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Transcript of On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square...

  • On Katos Square Root Problem

    Moritz Egert

    WIAS Berlin, February 11, 2015

  • T. Kato 1960s: Non-autonomous parabolic evo-lution equation

    ddt

    u(t)(x) + A(t)u(t)(x) = 0 (t > 0, x ),u(0) = u0 L2().

    I A(t) x (t , x)x via elliptic form a(t) : V() V() C.I u(t)(x) = etAu0(x) if A(t) = A for all t > 0.

    Kato Square Root Problem (1961)

    We do not know whether or not D(A1/2) = D(A1/2). This is perhaps not truein general. But the question is open even when A is regularly accretive. Inthis case it appears reasonable to suppose that both D(A1/2) and D(A1/2)coincide with D(a), where a is the regular sesquilinear form which defines A.

    I Counterexamples by Lions 1962, McIntosh 1982I Specialize to divergence-form operators.

  • T. Kato 1960s: Non-autonomous parabolic evo-lution equation

    ddt

    u(t)(x) + A(t)u(t)(x) = 0 (t > 0, x ),u(0) = u0 L2().

    I A(t) x (t , x)x via elliptic form a(t) : V() V() C.I u(t)(x) = etAu0(x) if A(t) = A for all t > 0.

    Kato Square Root Problem (1961)

    We do not know whether or not D(A1/2) = D(A1/2). This is perhaps not truein general. But the question is open even when A is regularly accretive. Inthis case it appears reasonable to suppose that both D(A1/2) and D(A1/2)coincide with D(a), where a is the regular sesquilinear form which defines A.

    I Counterexamples by Lions 1962, McIntosh 1982I Specialize to divergence-form operators.

  • T. Kato 1960s: Non-autonomous parabolic evo-lution equation

    ddt

    u(t)(x) + A(t)u(t)(x) = 0 (t > 0, x ),u(0) = u0 L2().

    I A(t) x (t , x)x via elliptic form a(t) : V() V() C.I u(t)(x) = etAu0(x) if A(t) = A for all t > 0.

    Kato Square Root Problem (1961)

    We do not know whether or not D(A1/2) = D(A1/2). This is perhaps not truein general. But the question is open even when A is regularly accretive. Inthis case it appears reasonable to suppose that both D(A1/2) and D(A1/2)coincide with D(a), where a is the regular sesquilinear form which defines A.

    I Counterexamples by Lions 1962, McIntosh 1982I Specialize to divergence-form operators.

  • T. Kato 1960s: Non-autonomous parabolic evo-lution equation

    ddt

    u(t)(x) + A(t)u(t)(x) = 0 (t > 0, x ),u(0) = u0 L2().

    I A(t) x (t , x)x via elliptic form a(t) : V() V() C.I u(t)(x) = etAu0(x) if A(t) = A for all t > 0.

    Kato Square Root Problem (1961)

    We do not know whether or not D(A1/2) = D(A1/2). This is perhaps not truein general. But the question is open even when A is regularly accretive. Inthis case it appears reasonable to suppose that both D(A1/2) and D(A1/2)coincide with D(a), where a is the regular sesquilinear form which defines A.

    I Counterexamples by Lions 1962, McIntosh 1982I Specialize to divergence-form operators.

  • Setup

    LetI Rd domain, D closed, L()I A accretive operator on L2()

    associated with

    W1,2D ()W1,2D () C, (u, v) 7

    u v .

    I A1/2 square root of A defined by e.g.

    A1/2u =1

    0

    t1/2A(t + A)1 dt .

    D

    Kato conjecture

    It holds D(A1/2) = W1,2D () with equivalent norms.

  • Why do we care about the Kato conjecture?

    Philosophy

    I Elliptic non-regularity results D(A) * W2,2().I Kato Conjecture optimal Sobolev regularity for A1/2.

    Ex. 1: Elliptic equations on Rd+

    2

    t2u(t)(x) + (x)u(t , x) = 0 (t > 0, x Rd ),

    u(0, x) = u0(x) W1,2(Rd ).

    I Solution u(t , x) = etA1/2

    u0(x).I Kato conjecture Rellich inequality tu|t=02 u02.

  • Why do we care about the Kato conjecture?

    Philosophy

    I Elliptic non-regularity results D(A) * W2,2().I Kato Conjecture optimal Sobolev regularity for A1/2.

    Ex. 1: Elliptic equations on Rd+

    2

    t2u(t)(x) + (x)u(t , x) = 0 (t > 0, x Rd ),

    u(0, x) = u0(x) W1,2(Rd ).

