J.L. Lions problem on the maximal regularity for non-autonomous - - PowerPoint PPT Presentation

J.L. Lions’ problem on the maximal regularity for non-autonomous equations

El Maati Ouhabaz, Univ. Bordeaux Marrakech, April 2018

Autonomous Equations

Consider the Cauchy problem ∂tu(t) + Au(t) = f(t), t ∈ [0, T], u(0) = 0. (1) A : D(A) ⊂ E → E is (minus) the generator of a holomorphic semigroup on E.

Autonomous Equations

Consider the Cauchy problem ∂tu(t) + Au(t) = f(t), t ∈ [0, T], u(0) = 0. (1) A : D(A) ⊂ E → E is (minus) the generator of a holomorphic semigroup on E.

Definition

Maximal Lp-regularity: f ∈ Lp(0, T, E) ⇒ ∃u ∈ W 1,p(0, T, E) ∩ Lp(0, T, D(A)) satisfying (1).

Autonomous Equations

Consider the Cauchy problem ∂tu(t) + Au(t) = f(t), t ∈ [0, T], u(0) = 0. (1) A : D(A) ⊂ E → E is (minus) the generator of a holomorphic semigroup on E.

Definition

Maximal Lp-regularity: f ∈ Lp(0, T, E) ⇒ ∃u ∈ W 1,p(0, T, E) ∩ Lp(0, T, D(A)) satisfying (1). ⇒ An apriori estimate: uLp(0,T,E) + ∂tuLp(0,T,E) + Au(.)Lp(0,T,E) ≤ CfLp(0,T,E).

Autonomous Equations

Consider the Cauchy problem ∂tu(t) + Au(t) = f(t), t ∈ [0, T], u(0) = 0. (1) A : D(A) ⊂ E → E is (minus) the generator of a holomorphic semigroup on E.

Definition

Maximal Lp-regularity: f ∈ Lp(0, T, E) ⇒ ∃u ∈ W 1,p(0, T, E) ∩ Lp(0, T, D(A)) satisfying (1). ⇒ An apriori estimate: uLp(0,T,E) + ∂tuLp(0,T,E) + Au(.)Lp(0,T,E) ≤ CfLp(0,T,E). Works by Da Prato-Grisvard, Dore-Venni, Lamberton, L. Weis, Kalton-Lancien, + . . . + . . . + . . .

de Simon(64): Always true if E = H: Hilbert space.

de Simon(64): Always true if E = H: Hilbert space. Dore-Veni(’87): E is UMD: Lp-MR holds if Ais ≤ Cew|s| ∀s ∈ R for some w < π/2.

de Simon(64): Always true if E = H: Hilbert space. Dore-Veni(’87): E is UMD: Lp-MR holds if Ais ≤ Cew|s| ∀s ∈ R for some w < π/2. Lamberton(’87): MR holds for sub-Markovian semigroups on E = Lq(Ω, µ), 1 < q < ∞.

de Simon(64): Always true if E = H: Hilbert space. Dore-Veni(’87): E is UMD: Lp-MR holds if Ais ≤ Cew|s| ∀s ∈ R for some w < π/2. Lamberton(’87): MR holds for sub-Markovian semigroups on E = Lq(Ω, µ), 1 < q < ∞. Hieber-Pr¨ uss(’97), Coulhon-Duong(2000), E = Lq(Ω, µ) + good upper bounds on the heat kernel of A.

de Simon(64): Always true if E = H: Hilbert space. Dore-Veni(’87): E is UMD: Lp-MR holds if Ais ≤ Cew|s| ∀s ∈ R for some w < π/2. Lamberton(’87): MR holds for sub-Markovian semigroups on E = Lq(Ω, µ), 1 < q < ∞. Hieber-Pr¨ uss(’97), Coulhon-Duong(2000), E = Lq(Ω, µ) + good upper bounds on the heat kernel of A.

• L. Weis(2001): E = Lq, MR is equivalent to R-boundedness of e−zA

(complex z ∈ Σθ): 1

• N
• j=0

rj(t)e−zjAfjEdt ≤ C 1

• N
• j=0

rj(t)fjEdt ∀fj ∈ E, ∀zj ∈ Σθ where (rj) is a sequence of independent {−1, 1}-valued random variables

• n [0, 1].

de Simon(64): Always true if E = H: Hilbert space. Dore-Veni(’87): E is UMD: Lp-MR holds if Ais ≤ Cew|s| ∀s ∈ R for some w < π/2. Lamberton(’87): MR holds for sub-Markovian semigroups on E = Lq(Ω, µ), 1 < q < ∞. Hieber-Pr¨ uss(’97), Coulhon-Duong(2000), E = Lq(Ω, µ) + good upper bounds on the heat kernel of A.

