Semigroups and Evolution Equations: Functional .Semigroups and evolution equations: Functional calculus,

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  • CHAPTER 1

    Semigroups and Evolution Equations: FunctionalCalculus, Regularity and Kernel Estimates

    Wolfgang ArendtAbteilung Angewandte Analysis, Universitt Ulm, 89069 Ulm, Germany,

    E-mail: arendt@mathematik.uni-ulm.de

    ContentsIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31. Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

    1.1. The algebraic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2. The Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3. Semigroups and Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4. More general C0-semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5. The inhomogeneous Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    2. Holomorphic semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1. Characterization of bounded holomorphic semigroups . . . . . . . . . . . . . . . . . . . . . . . . 72.2. Characterization of holomorphic semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3. Boundary groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4. The Gaussian semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5. The Dirichlet Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6. The Neumann Laplacian on C() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.7. Wentzell boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.8. Dynamic boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    3. Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.1. Exponential stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2. Ergodic semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3. Convergence and asymptotically almost periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4. Positive semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.5. Positive irreducible semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    4. Functional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.1. Sectorial operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2. The sum of commuting operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3. The elementary functional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.4. Fractional powers and BIP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.5. Bounded H-calculus for sectorial operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

    HANDBOOK OF DIFFERENTIAL EQUATIONSEvolutionary Equations, volume 1Edited by C.M. Dafermos and E. Feireisl 2004 Elsevier B.V. All rights reserved

    1

  • 2 W. Arendt

    4.6. Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.7. Groups and positive contraction semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

    5. Form methods and functional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.1. Bounded H-calculus on Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2. m-accretive operators on Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.3. Form methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.4. Form sums and Trotters product formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.5. The square root property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.6. Groups and cosine functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

    6. Fourier multipliers and maximal regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.1. Vector-valued Fourier series and periodic multipliers . . . . . . . . . . . . . . . . . . . . . . . . . 496.2. Maximal regularity via periodic multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

    7. Gaussian estimates and ultracontractivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597.1. The BeurlingDeny criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597.2. Extrapolating semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607.3. Ultracontractivity, kernels and Sobolev embedding . . . . . . . . . . . . . . . . . . . . . . . . . . 647.4. Gaussian estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

    8. Elliptic operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728.1. Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738.2. Positivity and irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738.3. Submarkov property: Dirichlet boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 748.4. Quasicontractivity in Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748.5. Gaussian estimates: real coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748.6. Complex second-order coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758.7. Further comments on Gaussian estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758.8. The square root property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768.9. The hyperbolic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

    8.10. Nondivergence form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778.11. Elliptic operators with Banach space-valued coefficients . . . . . . . . . . . . . . . . . . . . . . . 77

    Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

    Monographs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Research Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

    AbstractThis is a survey on recent developments of the theory of one-parameter semigroups and

    evolution equations with special emphasis on functional calculus and kernel estimates. Alsoother topics as asymptotic behavior for large time and holomorphic semigroups are discussed.As main application we consider elliptic operators with various boundary conditions.

  • Semigroups and evolution equations: Functional calculus, regularity and kernel estimates 3

    Introduction

    The theory of one-parameter semigroups provides a framework and tools to solve evolu-tionary problems. It is impossible to give an account of this rich and most active field.In this chapter we rather try to present a survey on a particular subject, namely functionalcalculus, maximal regularity and kernel estimates which, in our eyes, has seen a mostspectacular development, and which, so far, is not presented in book form. We commenton these three subjects:

    1. Functional calculus (Section 4). If A is a self-adjoint operator, one can define f (A)for all bounded complex-valued measurable functions defined on the spectrum of A. It wasMcIntosh who initiated and developed a theory of functional calculus for a less restrictedlarge class of operators, namely sectorial operators; i.e., operators whose spectrum is in-cluded in a sector and whose resolvent satisfies a certain estimate. Negative generatorsof bounded holomorphic semigroups are sectorial operators and are our main subject ofinvestigation. And indeed, for these operators f (A) can be defined for a large class ofholomorphic functions defined on a sector containing the spectrum. Taking f (z) = etzleads to the semigroup etA, the function f (z) = z to the fractional power of such anoperator A. One important reason to study functional calculus is the DoreVenni theorem.In its hypotheses functional calculus plays a role; the conclusion is the invertibility of thesum of two operators A and B . Thus, the DoreVenni theorem asserts that the equation

    Ax +Bx = y

    has a unique solution x D(A)D(B). To say that the solution is at the same time in bothdomains can be rephrased by saying that the solution has maximal regularity, a crucialproperty in many circumstances.

    2. Form methods (Section 5). On Hilbert space the functional calculus behaves particu-larly well as we show in Sections 4 and 5. Most interesting is the close connection withform methods. Basically, the following is true: an operator is associated with a form ifand only if it has a bounded H-calculus. Form methods, based on the fundamental LaxMilgram lemma, allow a most