes from the easy-to-handle characterization ofN-parabolic symbols. Thanks to the accurate analysis of the order structure of mixed order systems we do not have to determine the inverse matrix of the system. Furthermore we can avoid the application of

(I) Kalton-Weis Theorem (cf. [KW01]),

(II) theorems of Dore-Venni type (cf. [DV87], [PS90]), (III) Weis’s Theorem (cf. [Wei01]),

(IV) interpolation theory

in the proofs of well-posedness. Altogether we can give direct, algorithmic, and systematic proofs of well-posedness which avoid the introduction of reduced problems as well as other auxiliary problems.

(16)

(17)

Chapter1

Remarks about joint H ^∞ -calculus

1.1 Fundamentals about H

^∞

-calculus

The concept ofH^∞-calculus goes back to A. McIntosh, see for example [McI86] and [CDMY96]. Another comprehensive and more recent work about theH^∞-calculus can be found in [Haa06]. In this section we want to introduce the jointH^∞-calculus of a tuple of sectorial and bisectorial operators. This special version can be found in [DV05]. Here we present the basic notation and results from [DV05], which are important for this thesis.

Definition 1.1. Forθ∈(0, π)andδ∈(0,^π₂)we define the sector S_θ and the bisector Σ_δ by S_θ := {rexp(iα) :r∈R+, α∈(−θ, θ)},

Σδ := {rexp(iα) :r∈R\ {0}, α∈(π/2−δ, π/2 +δ)}.

LetΓϕ be the curve which is parameterized byγϕ:R\ {0} →C, r7→ |r|exp(−iϕsgn(r))forϕ∈(0, π).

The curveΓϕ is called admissible forSθ if 0< θ < ϕ. An admissible curve forΣδ is a curve of the form Γϕ∪(−Γϕ) withδ+^π₂ < ϕ < π.

Definition 1.2. Let X be a Banach space.

(i) An operatorT:D(T)⊆X →X is said to be sectorial if (I) D(T)andR(T) are dense in X

(II) there existsθ∈(0, π)such thatρ(T)⊇C\S_θ=−S_π−θ and sup

λ∈C\S_θ0

kλ(λ−T)⁻¹kL(X)<∞ (1.1)

for allθ⁰ ∈(θ, π).

If T is sectorial, we define the spectral angleϕ_T as the infimum of all angles θ∈(0, π) such that (1.1) holds for all θ⁰ ∈(θ, π).

(ii) An operatorT:D(T)⊆X →X is said to be bisectorial if (I) D(T)andR(T) are dense in X

(II) there existsδ∈(0, π/2)such that ρ(T)⊇C\Σ_δ =−S^π

2−δ∪S^π

2−δ and sup

λ∈C\Σ_δ0

kλ(λ−T)⁻¹k_L(X)<∞ (1.2)

for allδ⁰ ∈(δ, π/2).

7

(18)

If T is bisectorial, we define the spectral angle ϕ^(bi)_T as the infimum of all angles δ∈(0, π/2) such that (1.2) holds for all δ⁰∈(δ, π/2).

Remark 1.3. Note that every bisectorial operatorT is also sectorial with π/2≤ϕT ≤π/2 +ϕ^(bi)_T . If an operator provides resolvent estimates on a reflexive Banach space, we can conclude the density ofR(T)by injectivity. This useful tool is concretized by the next lemma.

Lemma 1.4 ([KW04, Proposition 15.2 c)]). Let T be a linear operator on a reflexive Banach space X with domainD(T)such that there exists θ∈(0, π)withρ(T)⊆C\Sθ and

sup

λ∈C\S_θ0

kλ(λ−T)⁻¹k_L(X)<∞ for allθ⁰∈(θ, π). Then we can decomposeX into

X = ker(T)⊕R(T).

Definition 1.5. By virtue of the appearance of various spaces we often want to make clear in which space an integral or limit is understood. Hence we write

Z

. . . dx_[X] and _[X] lim

n→∞. . . to indicate the convergence in the spaceX.

The essential part of the next result can be found in [KW04, Proposition 15.2 a)]. For the sake of completeness we state a modification, which is suitable for our purposes.

Lemma 1.6. Let A:D(A)⊆X→X be a linear operator such that0 and−∞are accumulation points of ρ(A). Let(zn)n∈N,(wn)n∈N⊆ρ(A) withzn→0,Rewn→ −∞,Imwn →0, and

sup

n∈N

kzn(zn−A)⁻¹kL(X)<∞, sup

n∈N

kwn(wn−A)⁻¹kL(X)<∞. (1.3) Then we have

R(A) = {x∈X:[X] lim

n→∞zn(zn−A)⁻¹x= 0}, D(A) = {x∈X:_[X] lim

n→∞w_n(w_n−A)⁻¹x=x}.