• L. Weis(2001): E = Lq, MR is equivalent to R-boundedness of e−zA

(complex z ∈ Σθ): 1

• N
• j=0

rj(t)e−zjAfjEdt ≤ C 1

• N
• j=0

rj(t)fjEdt ∀fj ∈ E, ∀zj ∈ Σθ where (rj) is a sequence of independent {−1, 1}-valued random variables

• n [0, 1].

Kalton-Lancien(2000): ”negative results”.

Non-autonomous Equations

Consider the Cauchy problem (NACP) ∂tu(t) + A(t)u(t) = f(t), t ∈ [0, T], u(0) = u0. A(t) : D(A(t)) ⊂ E → E · · ·

Non-autonomous Equations

Consider the Cauchy problem (NACP) ∂tu(t) + A(t)u(t) = f(t), t ∈ [0, T], u(0) = u0. A(t) : D(A(t)) ⊂ E → E · · ·

Definition

Maximal Lp-regularity: f ∈ Lp(0, T, E) ⇒ ∃u ∈ W 1,p(0, T, E), t → A(t)u(t) ∈ Lp(0, T, E) unique which satisfies (NACP) in Lp − sense.

Non-autonomous Equations

Consider the Cauchy problem (NACP) ∂tu(t) + A(t)u(t) = f(t), t ∈ [0, T], u(0) = u0. A(t) : D(A(t)) ⊂ E → E · · ·

Definition

Maximal Lp-regularity: f ∈ Lp(0, T, E) ⇒ ∃u ∈ W 1,p(0, T, E), t → A(t)u(t) ∈ Lp(0, T, E) unique which satisfies (NACP) in Lp − sense. Works by: H. Amann, M. Giga, Y. Giga, H. Sohr, Pr¨ uss-Schnaubelt, Arendt-Chill-Fornaro-Poupaud, Batty-Chill-Srivastava, . . . assuming: D(A(t)) = D(A(0)) = D + continuity of t → A(t)u.

J.L. Lions’ theorems

Assumptions-Notations: H, V Hilbert spaces, V ⊂ H continuously and densely, and a(t, ·, ·) : V × V → C sesquilinear forms s.t. :

• |a(t, u, v)| ≤ MuVvV, u, v ∈ V, t ∈ [0, T];
• Rea(t, u, u) ≥ δu2

V − ku2 H,

• t → a(t, u, v) measurable for all u, v ∈ V.

Denote by A(t) the associated operator with the form a(t, ., .).

J.L. Lions’ theorems

Assumptions-Notations: H, V Hilbert spaces, V ⊂ H continuously and densely, and a(t, ·, ·) : V × V → C sesquilinear forms s.t. :

• |a(t, u, v)| ≤ MuVvV, u, v ∈ V, t ∈ [0, T];
• Rea(t, u, u) ≥ δu2

V − ku2 H,

• t → a(t, u, v) measurable for all u, v ∈ V.

Denote by A(t) the associated operator with the form a(t, ., .). Example: a(t, u, v) =

• k,l
• Ω

akl(t, x)∂lu∂kv dx, W 1,2 (Ω) ⊂ V ⊂ W 1,2(Ω) A(t) = −

• k,l

∂k(akl(t, x)∂l) + boundary conditions given byV.

• If V = W 1,2

(Ω) then we have the Dirichlet boundary conditions.

• If V = W 1,2(Ω) then we have Neumann type boundary conditions.

Theorem (J.L. Lions)

For u0 ∈ H, the non-autonomous Cauchy problem (NACP) has maximal L2-regularity in the dual space V ′.

Theorem (J.L. Lions)

For u0 ∈ H, the non-autonomous Cauchy problem (NACP) has maximal L2-regularity in the dual space V ′. Note however that working in V ′ is less interesting: when dealing with boundary value problems, one has to work in H = L2 in order to identify the boundary conditions.

Theorem (J.L. Lions)

For u0 ∈ H, the non-autonomous Cauchy problem (NACP) has maximal L2-regularity in the dual space V ′. Note however that working in V ′ is less interesting: when dealing with boundary value problems, one has to work in H = L2 in order to identify the boundary conditions.