    I Solution u(t , x) = etA1/2

    u0(x).I Kato conjecture Rellich inequality tu|t=02 u02.

  • Why do we care about the Kato conjecture?

    Philosophy

    I Elliptic non-regularity results D(A) * W2,2().I Kato Conjecture optimal Sobolev regularity for A1/2.

    Ex. 1: Elliptic equations on Rd+

    2

    t2u(t)(x) + (x)u(t , x) = 0 (t > 0, x Rd ),

    u(0, x) = u0(x) W1,2(Rd ).

    I Solution u(t , x) = etA1/2

    u0(x).I Kato conjecture Rellich inequality tu|t=02 u02.

  • Ex. 2: Maximal parabolic regularity (e.g. Haller-Dintelmann-Rehberg)

    I In Lp-setting study parabolic equation{ ddt

    u(t) + Au(t) = f (0 < t < T ),

    u(0) = 0.

    I Goal: Transport Max. Reg. from Lp() to W1,pD ().

    I Lp-Kato conjecture ( >)1/2 : W1,p

    D () Lp() isom.

    I Adjoint ( )1/2 : Lp()W1,pD () isomorphism thatcommutes with parabolic solution operator( d

    dt+ A

    )1.

    Many further examples, e.g. Cauchy-Integral along Lipschitz curve,hyperbolic wave equations, . . . .

  • Ex. 2: Maximal parabolic regularity (e.g. Haller-Dintelmann-Rehberg)

    I In Lp-setting study parabolic equation{ ddt

    u(t) + Au(t) = f (0 < t < T ),

    u(0) = 0.

    I Goal: Transport Max. Reg. from Lp() to W1,pD ().

    I Lp-Kato conjecture ( >)1/2 : W1,p

    D () Lp() isom.

    I Adjoint ( )1/2 : Lp()W1,pD () isomorphism thatcommutes with parabolic solution operator( d

    dt+ A

    )1.

    Many further examples, e.g. Cauchy-Integral along Lipschitz curve,hyperbolic wave equations, . . . .

  • Ex. 2: Maximal parabolic regularity (e.g. Haller-Dintelmann-Rehberg)

    I In Lp-setting study parabolic equation{ ddt

    u(t) + Au(t) = f (0 < t < T ),

    u(0) = 0.

    I Goal: Transport Max. Reg. from Lp() to W1,pD ().

    I Lp-Kato conjecture ( >)1/2 : W1,p

    D () Lp() isom.

    I Adjoint ( )1/2 : Lp()W1,pD () isomorphism thatcommutes with parabolic solution operator( d

    dt+ A

    )1.

    Many further examples, e.g. Cauchy-Integral along Lipschitz curve,hyperbolic wave equations, . . . .

  • Ex. 2: Maximal parabolic regularity (e.g. Haller-Dintelmann-Rehberg)

    I In Lp-setting study parabolic equation{ ddt

    u(t) + Au(t) = f (0 < t < T ),

    u(0) = 0.

    I Goal: Transport Max. Reg. from Lp() to W1,pD ().

    I Lp-Kato conjecture ( >)1/2 : W1,p

    D () Lp() isom.

    I Adjoint ( )1/2 : Lp()W1,pD () isomorphism thatcommutes with parabolic solution operator( d

    dt+ A

    )1.

    Many further examples, e.g. Cauchy-Integral along Lipschitz curve,hyperbolic wave equations, . . . .

  • Positive answers

    Self-adjoint operators

    XWhole space = Rd

    X

    I d = 1: Coifman - McIntosh - Meyer 82.

    I d 2: Auscher-Hofmann-Lacey-McIntosh-Tchamitchian 01,Axelsson-Keith-McIntosh 06.

    Bounded domainsI Lipschitz, D {, }: Auscher-Tchamitchian 03, 01 (p 6= 2).I smooth, smooth D ! \ D interface: Axelsson-Keith-McIntosh 06.I Lipschitz around \ D:

    Auscher-Badr-Haller-Dintelmann-Rehberg 12 (p 6= 2).

  • Positive answers

    Self-adjoint operatorsX

    Whole space = Rd

    X

    I d = 1: Coifman - McIntosh - Meyer 82.

    I d 2: Auscher-Hofmann-Lacey-McIntosh-Tchamitchian 01,Axelsson-Keith-McIntosh 06.

