On Kato’s Square Root Problem
Moritz Egert
WIAS Berlin, February 11, 2015
T. Kato 1960s: Non-autonomous parabolic evolution equation

d
u(t)(x) + A(t)u(t)(x) = 0
(t > 0, x ∈ Ω),
dt

u(0) = u ∈ L2 (Ω).
0
I
I
A(t) ∼ − ∇x · µ(t, x)∇x via elliptic form a(t) : V(Ω) × V(Ω) → C.
u(t)(x) = e−tA u0 (x) if A(t) = A for all t > 0.
T. Kato 1960s: Non-autonomous parabolic evolution equation

d
u(t)(x) + A(t)u(t)(x) = 0
(t > 0, x ∈ Ω),
dt

u(0) = u ∈ L2 (Ω).
0
I
I
A(t) ∼ − ∇x · µ(t, x)∇x via elliptic form a(t) : V(Ω) × V(Ω) → C.
u(t)(x) = e−tA u0 (x) if A(t) = A for all t > 0.
T. Kato 1960s: Non-autonomous parabolic evolution equation

d
u(t)(x) + A(t)u(t)(x) = 0
(t > 0, x ∈ Ω),
dt

u(0) = u ∈ L2 (Ω).
0
I
I
A(t) ∼ − ∇x · µ(t, x)∇x via elliptic form a(t) : V(Ω) × V(Ω) → C.
u(t)(x) = e−tA u0 (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(A∗1/2 ). This is perhaps not true
in general. But the question is open even when A is regularly accretive. In
this case it appears reasonable to suppose that both D(A1/2 ) and D(A∗1/2 )
coincide with D(a), where a is the regular sesquilinear form which defines A.”
T. Kato 1960s: Non-autonomous parabolic evolution equation

d
u(t)(x) + A(t)u(t)(x) = 0
(t > 0, x ∈ Ω),
dt

u(0) = u ∈ L2 (Ω).
0
I
I
A(t) ∼ − ∇x · µ(t, x)∇x via elliptic form a(t) : V(Ω) × V(Ω) → C.
u(t)(x) = e−tA u0 (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(A∗1/2 ). This is perhaps not true
in general. But the question is open even when A is regularly accretive. In
this case it appears reasonable to suppose that both D(A1/2 ) and D(A∗1/2 )
coincide with D(a), where a is the regular sesquilinear form which defines A.”
I
I
Counterexamples by Lions 1962, Mc Intosh 1982
Specialize to divergence-form operators.
Setup
Let
I
Ω ⊆ Rd domain, D ⊆ ∂ Ω closed, µ ∈ L∞ (Ω)
A ∼ −∇ · µ∇ accretive operator on L2 (Ω)
associated with
Z
1,2
W1,2
(Ω)
×
W
(u, v ) 7→
µ∇u · ∇v .
D
D (Ω) → C,
I
Ω
I
A1/2 square root of A defined by e.g.
Z ∞
1
A1/2 u =
t −1/2 A(t + A)−1 dt.
π 0
Kato conjecture
It holds D(A1/2 ) = W1,2
D (Ω) with equivalent norms.
D
Why do we care about the Kato conjecture?
Philosophy
I
Elliptic non-regularity results D(A) * W2,2 (Ω).
I
Kato Conjecture ∼ optimal Sobolev regularity for A1/2 .
Why do we care about the Kato conjecture?
Philosophy
I
Elliptic non-regularity results D(A) * W2,2 (Ω).
I
Kato Conjecture ∼ optimal Sobolev regularity for A1/2 .
Ex. 1: Elliptic equations on Rd+



∂2
d
u(t)(x)
+
∇
·
µ(x)∇u(t,
x)
=
0
(t
>
0,
x
∈
R
),
2
∂t
u(0, x) = u0 (x) ∈ W1,2 (Rd ).
1/2
I
Solution u(t, x) = e−tA
I
Kato conjecture ∼ Rellich inequality “k∂t u|t=0 k2 ∼ k∇u0 k2 ”.
u0 (x).
Why do we care about the Kato conjecture?
Philosophy
I
Elliptic non-regularity results D(A) * W2,2 (Ω).
I
Kato Conjecture ∼ optimal Sobolev regularity for A1/2 .
Ex. 1: Elliptic equations on Rd+