Theorem (J.L. Lions)

• If t → a(t, u, v) is C1 and a(t, ., .) are symmetric then (NACP) with u0 = 0 has

maximal L2-regularity in H.

• If t → a(t, u, v) is C2 and a(t, ., .) are symmetric then (NACP) with

u0 ∈ D(A(0)) has maximal L2-regularity in H.

J.L. Lions’ problem (1961)

Problem 1: Does maximal L2-regularity hold in H without C1 assumption on t → a(t, u, v) when u0 = 0 ?

J.L. Lions’ problem (1961)

Problem 1: Does maximal L2-regularity hold in H without C1 assumption on t → a(t, u, v) when u0 = 0 ? Problem 2: Does maximal L2-regularity hold for all u0 ∈ D(A(0)) when t → a(t, u, v) is C1 ?

J.L. Lions’ problem (1961)

Problem 1: Does maximal L2-regularity hold in H without C1 assumption on t → a(t, u, v) when u0 = 0 ? Problem 2: Does maximal L2-regularity hold for all u0 ∈ D(A(0)) when t → a(t, u, v) is C1 ? Bardos (1971): u0 ∈ V is allowed provided D(A(t)1/2) = V and strong regularity of A(t) with respect to t.

J.L. Lions’ problem (1961)

Problem 1: Does maximal L2-regularity hold in H without C1 assumption on t → a(t, u, v) when u0 = 0 ? Problem 2: Does maximal L2-regularity hold for all u0 ∈ D(A(0)) when t → a(t, u, v) is C1 ? Bardos (1971): u0 ∈ V is allowed provided D(A(t)1/2) = V and strong regularity of A(t) with respect to t.

Theorem (Ou-Spina, J.D.E 2010)

Suppose t → a(t, u, v) is H¨

• lder continuous in the sense: for some α > 1

2,

|a(t, u, v) − a(s, u, v)| ≤ K|t − s|αuVvV for all s, t ∈ [0, T] and u, v ∈ V. Then (NACP) has maximal Lp-regularity in H when u0 = 0. ⇒ partial answer to Problem 1.

Theorem (Haak-Ou, Math. Ann. 2015)

Suppose that |a(t, u, v) − a(s, u, v)| ≤ ω(|t − s|)uVvV with ω : [0, T] → [0, ∞) a non-decreasing function such that T ω(t) t3/2 dt < ∞. Then the non autonomous Cauchy problem (NACP), with u0 = 0, has maximal Lp-regularity in H for all p ∈ (1, ∞). If in addition ω satisfies T ω(t)p t

1+p 2

dt < ∞ then (NACP) has maximal Lp-regularity for all u0 ∈ (H, D(A(0)))1−1/p,p. In the particular case p = 2, maximal L2-regularity holds for all u0 ∈ D(A(0)1/2) if t → a(t, u, v) is piecewise α- H¨

• lder continuous for some

α > 1/2. This gives a complete answer to Problem 2 by Lions.

Corollary

Suppose that the form a is piecewise α-H¨

• lder continuous for some α > 1/2.

That is, there exist t0 = 0 < t1 < ... < tk = τ such that on each interval (ti, ti+1) the form is the restriction of a α–H¨

• lder continuous form on [ti, ti+1]. Assume

in addition that at the discontinuity points, we have D((δ + A(t−

j ))1/2) = D((δ + A(t+ j ))1/2). Then (NACP) has maximal

L2–regularity for all u0 ∈ D((δ + A(0))1/2) and there exists a positive constant C such that uW 1

2 (0,τ;H) + A(·)u(·)L2(0,τ;H) ≤ C

• fL2(0,τ;H) + u0D((δ+A(0))1/2)
• .

Corollary

Suppose that the form a is piecewise α-H¨

• lder continuous for some α > 1/2.

That is, there exist t0 = 0 < t1 < ... < tk = τ such that on each interval (ti, ti+1) the form is the restriction of a α–H¨

• lder continuous form on [ti, ti+1]. Assume

in addition that at the discontinuity points, we have D((δ + A(t−

j ))1/2) = D((δ + A(t+ j ))1/2). Then (NACP) has maximal

L2–regularity for all u0 ∈ D((δ + A(0))1/2) and there exists a positive constant C such that uW 1

2 (0,τ;H) + A(·)u(·)L2(0,τ;H) ≤ C

• fL2(0,τ;H) + u0D((δ+A(0))1/2)
• .