Proof. First, we want to mention that

{x∈X: lim

n→∞zn(zn−A)⁻¹x= 0}, {x∈X:_[X] lim

n→∞w_n(w_n−A)⁻¹x=x}

are closed subsets ofX due to (1.3).

(i) Letx∈R(A)andy∈D(A)withx=Ay. With this we get

zn(zn−A)⁻¹x = znA(zn−A)⁻¹y=zn(A−zn)(zn−A)⁻¹y+z_n²(zn−A)⁻¹y

= −zny+z²_n(zn−A)⁻¹y→0 due to the boundedness in (1.3). So we have proved

R(A)⊆ {x∈X: lim

n→∞z_n(z_n−A)⁻¹x= 0}

according to the closedness.

Letx∈X withlim_n→∞zn(zn−A)⁻¹x= 0. Then we derive x=zn(zn−A)⁻¹x−A(zn−A)⁻¹x and thereforex=−lim_n→∞A(zn−A)⁻¹x∈R(A).

(19)

1.1: Fundamentals aboutH^∞-calculus 9

θ σ(T)

Γϕ

Figure 1.1: Spectrum, S_θ, and admissable curveΓ_ϕ

δ σ(T)

Γ_ϕ Γ_−ϕ

Figure 1.2: Spectrum, Σδ, and admissable curveΓϕ∪Γ_−ϕ

(ii) Due tow_n(w_n−A)⁻¹x∈D(A)for allx∈X the inclusion {x∈X:_[X] lim

n→∞wn(wn−A)⁻¹x=x} ⊆D(A) is obvious.

Letx∈D(A). Then we getwn(wn−A)⁻¹x−x=w_n⁻¹·wn(wn−A)⁻¹Ax→0 by (1.3). We now derive the assertion by the same arguments as in part (i).

Definition 1.7. Let X be a Banach space and

Tk:D(Tk)⊆X→X, k= 1, . . . , N linear operators. The operator tupleT:= (T1, . . . , TN)is called admissible if

(i) each of the operatorsT_k (k= 1, . . . , N) is sectorial or bisectorial (ii) for everyj, k= 1, . . . , N the resolvents ofTj andTk commute.

Remark 1.8 ([Kat76, Theorem 6.5]). Let X be a Banach space and T:D(T)⊆X → X,

S:D(S)⊆X → X

linear operators. If there existλ0∈ρ(T)andµ0∈ρ(S)with(λ0−T)⁻¹(µ0−S)⁻¹= (µ0−S)⁻¹(λ0−T)⁻¹, then we already have(λ−T)⁻¹(µ−S)⁻¹= (µ−S)⁻¹(λ−T)⁻¹ for all λ∈ρ(T)andµ∈ρ(S).

To fix the notation in this chapter we introduce the following abbreviations. Let the operator tuple T:= (T₁, . . . , T_N)be admissible andθ_k ≥ϕ_T_korδ_k≥ϕ^(bi)_T

k (according to the case). DefineΩ :=QN k=1Ω_k where each Ω_k is of the same type as S_θ_k or Σ_δ_k (according to the case) with a greater angle. Let Γ := QN

k=1Γ^k where each Γ^k is an admissible curve for S_θ_k or Σ_δ_k (according to the case) contained in Ω_k. Let B be the commutator of {(λ−T_k)⁻¹: k ∈ {1, . . . , N}, λ ∈ ρ(T_k)} ⊆ L(X), that is, the closed subalgebra consisting of all bounded operators that commute with all resolvents (λ−T_k)⁻¹, k∈ {1, . . . , N},λ∈ρ(Tk).

(20)

Definition 1.9. Let Y be a Banach space. We define the following spaces of holomorphic functions:

(i) H(Ω, Y)the vector space of Y-valued holomorphic functions onΩ.

(ii) H^∞(Ω, Y)the Banach space ofY-valued bounded holomorphic functions onΩ.

(iii) H_R^∞(Ω, Y) :={f ∈H^∞(Ω, Y) :f(Ω)isR-bounded}.