    Bounded domainsI Lipschitz, D {, }: Auscher-Tchamitchian 03, 01 (p 6= 2).I smooth, smooth D ! \ D interface: Axelsson-Keith-McIntosh 06.I Lipschitz around \ D:

    Auscher-Badr-Haller-Dintelmann-Rehberg 12 (p 6= 2).

  • Positive answers

    Self-adjoint operatorsXWhole space = Rd

    X

    I d = 1: Coifman - McIntosh - Meyer 82.

    I d 2: Auscher-Hofmann-Lacey-McIntosh-Tchamitchian 01,Axelsson-Keith-McIntosh 06.

    Bounded domainsI Lipschitz, D {, }: Auscher-Tchamitchian 03, 01 (p 6= 2).I smooth, smooth D ! \ D interface: Axelsson-Keith-McIntosh 06.I Lipschitz around \ D:

    Auscher-Badr-Haller-Dintelmann-Rehberg 12 (p 6= 2).

  • Positive answers

    Self-adjoint operatorsXWhole space = Rd X

    I d = 1: Coifman - McIntosh - Meyer 82.

    I d 2: Auscher-Hofmann-Lacey-McIntosh-Tchamitchian 01,Axelsson-Keith-McIntosh 06.

    Bounded domainsI Lipschitz, D {, }: Auscher-Tchamitchian 03, 01 (p 6= 2).I smooth, smooth D ! \ D interface: Axelsson-Keith-McIntosh 06.I Lipschitz around \ D:

    Auscher-Badr-Haller-Dintelmann-Rehberg 12 (p 6= 2).

  • Positive answers

    Self-adjoint operatorsXWhole space = Rd X

    I d = 1: Coifman - McIntosh - Meyer 82.

    I d 2: Auscher-Hofmann-Lacey-McIntosh-Tchamitchian 01,Axelsson-Keith-McIntosh 06.

    Bounded domainsI Lipschitz, D {, }: Auscher-Tchamitchian 03, 01 (p 6= 2).I smooth, smooth D ! \ D interface: Axelsson-Keith-McIntosh 06.I Lipschitz around \ D:

    Auscher-Badr-Haller-Dintelmann-Rehberg 12 (p 6= 2).

  • Kato for mixed boundary conditions

    Theorem (E.-Haller-Dintelmann-Tolksdorf 14)Suppose

    I Rd bounded d-Ahlfors regular domain,

    I D closed and (d 1)-Ahlfors regular,

    I Lipschitz around \ D.

    Then

    D(A1/2) = W1,2D () with A1/2u2 u2.

    I First formulated by J.-L. Lions 1962.I For rough (= L) coefficients new even on Lipschitz domains.

  • Some ideas of the proof

    1 First-order approach via perturbed Dirac operators la AKM 06,H-functional calculus.

    2 Getting rid of the coefficients via

    Reduction-Theorem (E.-Haller-Dintelmann-Tolksdorf 14)

    In essence, the following holds: If D((V)s

    ) H2s,2() for

    some s > 12 , then D(A1/2) = W1,2D ().

    Extrapolate Kato for V = Kato property for general Ageometry, potential theoryharmonic analysis

  • Some ideas of the proof

    1 First-order approach via perturbed Dirac operators la AKM 06,H-functional calculus.

    2 Getting rid of the coefficients via

    Reduction-Theorem (E.-Haller-Dintelmann-Tolksdorf 14)

    In essence, the following holds: If D((V)s

    ) H2s,2() for

    some s > 12 , then D(A1/2) = W1,2D ().

    Extrapolate Kato for V = Kato property for general Ageometry, potential theoryharmonic analysis

  • Some ideas of the proof

    1 First-order approach via perturbed Dirac operators la AKM 06,H-functional calculus.

    2 Getting rid of the coefficients via

    Reduction-Theorem (E.-Haller-Dintelmann-Tolksdorf 14)

    In essence, the following holds: If D((V)s

    ) H2s,2() for

    some s > 12 , then D(A1/2) = W1,2D ().

    Extrapolate Kato for V = Kato property for general A

    geometry, potential theoryharmonic analysis

  • Some ideas of the proof

    1 First-order approach via perturbed Dirac operators la AKM 06,H-functional calculus.

    2 Getting rid of the coefficients via

    Reduction-Theorem (E.-Haller-Dintelmann-Tolksdorf 14)

    In essence, the following holds: If D((V)s

    ) H2s,2() for

    some s > 12 , then D(A1/2) = W1,2D ().