The idea is to apply the previous theorem on each sub-interval (ti, ti+1), prove u(ti+1) ∈ D((δ + A(t−

j ))1/2) = D((δ + A(t+ j ))1/2) and glue the corresponding

solutions. The result in the corollary does NOT hold if D((δ + A(t−

j ))1/2) = D((δ + A(t+ j ))1/2) ! Observation due to D. Dier.

Theorem (Ou., Arch. Math 2015)

Suppose that for some β, γ ∈ [0, 1] |a(t, u, v) − a(s, u, v)| ≤ ω(|t − s|)u[H,V]βv[H,V]γ where ω : [0, T] → [0, ∞) is a non-decreasing function such that T ω(t) t1+ γ

2 dt < ∞.

Then (NACP) with u0 = 0 has maximal Lp-regularity in H for all p ∈ (1, ∞). If in addition, T w(t)p t

1 2 (β+pγ) dt < ∞

then (NACP) has maximal Lp-regularity in H for all u0 ∈ (H, D(A(0)))1− 1

p ,p.

A Related result by Arendt and Monniaux (Math. Nach. 2016)

• S. Fackler’s negative result:

Theorem (Fackler, AIHP 2016)

There exist (even symmetric) forms such that t → a(t, u, v) is 1

2-H¨

• lder

continuous such that the corresponding non-autonomous Cauchy problem (NACP) does NOT have maximal L2-regularity. This proves that for the remaining part Cα for α ≤ 1/2 in Lions’ problem (Problem 1 above) the answer is no in general. In particular, our previous result with t → a(t, u, v) is piecewice Cα for some α > 1/2 is sharp.

Theorem (Achache-Ou, Studia Math. 2018)

Suppose that for some β, γ ∈ [0, 1] |a(t, u, v) − a(s, u, v)| ≤ ω(|t − s|)u[H,V]βv[H,V]γ, where ω : [0, T] → [0, ∞) is a non-decreasing function such that : T ω(t) t1+ γ

2 dt < ∞.

Let B(t), P(t) be bounded operators on H such that t → B(t) is continuous on [0, T] with values in L(H) and Re(B(t)−1x, x) ≥ δx2

• H. Then the Cauchy

problem u′(t) + B(t)A(t)u(t) + P(t)u(t) = f(t), u(0) = 0 has maximal Lp-regularity in H for all p ∈ (1, ∞). If in addition, T ω(t)p t

1 2 (β+pγ) dt < ∞

then: u′(t) + B(t)A(t)u(t) + P(t)u(t) = f(t), u(0) = u0 has maximal Lp-regularity in H provided u0 ∈ (H, D(A(0)))1− 1

p ,p.

Some Ideas of Proof of Theorem (Haak-Ou.):

Some Ideas of Proof of Theorem (Haak-Ou.): Set A(t) : V → V ′, a(t, u, v) = < A(t)u, v >V ′,V.

Some Ideas of Proof of Theorem (Haak-Ou.): Set A(t) : V → V ′, a(t, u, v) = < A(t)u, v >V ′,V. Then the solution u(t) ∈ D(A(t)) exists in V ′ by Lions’ theorem. In addition A(t)u(t) = (QA(·)u(·))(t) + (Lf)(t) + (Ru0)(t), where (Qg)(t) := t

0 A(t)e−(t−s)A(t)(A(t) − A(s)) A(s)−1g(s) ds

(Lg)(t) := A(t) t

0 e−(t−s)A(t)g(s) ds

(Ru0)(t) := A(t)e−tA(t)u0.

Some Ideas of Proof of Theorem (Haak-Ou.): Set A(t) : V → V ′, a(t, u, v) = < A(t)u, v >V ′,V. Then the solution u(t) ∈ D(A(t)) exists in V ′ by Lions’ theorem. In addition A(t)u(t) = (QA(·)u(·))(t) + (Lf)(t) + (Ru0)(t), where (Qg)(t) := t

0 A(t)e−(t−s)A(t)(A(t) − A(s)) A(s)−1g(s) ds

(Lg)(t) := A(t) t

0 e−(t−s)A(t)g(s) ds

(Ru0)(t) := A(t)e−tA(t)u0. One has the estimates: A(t)e−(t−s)A(t)(A(t) − A(s)) A(s)−1g(s)H ≤ A(t)e−(t−s)A(t)V ′−H A(t) − A(s)V−V ′ A(s)−1g(s)V ≤ C(t − s)−3/2ω(t − s)g(s)H.