(iv) H₀^∞(Ω, Y) :=n

f ∈H(Ω, Y) :∃C, s >0∀z∈Ω :kf(z)kY ≤CQN

k=1 min{|zk|,|zk|⁻¹}so . (v) H_P(Ω, Y) :=n

f ∈H(Ω, Y) :∃C, s >0∀z∈Ω :kf(z)kY ≤CQN

k=1 max{|zk|,|zk|⁻¹}^so .

Definition 1.10(Operator-valuedH^∞-calculus).LetA ⊆ Bbe a closed subalgebra andf ∈H₀^∞(Ω,A).

Then we set

f(T) := 1 (2πi)^N

Z

Γ

f(z)

N

Y

k=1

(z_k−T_k)⁻¹dz_[L(X)]∈ A.

Note that f(T)does not depend on Γby Cauchy’s integral theorem.

Lemma 1.11. Forn∈Nthe function

Ψn,N: (C\ {−n,−1/n})^N →C, z7→

N

Y

k=1

n²zk

(1 +nzk)(n+zk) has the following properties:

(i) We have Ψ_n,N ∈H₀^∞(Ω).

(ii) For any Banach spaceY and all f ∈H_P(Ω, Y) there existsm∈N0 such that Ψ^m_n,Nf ∈H₀^∞(Ω, Y) for all n∈N.

(iii) The operator Ψn,N(T) is injective.

Definition 1.12 (H_P-calculus). Let f ∈ H_P(Ω,B) and m ∈ N0 such that Ψ^mf ∈ H₀^∞(Ω,B) where Ψ := Ψ_1,N. Then we set

f(T)x:= Ψ(T)^−m(Ψ^mf)(T)x, x∈D(f(T)) with domainD(f(T)) :={x∈X: (Ψ^mf)(T)x∈R(Ψ(T)^m)}.

Lemma 1.13. Every bisectorial operator A is also sectorial. So we can define f(A) by using the two different curves Γ_s andΓ_bi but both interpretations are equal i.e.f(A_s) =f(A_bi)for allf ∈H^∞(S_θ,B).

Proof. This can be proved easily by Cauchy’s integral theorem.

Remark 1.14. (i) The definition of the domain in Definition 1.12 is not constructive and it is not easy to determine D(T) in a concrete situation.

(ii) Let f ∈ H^∞(Ω,B) with f(T) ∈L(X). Then we have Ψn,Nf ∈ H₀^∞(Ω,B)for all n ∈N. For all x∈X we can represent f(T)by

f(T)x=_[X] lim

n→∞(Ψn,Nf)(T)x.

This is the so-called convergence lemma, cf. [DV05, Theorem 4.7] and [Haa06, Proposition 5.1.4]

for example.

(iii) For all x∈X we have Ψn,N(T)x→x,n→ ∞, inX.

(21)

1.1: Fundamentals aboutH^∞-calculus 11 Definition 1.15. Let A be a closed subalgebra of B. The operator tuple T = (T1, . . . , TN) admits a bounded (joint) H^∞(Ω,A)-calculus if there exists C > 0 such that kf(T)kL(X) ≤ Ckfk∞ for all f ∈H₀^∞(Ω,A). The tuple Tadmits anR-bounded (joint)H^∞(Ω,A)-calculus if

{f(T) :f ∈H₀^∞(Ω,A), kfk_∞≤1} ⊆L(X)

isR-bounded. In Appendix B we state the basic definitions and properties of R-bounded families.

Lemma 1.16. LetAbe a closed subalgebra ofB. IfTadmits anR-bounded (joint)H^∞(Ω,A)-calculus, then

{f(T) :f ∈H^∞(Ω,A),kfk_∞≤C} ⊆L(X) is alsoR-bounded for allC >0 and we have

Rp({f(T) :f ∈H^∞(Ω,A),kfk∞≤C})≤C· Rp({g(T) :g∈H₀^∞(Ω,A),kgk∞≤1}).

Proof. We trivially have

M :={(1/C·f)(T) :f ∈H₀^∞(Ω,A),kfk∞≤C} ⊆ {g(T) :g∈H₀^∞(Ω,A),kgk∞≤1}.

Hence M is also R-bounded with R_p(M)≤ R_p({g(T) :g ∈H₀^∞(Ω,A),kgk_∞≤1}). Remark B.14 (iii) and (v), and Remark 1.14 (ii) then yield the claim.

