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

date post

11-Apr-2018Category

## Documents

view

226download

5

Embed Size (px)

### 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)

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.

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.

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.