Some Ideas of Proof of Theorem (Haak-Ou.): Set A(t) : V → V ′, a(t, u, v) = < A(t)u, v >V ′,V. Then the solution u(t) ∈ D(A(t)) exists in V ′ by Lions’ theorem. In addition A(t)u(t) = (QA(·)u(·))(t) + (Lf)(t) + (Ru0)(t), where (Qg)(t) := t

0 A(t)e−(t−s)A(t)(A(t) − A(s)) A(s)−1g(s) ds

(Lg)(t) := A(t) t

0 e−(t−s)A(t)g(s) ds

(Ru0)(t) := A(t)e−tA(t)u0. One has the estimates: A(t)e−(t−s)A(t)(A(t) − A(s)) A(s)−1g(s)H ≤ A(t)e−(t−s)A(t)V ′−H A(t) − A(s)V−V ′ A(s)−1g(s)V ≤ C(t − s)−3/2ω(t − s)g(s)H. The assumption T

ω(t) t3/2 dt < ∞ implies that I − Q is invertible on L2(0, T; H).

Some Ideas of Proof of Theorem (Haak-Ou.): Set A(t) : V → V ′, a(t, u, v) = < A(t)u, v >V ′,V. Then the solution u(t) ∈ D(A(t)) exists in V ′ by Lions’ theorem. In addition A(t)u(t) = (QA(·)u(·))(t) + (Lf)(t) + (Ru0)(t), where (Qg)(t) := t

0 A(t)e−(t−s)A(t)(A(t) − A(s)) A(s)−1g(s) ds

(Lg)(t) := A(t) t

0 e−(t−s)A(t)g(s) ds

(Ru0)(t) := A(t)e−tA(t)u0. One has the estimates: A(t)e−(t−s)A(t)(A(t) − A(s)) A(s)−1g(s)H ≤ A(t)e−(t−s)A(t)V ′−H A(t) − A(s)V−V ′ A(s)−1g(s)V ≤ C(t − s)−3/2ω(t − s)g(s)H. The assumption T

ω(t) t3/2 dt < ∞ implies that I − Q is invertible on L2(0, T; H).

L is a pseudo-differential operator Lf(t) = F−1 ξ → σ(t, ξ)F(ξ)

• ,

with operator-valued symbol σ(t, ξ) =    A(0)(iξ + A(0))−1 if t < 0 A(t)(iξ + A(t))−1 if 0 ≤ t ≤ η A(η)(iξ + A(η))−1 if t > η

L is bounded on L2(R; H) by the following theorem (due to Muramatu and Nagase ’81 in the scalar case H = R).

L is bounded on L2(R; H) by the following theorem (due to Muramatu and Nagase ’81 in the scalar case H = R).

Theorem

Let T be a pseudo-differential operator with symbol σ(t, ξ). Suppose that there exists a non-decreasing function w : [0, ∞) → [0, ∞) such that ∂α

ξ σ(x, ξ)L(H) ≤ Cα < ξ >−|α|

∂α

ξ σ(x, ξ) − ∂α ξ σ(x′, ξ)L(H) ≤ Cα < ξ >−|α| ω(|x − x′|)

for all |α| ≤ [ n

2] + 2 and some positive constant Cα. Suppose in addition that

1 ω(t)2 dt t < ∞, then T is a bounded operator on L2(R; H).

L is bounded on L2(R; H) by the following theorem (due to Muramatu and Nagase ’81 in the scalar case H = R).

Theorem

Let T be a pseudo-differential operator with symbol σ(t, ξ). Suppose that there exists a non-decreasing function w : [0, ∞) → [0, ∞) such that ∂α

ξ σ(x, ξ)L(H) ≤ Cα < ξ >−|α|

∂α

ξ σ(x, ξ) − ∂α ξ σ(x′, ξ)L(H) ≤ Cα < ξ >−|α| ω(|x − x′|)

for all |α| ≤ [ n

2] + 2 and some positive constant Cα. Suppose in addition that

1 ω(t)2 dt t < ∞, then T is a bounded operator on L2(R; H). We obtain (I − Q)−1(Lf) is bounded on L2(0, T; H). Boundedness on Lp(0, T; H) is obtained by Calder´

• n-Zygmund theory.