Definition 1.17. LetT be a sectorial or bisectorial operator andΩϕ:=Sϕ orΩϕ:= Σϕaccording to the case. Then we define theH^∞-angleϕ^∞_T (respectively,ϕ^∞,(bi)_T ) as the infimum of allϕsuch thatT admits a boundedH^∞(Ωϕ)-calculus. In analogy we define theR-H^∞-angleϕ^R,∞_T (respectively, ϕ^R,∞,(bi)_T ) as the infimum of allϕsuch thatT admits anR-boundedH^∞(Ωϕ)-calculus.

Remark 1.18. (i) In general we cannot define an analog to the spectral angle or H^∞-angle in the case of more then one operator (i.e.N > 1), because the angles of the sectors or bisectors can be correlated.

(ii) We haveϕ_T ≤ϕ^∞_T ≤ϕ^R,∞_T (respectively, ϕ^(bi)_T ≤ϕ^∞,(bi)_T ≤ϕ^R,∞,(bi)_T ).

Theorem 1.19([DV05, Theorem 4.5]). Let f, g∈HP(Ω,B). Then the following assertions hold:

(i) f(T)is a closed operator with dense domain.

(ii) We have

f(T)g(T) ⊆ (f g)(T), f(T) +g(T) ⊆ (f+g)(T) where the domains are given by

D(f(T)g(T)) := {x∈D(g(T)) :g(T)x∈D(f(T))}, D(f(T) +g(T)) := f(T)∩g(T).

(iii) If g(T)∈L(X), then (f g)(T) =f(T)g(T)and(f+g)(T) =f(T) +g(T).

Theorem 1.20 ([DV05, Theorem 4.9]). Let A be a closed subalgebra of B. Then the following statements are equivalent:

(i) f(T)∈L(X)for all f ∈H^∞(Ω,A).

(ii) Thas a bounded jointH^∞(Ω,A)-calculus.

Next, we state the Kalton-Weis Theorem [KW01, Theorem 4.4] in a version of G. Dore and A. Venni [DV05, Theorem 6.7], which generalizes the situation to tuple of sectorial and bisectorial operators.

(22)

Theorem 1.21 (Kalton-Weis Theorem). Let Ω⁰ be a set of the same type as Ω but with smaller angles. IfTadmits a bounded scalarH^∞(Ω⁰)-calculus, then we haveg(T)∈L(X)for allg∈H_R^∞(Ω,B).

In particular, there exists C=C(T)>0with

kg(T)kL(X)≤C· R2(g(Ω)).

The next two lemmas are very helpful in the following because they clarify some compatibility properties of theH^∞-calculus.

Lemma 1.22 (Iterative calculus). Let T := (T1, . . . , TN) be an admissible tuple and let B be the commutator of the resolvents of the operators T_k, k ∈ {1, . . . , N}. For all f ∈ H₀^∞(Ω) we then have g ∈ H₀^∞(Ω⁰,B)where g(z⁰) :=f(T₁, z⁰), z⁰ ∈ Ω⁰ :=QN

k=2Ω_k. In particular, we get f(T) = g(T⁰) with T⁰:= (T₂, . . . , T_N).

Proof. We obviously have f(·, z⁰) ∈ H₀^∞(Ω₁) for all z⁰ ∈ Ω⁰ and g(z⁰) := f(T₁, z⁰) ∈ B due to Defini- tion 1.10. Then Morera’s theorem and Fubini’s theorem yield the holomorphy of the function g and even

g∈H₀^∞(Ω⁰,B).

Henceg(T⁰)is meaningful. Due to the boundedness of the resolvents we obtain g(T⁰) =

Z

Γ⁰

g(z⁰)

N

Y

k=2

(z_k−T_k)⁻¹dz_[L(X)]⁰

= Z

Γ⁰

Z

Γ1

f(z)(z1−T1)⁻¹dz_1[L(X)]

^N Y

k=2

(zk−Tk)⁻¹dz_[L(X)]⁰

= Z

Γ⁰

Z

Γ1

f(z)(z1−T1)⁻¹

N

Y

k=2

(zk−Tk)⁻¹dz_1[L(X)]dz_[L(X)]⁰ =f(T).

Lemma 1.23. Let f ∈HP(Ω⁰,B)withΩ⁰ :=QJ

k=1Ωk,J ∈ {1, . . . , N}. Defining g: Ω→ B, z7→f(z1, . . . , zJ)

we getg∈HP(Ω,B)andg(T) =f(T⁰)with T⁰:= (T1, . . . , TJ).

Proof. The easy proof forJ = 1can be found in [DV05, Theorem 4.12]. It is trivial to extend this to get the result of Lemma 1.23.