∂2
d
u(t)(x)
+
∇
·
µ(x)∇u(t,
x)
=
0
(t
>
0,
x
∈
R
),
2
∂t
u(0, x) = u0 (x) ∈ W1,2 (Rd ).
1/2
I
Solution u(t, x) = e−tA
I
Kato conjecture ∼ Rellich inequality “k∂t u|t=0 k2 ∼ k∇u0 k2 ”.
u0 (x).
Ex. 2: Maximal parabolic regularity (e.g. Haller-Dintelmann-Rehberg)
I
In Lp -setting study parabolic equation
(
I
d
u(t) + Au(t) = f
dt
u(0) = 0.
(0 < t < T ),
Goal: Transport Max. Reg. from Lp (Ω) to W−1,p
(Ω).
D
Ex. 2: Maximal parabolic regularity (e.g. Haller-Dintelmann-Rehberg)
I
In Lp -setting study parabolic equation
(
I
d
u(t) + Au(t) = f
dt
u(0) = 0.
(0 < t < T ),
Goal: Transport Max. Reg. from Lp (Ω) to W−1,p
(Ω).
D
0
I Lp -Kato
conjecture ∼ (−∇ ·
µ> ∇)1/2
:
1,p0
WD (Ω)
0
→ Lp (Ω) isom.
Ex. 2: Maximal parabolic regularity (e.g. Haller-Dintelmann-Rehberg)
I
In Lp -setting study parabolic equation
(
I
(0 < t < T ),
Goal: Transport Max. Reg. from Lp (Ω) to W−1,p
(Ω).
D
0
I Lp -Kato
I
d
u(t) + Au(t) = f
dt
u(0) = 0.
conjecture ∼ (−∇ ·
µ> ∇)1/2
:
1,p0
WD (Ω)
0
→ Lp (Ω) isom.
Adjoint (−∇ · µ∇)1/2 : Lp (Ω) → WD−1,p (Ω) isomorphism that
commutes with parabolic solution operator
d
−1
+A
.
dt
Ex. 2: Maximal parabolic regularity (e.g. Haller-Dintelmann-Rehberg)
I
In Lp -setting study parabolic equation
(
I
(0 < t < T ),
Goal: Transport Max. Reg. from Lp (Ω) to W−1,p
(Ω).
D
0
I Lp -Kato
I
d
u(t) + Au(t) = f
dt
u(0) = 0.
conjecture ∼ (−∇ ·
µ> ∇)1/2
:
1,p0
WD (Ω)
0
→ Lp (Ω) isom.
Adjoint (−∇ · µ∇)1/2 : Lp (Ω) → WD−1,p (Ω) isomorphism that
commutes with parabolic solution operator
d
−1
+A
.
dt
Many further examples, e.g. Cauchy-Integral along Lipschitz curve,
hyperbolic wave equations, . . . .
Positive answers
Self-adjoint operators
Positive answers
Self-adjoint operators
X
Positive answers
Self-adjoint operators
X
Whole space Ω = Rd
I
I
d = 1: Coifman - Mc Intosh - Meyer ’82.
d ≥ 2: Auscher-Hofmann-Lacey-Mc Intosh-Tchamitchian ’01,
Axelsson-Keith-Mc Intosh ’06.
Positive answers
Self-adjoint operators
Whole space Ω = Rd
I
I
X
X
d = 1: Coifman - Mc Intosh - Meyer ’82.
d ≥ 2: Auscher-Hofmann-Lacey-Mc Intosh-Tchamitchian ’01,
Axelsson-Keith-Mc Intosh ’06.
Positive answers
Self-adjoint operators
Whole space Ω = Rd
I
I
X
X
d = 1: Coifman - Mc Intosh - Meyer ’82.
d ≥ 2: Auscher-Hofmann-Lacey-Mc Intosh-Tchamitchian ’01,
Axelsson-Keith-Mc Intosh ’06.
Bounded domains
I
Ω Lipschitz, D ∈ {∅, ∂ Ω}: Auscher-Tchamitchian ’03, ’01 (p 6= 2).
I
Ω smooth, smooth D ! ∂ Ω \ D interface: Axelsson-Keith-Mc Intosh ’06.
I
Ω Lipschitz around ∂ Ω \ D:
Auscher-Badr-Haller-Dintelmann-Rehberg ’12 (p 6= 2).
Kato for mixed boundary conditions
Theorem (E.-Haller-Dintelmann-Tolksdorf ’14)
Suppose
I
Ω ⊆ Rd bounded d-Ahlfors regular domain,
I
D ⊆ ∂ Ω closed and (d − 1)-Ahlfors regular,
I
Ω Lipschitz around ∂ Ω \ D.
Then
D(A1/2 ) = W1,2
D (Ω)
with kA1/2 uk2 ∼ k∇uk2 .
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 á la AKM ’06,
H∞ -functional calculus.
Some ideas of the proof
1
First-order approach via perturbed Dirac operators á la AKM ’06,
H∞ -functional calculus.
2
Getting rid of the coefficients via
Reduction-Theorem (E.-Haller-Dintelmann-Tolksdorf ’14)
s
In essence, the following holds: If D (−∆V ) ,→ H2s,2 (Ω) for
some s > 12 , then D(A1/2 ) = W1,2
D (Ω).
Some ideas of the proof
1
First-order approach via perturbed Dirac operators á la AKM ’06,
H∞ -functional calculus.
2
Getting rid of the coefficients via
Reduction-Theorem (E.-Haller-Dintelmann-Tolksdorf ’14)
s
