Semigroups and Evolution Equations: Functional Calculus ... maria.lancia/Didattica...¢  Semigroups and

  • View
    0

  • Download
    0

Embed Size (px)

Text of Semigroups and Evolution Equations: Functional Calculus ... maria.lancia/Didattica...¢ ...

  • CHAPTER 1

    Semigroups and Evolution Equations: Functional Calculus, Regularity and Kernel Estimates

    Wolfgang Arendt Abteilung Angewandte Analysis, Universität Ulm, 89069 Ulm, Germany,

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

    Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

    1.1. The algebraic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2. The Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3. Semigroups and Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4. More general C0-semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5. The inhomogeneous Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    2. Holomorphic semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1. Characterization of bounded holomorphic semigroups . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2. Characterization of holomorphic semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3. Boundary groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4. The Gaussian semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5. The Dirichlet Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.6. The Neumann Laplacian on C(�Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.7. Wentzell boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.8. Dynamic boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    3. Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1. Exponential stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2. Ergodic semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3. Convergence and asymptotically almost periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.4. Positive semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.5. Positive irreducible semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    4. Functional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.1. Sectorial operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2. The sum of commuting operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.3. The elementary functional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.4. Fractional powers and BIP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.5. Bounded H∞-calculus for sectorial operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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

    1

  • 2 W. Arendt

    4.6. Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.7. Groups and positive contraction semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

    5. Form methods and functional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.1. Bounded H∞-calculus on Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.2. m-accretive operators on Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.3. Form methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.4. Form sums and Trotter’s product formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.5. The square root property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.6. Groups and cosine functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

    6. Fourier multipliers and maximal regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.1. Vector-valued Fourier series and periodic multipliers . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.2. Maximal regularity via periodic multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

    7. Gaussian estimates and ultracontractivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7.1. The Beurling–Deny criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7.2. Extrapolating semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 7.3. Ultracontractivity, kernels and Sobolev embedding . . . . . . . . . . . . . . . . . . . . . . . . . . 64 7.4. Gaussian estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

    8. Elliptic operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 8.1. Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 8.2. Positivity and irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 8.3. Submarkov property: Dirichlet boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 74 8.4. Quasicontractivity in Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 8.5. Gaussian estimates: real coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 8.6. Complex second-order coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 8.7. Further comments on Gaussian estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 8.8. The square root property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 8.9. The hyperbolic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

    8.10. Nondivergence form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8.11. Elliptic operators with Banach space-valued coefficients . . . . . . . . . . . . . . . . . . . . . . . 77

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

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

    Abstract This 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. Also other 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 functional calculus, maximal regularity and kernel estimates which, in our eyes, has seen a most spectacular development, and which, so far, is not presented in book form. We comment on 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 was McIntosh who initiated and developed a theory of functional calculus for a less restricted large 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 generators of bounded holomorphic semigroups are sectorial operators and are our main subject of investigation. And indeed, for these operators f (A) can be defined for a large class of holomorphic functions defined on a sector containing the spectrum. Taking f (z) = e−tz leads to the semigroup e−tA, the function f (z) = zα to the fractional power of such an operator A. One important reason to study functional calculus is the Dore–Venni theorem. In its hypotheses functional calculus plays a role; the conclusion is the invertibility of the sum of two operators A and B . Thus, the Dore–Venni 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 both domains can be rephrased by saying that the solution has “maximal regularity”, a crucial property 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 with