To determine the domain of operators given by the H_P-calculus we state some useful results in the next two lemmas.

Lemma 1.24. LetTadmit a boundedH^∞(Ω)-calculus. Forf1, f2∈HP(Ω)we have the following results:

(i) If there existsC >0with|f1(z)| ≤C|f2(z)| 6= 0for allz∈Ω, then we haveD(f2(T))⊆D(f1(T)), R(f1(T))⊆R(f2(T)), and

kf1(T)xkX≤C⁰kf2(T)xkX, x∈D(f2(T)) for someC⁰ >0.

(ii) If there exist C₁, C₂>0 such thatC₁· |f₂(z)| ≤ |f₁(z)| ≤C₂· |f₂(z)| andf₁(z), f₂(z)6= 0 for all z∈Ω, then we have D(f₂(T)) =D(f₁(T)),R(f₁(T)) =R(f₂(T))and

kf₂(T)xk_X≤Ckf₁(T)xk_X ≤C⁰kf₂(T)xk_X, x∈D(f₂(T)) for someC, C⁰>0.

(23)

1.2:H^∞-calculus, interpolation, and isomorphisms 13

Proof. (i) We have f1/f2∈H^∞(Ω)by assumption. Hence we get f1(T) =

f1

f₂ ·f2

(T)⊇

f1

f₂

(T)f2(T)

(cf. Theorem 1.19 (ii)) and (f₁/f₂)(T) ∈ L(X). So we obtain the claimed relation between the domains as well as the claimed estimate. On the other hand we have

f1(T) =

f2· f1

f₂

(T) =f2(T) f1

f₂

(T)

(cf. Theorem 1.19 (iii)), which yields the claimed relation between the ranges.

(ii) This follows immediately from (i).

Lemma 1.25. Let g, h∈HP(Ω) with g(z), h(z)6= 0for all z∈Ωandg⁻¹, h⁻¹∈H^∞(Ω). If we define f(z) :=g(z)·h(z),z∈Ω, we can conclude

f(T) =g(T)h(T) =h(T)g(T), which especially yieldsD(f(T)) =D(g(T)h(T)) =D(h(T)g(T)).

Proof. Without loss of generality we only showf(T) =g(T)h(T). From Theorem 1.19 we already know f(T)⊇g(T)h(T), f⁻¹(T) =g⁻¹(T)h⁻¹(T) =h⁻¹(T)g⁻¹(T),

which yieldD(g(T)h(T))⊆D(f(T)) =R(f⁻¹(T)) =R(g⁻¹(T)h⁻¹(T)) =R(h⁻¹(T)g⁻¹(T)). One can easily showR(h⁻¹(T)g⁻¹(T))⊆D(g(T)h(T)), which ends the proof.

1.2 H

^∞

-calculus, interpolation, and isomorphisms

Proposition 1.26 (H^∞-calculus and isomorphisms). Let X, Y be Banach spaces such that there exists an isomorphismΦ∈LIsom(Y, X). For eachk= 1, . . . , N let

Tk:D(Tk)⊆X →X be an arbitrary linear operator and define

Sk:D(Sk)⊆Y →Y, y7→Φ⁻¹TkΦy

whereD(S_k) := Φ⁻¹(D(T_k)). If we defineT:= (T₁, . . . , T_N)andS:= (S₁, . . . , S_N), we get the following assertions:

(i) IfT_k is sectorial (respectively, bisectorial), then the operator S_k is also sectorial (respectively, bisectorial) with ϕ_S_k =ϕ_T_k (respectively, ϕ^(bi)_S

k =ϕ^(bi)_T

k ). In particular, we have ρ(S_k) = ρ(T_k) and (λ−Sk)⁻¹= Φ⁻¹(λ−Tk)⁻¹Φfor allλ∈ρ(Tk). If Tis admissible, thenSis also admissible.

(ii) IfThas a bounded jointH^∞(Ω)-calculus, thenSalso has a bounded jointH^∞(Ω)-calculus and the representation

f(S) = Φ⁻¹f(T)Φ holds for allf ∈H_P(Ω).

(iii) IfThas anR-bounded jointH^∞(Ω)-calculus, thenSalso has anR-bounded jointH^∞(Ω)-calculus.