In essence, the following holds: If D (−∆V ) ,→ H2s,2 (Ω) for
some s > 12 , then D(A1/2 ) = W1,2
D (Ω).
Extrapolate Kato for −∆V =⇒ Kato property for general A
Some ideas of the proof
1
First-order approach via perturbed Dirac operators á la AKM ’06,
H∞ -functional calculus.
2
Getting rid of the coefficients via
Reduction-Theorem (E.-Haller-Dintelmann-Tolksdorf ’14)
s
In essence, the following holds: If D (−∆V ) ,→ H2s,2 (Ω) for
some s > 12 , then D(A1/2 ) = W1,2
D (Ω).
Extrapolate Kato for −∆V =⇒ Kato property for general A
geometry, potential theory ! harmonic analysis
3
D((−∆V )1/2 ) = W1,2
D (Ω) by self-adjointness. Extrapolate by
Sneiberg’s stability theorem and the following result.
Theorem (E.-Haller-Dintelmann-Tolksdorf ’14)
Let θ ∈ (0, 1) and s0 , s1 ∈ ( 12 , 32 ). Put sθ := (1 − θ)s0 + θs1 . Then,
1,2
W1,2
(Ω)
=
H
D
D (Ω)
s0 ,2
s1 ,2
HD (Ω), HD (Ω) θ = HsDθ ,2 (Ω).
(
2
1,2
L (Ω), HD (Ω) θ =
1
Hθ,2
(Ω),
if
θ
>
D
2,
Hθ,2 (Ω), if θ < 12 .
3
D((−∆V )1/2 ) = W1,2
D (Ω) by self-adjointness. Extrapolate by
Sneiberg’s stability theorem and the following result.
Theorem (E.-Haller-Dintelmann-Tolksdorf ’14)
Let θ ∈ (0, 1) and s0 , s1 ∈ ( 12 , 32 ). Put sθ := (1 − θ)s0 + θs1 . Then,
1,2
W1,2
(Ω)
=
H
D
D (Ω)
s0 ,2
s1 ,2
HD (Ω), HD (Ω) θ = HsDθ ,2 (Ω).
(
2
1,2
L (Ω), HD (Ω) θ =
4
1
Hθ,2
(Ω),
if
θ
>
D
2,
Hθ,2 (Ω), if θ < 12 .
1
In fact, D((−∆V )s ) = H2s,2
(Ω)
for
|
D
2 − s| < ε.
Elliptic BVPs on cylindrical domains
Elliptic mixed BVP
−divt,x µ(x)∇t,x U = 0
(R+ × Ω)
U=0
(R+ × D)
∂νµ U = 0
(R+ × N)
∂νµ U = f ∈ L2
({0} × Ω)
Elliptic BVPs on cylindrical domains
Elliptic mixed BVP
−divt,x µ(x)∇t,x U = 0
(R+ × Ω)
U=0
(R+ × D)
∂νµ U = 0
(R+ × N)
∂νµ U = f ∈ L2
l
∂νµ U
F ∼
∇x U
({0} × Ω)
First order equation
∗
0
(−∇V )
∂t F +
BF = 0
−∇V
0
|
{z
}
D
F (0)⊥ = f
(t > 0)
Elliptic BVPs on cylindrical domains
Elliptic mixed BVP
−divt,x µ(x)∇t,x U = 0
(R+ × Ω)
U=0
(R+ × D)
∂νµ U = 0
(R+ × N)
∂νµ U = f ∈ L2
l
∂νµ U
F ∼
∇x U
({0} × Ω)
First order equation
∗
0
(−∇V )
∂t F +
BF = 0
−∇V
0
|
{z
}
(t > 0)
D
F (0)⊥ = f
L2loc (R+ ; L2 (Ω)) 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) = e−tDB F (0)
(t ≥ 0).
∂ U
Via F ∼ νµ
these functions are in one-to-one correspon∇x U
dence with weak solutions U such that
f∗ (|∇t,x U|) ∈ L2 (R+ × Ω).
N
2
If µ is either block-diagonal or Hermitean, then for each f ∈ L2 (Ω)
there exists a unique such solution u.
Thank you for your attention!
Kato
Square
Root
Problem
Literature
[1] T. K ATO. Fractional powers of dissipative operators. J. Math. Soc. Japan 13
(1961), 246–274.
[2] P. AUSCHER, S. H OFMANN, M. L ACEY, A. M C I NTOSH, and P. T CHAMITCHIAN. The
solution of the Kato square root problem for second order elliptic operators on Rn .
Ann. of Math. (2) 156 (2002), no. 2, 633–654.
[3] A. A XELSSON, S. K EITH, and A. M C I NTOSH. Quadratic estimates and functional
calculi of perturbed Dirac operators. Invent. Math. 163 (2006), no. 3, 455–497.
[4] M. E GERT, R. H ALLER -D INTELMANN, and P. TOLKSDORF. The Kato Square Root
Problem follows from an extrapolation property of the Laplacian. Submitted.
[5] M. E GERT, R. H ALLER -D INTELMANN, and P. TOLKSDORF. The Kato Square Root
Problem for mixed boundary conditions. J. Funct. Anal. 267 (2014), no. 5,
1419–1461.
[6] P. AUSCHER, A. A XELSSON, and A. M C I NTOSH. Solvability of elliptic systems with
square integrable boundary data. Ark. Mat. 48 (2010), no. 2, 253–287.