    Extrapolate Kato for V = Kato property for general Ageometry, potential theory ! harmonic analysis

  • 3 D((V)1/2) = W1,2D () by self-adjointness. Extrapolate bySneibergs stability theorem and the following result.

    Theorem (E.-Haller-Dintelmann-Tolksdorf 14)

    Let (0,1) and s0, s1 (12 ,32). Put s := (1 )s0 + s1. Then,

    W1,2D () = H1,2D ()[

    Hs0,2D (),Hs1,2D ()

    ]

    = Hs,2D ().

    [L2(),H1,2D ()

    ]

    =

    {H,2D (), if >

    12 ,

    H,2(), if < 12 .

    4 In fact, D((V)s) = H2s,2D () for |12 s| < .

  • 3 D((V)1/2) = W1,2D () by self-adjointness. Extrapolate bySneibergs stability theorem and the following result.

    Theorem (E.-Haller-Dintelmann-Tolksdorf 14)

    Let (0,1) and s0, s1 (12 ,32). Put s := (1 )s0 + s1. Then,

    W1,2D () = H1,2D ()[

    Hs0,2D (),Hs1,2D ()

    ]

    = Hs,2D ().

    [L2(),H1,2D ()

    ]

    =

    {H,2D (), if >

    12 ,

    H,2(), if < 12 .

    4 In fact, D((V)s) = H2s,2D () for |12 s| < .

  • Elliptic BVPs on cylindrical domains

    Elliptic mixed BVP

    divt ,x(x)t ,xU = 0 (R+ )U = 0 (R+ D)

    U = 0 (R+ N)U = f L2 ({0} )

    l F [UxU]

    First order equation

    tF +[

    0 (V)V 0

    ]

    D

    BF = 0 (t > 0)

    F (0) = f L2loc(R+; L2()) setting

  • Elliptic BVPs on cylindrical domains

    Elliptic mixed BVP

    divt ,x(x)t ,xU = 0 (R+ )U = 0 (R+ D)

    U = 0 (R+ N)U = f L2 ({0} )

    l F [UxU]

    First order equation

    tF +[

    0 (V)V 0

    ]

    D

    BF = 0 (t > 0)

    F (0) = f

    L2loc(R+; L2()) setting

  • Elliptic BVPs on cylindrical domains

    Elliptic mixed BVP

    divt ,x(x)t ,xU = 0 (R+ )U = 0 (R+ D)

    U = 0 (R+ N)U = f L2 ({0} )

    l F [UxU]

    First order equation

    tF +[

    0 (V)V 0

    ]

    D

    BF = 0 (t > 0)

    F (0) = f L2loc(R+; L2()) setting

  • Semigroup solutions via DB-formalism

    DB has bounded H-calculus on H = R(DB) (Kato Technology).

    Theorem (Auscher-E. 14)1 For every F (0) H+ := 1C+(DB)H a solution to the first-order

    system is

    F (t) = etDBF (0) (t 0).

    Via F [UxU

    ]these functions are in one-to-one correspon-

    dence with weak solutions U such that

    N(|t ,xU|) L2(R+ ).

    2 If is either block-diagonal or Hermitean, then for each f L2()there exists a unique such solution u.

  • Thank you for your attention!

    Kato Square Root Problem

  • Literature

    [1] T. KATO. Fractional powers of dissipative operators. J. Math. Soc. Japan 13(1961), 246274.

    [2] P. AUSCHER, S. HOFMANN, M. LACEY, A. MC INTOSH, and P. TCHAMITCHIAN. Thesolution of the Kato square root problem for second order elliptic operators on Rn.Ann. of Math. (2) 156 (2002), no. 2, 633654.

    [3] A. AXELSSON, S. KEITH, and A. MC INTOSH. Quadratic estimates and functionalcalculi of perturbed Dirac operators. Invent. Math. 163 (2006), no. 3, 455497.

    [4] M. EGERT, R. HALLER-DINTELMANN, and P. TOLKSDORF. The Kato Square RootProblem follows from an extrapolation property of the Laplacian. Submitted.

    [5] M. EGERT, R. HALLER-DINTELMANN, and P. TOLKSDORF. The Kato Square RootProblem for mixed boundary conditions. J. Funct. Anal. 267 (2014), no. 5,14191461.

    [6] P. AUSCHER, A. AXELSSON, and A. MC INTOSH. Solvability of elliptic systems withsquare integrable boundary data. Ark. Mat. 48 (2010), no. 2, 253287.