(24)

Proof. (i) Without loss of generality let Tk be a sectorial operator for all k = 1, . . . , N. Then we obviously have ρ(T_k) = ρ(S_k)and(λ−S_k)⁻¹ = Φ⁻¹(λ−T_k)⁻¹Φfor allλ∈ρ(T_k). From this we easily obtain forθ∈(0, π)withρ(T_k) =ρ(S_k)⊇C\S_θ

sup

λ∈S_θ0

kλ(λ−Sk)⁻¹k_L(Y₎≤C sup

λ∈S_θ0

kλ(λ−Tk)⁻¹k_L(X)<∞

for allθ⁰∈(θ, π). The density ofD(Sk)andR(Sk)is obvious. Therefore,Sk is also sectorial with ϕ_T_k =ϕ_S_k. The claimed admissibility ofSis obvious.

(ii) Forf ∈H₀^∞(Ω)we directly derive from (i) f(S)y = 1

(2πi)^N Z

Γ

f(z)

N

Y

k=1

(zk−Sk)⁻¹ydz[Y] = 1 (2πi)^N

Z

Γ

f(z)

N

Y

k=1

Φ⁻¹(λ−Tk)⁻¹Φydz[Y]

= 1

(2πi)^NΦ⁻¹

"

Z

Γ

f(z)

N

Y

k=1

(λ−T_k)⁻¹dz_[L(X)]

#

Φy= Φ⁻¹f(T)Φy

for all y ∈ Y. Hence we have kf(S)k_L(Y₎ ≤ Ckf(T)k_L(X) ≤ Ckfk_∞ for all f ∈ H₀^∞(Ω). This proves thatSadmits a bounded joint H^∞(Ω)-calculus.

Let f ∈ H_P(Ω) and m ∈ N0 such that Ψ^mf ∈ H₀^∞(Ω). Then we have Ψ(S) = Φ⁻¹Ψ(T)Φand (Ψ^mf)(S) = Φ⁻¹(Ψ^mf)(T)Φdue toΨ∈H₀^∞(Ω). We then derive

f(S) = Ψ(S)^−m(Ψ^mf)(S) = Φ⁻¹Ψ(T)^−mΦΦ⁻¹(Ψ^mf)(T)Φ

= Φ⁻¹f(T)Φ.

(iii) This easily follows from the representation in (ii) and Remark B.14 (iii).

Theorem 1.27 (H^∞-calculus and interpolation). Let F ∈ {(·,·)θ,p,[·,·]θ}, θ ∈ (0,1), p∈ (1,∞), and let{X0, X1} be an interpolation couple. Additionally, let

Tk:D(Tk)⊆X0→X0,

S_k:D(S_k)⊆X₁→X₁, k= 1, . . . , N

be linear operators with the compatibility conditions Tkx=Skxfor allx∈D(Tk)∩D(Sk)and

(λ−T_k)⁻¹x= (λ−S_k)⁻¹x, λ∈ρ(T_k)∩ρ(S_k), x∈X₀∩X₁. (1.4) Moreover we define fork= 1, . . . , N the interpolated operators

A_k:D(A_k)⊆ F({X0, X₁})→ F({X0, X₁}), D(A_k) :=F({D(Tk), D(S_k)})

with Akx:=Tkx0+Skx1 for x=x0+x1 ∈D(Ak) ,→ D(Tk) +D(Sk)with x0 ∈ D(Tk), x1 ∈ D(Sk).

Note that we equipD(Tk)andD(Sk)with the graph norm. Then we have the following results:

(i) Letk= 1, . . . , N. IfTk andSk are both sectorial (respectively, bisectorial), thenAk is also sectorial (respectively, bisectorial) with ϕA_k ≤max{ϕT_k, ϕS_k} (respectively, ϕ^(bi)_A

k ≤max{ϕ^(bi)_T

k , ϕ^(bi)_S

k }). If T andSare both admissible, then Ais also admissible.

(ii) LetT:= (T1, . . . , TN)andS:= (S1, . . . , SN)be tuples of sectorial and bisectorial operators such that T_k andS_k are of the same type. IfTandSadmit a boundedH^∞(Ω)-calculus, then the interpolated tuple A:= (A₁, . . . , A_N)also admits a boundedH^∞(Ω)-calculus and we have the representation

f(A)x=f(T)x0+f(S)x1, x∈ F({X0, X1}), x=x0+x1, xi∈Xi

for all f ∈H^∞(Ω). ForN = 1we even haveϕ^∞_A

1 ≤max{ϕ^∞_T₁, ϕ^∞_S

1} (respectively, ϕ^∞,(bi)_A

1 ≤max{ϕ^∞,(bi)_T

1 , ϕ^∞,(bi)_S

1 }).

General parabolic mixed order systems in Lp and applications