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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, . . . .

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).

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).

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).

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).

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

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

 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.

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

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

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

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