L is bounded on L2(R; H) by the following theorem (due to Muramatu and Nagase ’81 in the scalar case H = R).

Theorem

Let T be a pseudo-differential operator with symbol σ(t, ξ). Suppose that there exists a non-decreasing function w : [0, ∞) → [0, ∞) such that ∂α

ξ σ(x, ξ)L(H) ≤ Cα < ξ >−|α|

∂α

ξ σ(x, ξ) − ∂α ξ σ(x′, ξ)L(H) ≤ Cα < ξ >−|α| ω(|x − x′|)

for all |α| ≤ [ n

2] + 2 and some positive constant Cα. Suppose in addition that

1 ω(t)2 dt t < ∞, then T is a bounded operator on L2(R; H). We obtain (I − Q)−1(Lf) is bounded on L2(0, T; H). Boundedness on Lp(0, T; H) is obtained by Calder´

• n-Zygmund theory. For u0 = 0, consider the

difference with the case t = 0 and use functional calculus A(t)e−tA(t)u0 − A(0)e−tA(0)u0 = 1 2πi

• Γ

ze−tz R(z, A(t)) − R(z, A(0))

• dz

Fractional Sobolev regularity

Recall that g ∈ L2(I; X) is in the fractional Sobolev space Hα(I; X) if

• I×I

f(t) − f(s)2

X

|t − s|2α+1 dsdt < ∞. If t → A(t) ∈ H

1 2 +ε(0, τ; L(V, V ′)) then maximal L2-regularity holds in H

(Dier-Zacher, J.E.E. 2017).

Fractional Sobolev regularity

Recall that g ∈ L2(I; X) is in the fractional Sobolev space Hα(I; X) if

• I×I

f(t) − f(s)2

X

|t − s|2α+1 dsdt < ∞. If t → A(t) ∈ H

1 2 +ε(0, τ; L(V, V ′)) then maximal L2-regularity holds in H

(Dier-Zacher, J.E.E. 2017). A Banach space version (instead of H) was proved by Fackler.

Fractional Sobolev regularity

Recall that g ∈ L2(I; X) is in the fractional Sobolev space Hα(I; X) if

• I×I

f(t) − f(s)2

X

|t − s|2α+1 dsdt < ∞. If t → A(t) ∈ H

1 2 +ε(0, τ; L(V, V ′)) then maximal L2-regularity holds in H

(Dier-Zacher, J.E.E. 2017). A Banach space version (instead of H) was proved by Fackler. If A(t) = −

k,j ∂k

• akj(t, x)∂j
• n H = L2(Rn) with t → akj(t, x) satisfy a

certain BMO − H1/2 condition then maximal L2-regularity holds (Auscher-Egert, Archiv Math. 2016).

Fractional Sobolev regularity

Recall that g ∈ L2(I; X) is in the fractional Sobolev space Hα(I; X) if

• I×I

f(t) − f(s)2

X

|t − s|2α+1 dsdt < ∞. If t → A(t) ∈ H

1 2 +ε(0, τ; L(V, V ′)) then maximal L2-regularity holds in H

(Dier-Zacher, J.E.E. 2017). A Banach space version (instead of H) was proved by Fackler. If A(t) = −

k,j ∂k

• akj(t, x)∂j
• n H = L2(Rn) with t → akj(t, x) satisfy a

certain BMO − H1/2 condition then maximal L2-regularity holds (Auscher-Egert, Archiv Math. 2016). Fackler’s counter-example actually shows that A(.) ∈ W 1/2,p(0, τ; L(V, V ′)) for p > 2 does NOT imply maximal L2-regularity (Arendt-Dier-Fackler, Archiv. Math. 2017).

Fractional Sobolev regularity

Recall that g ∈ L2(I; X) is in the fractional Sobolev space Hα(I; X) if

• I×I

f(t) − f(s)2

X

|t − s|2α+1 dsdt < ∞. If t → A(t) ∈ H

1 2 +ε(0, τ; L(V, V ′)) then maximal L2-regularity holds in H

(Dier-Zacher, J.E.E. 2017). A Banach space version (instead of H) was proved by Fackler. If A(t) = −

k,j ∂k

• akj(t, x)∂j
• n H = L2(Rn) with t → akj(t, x) satisfy a

certain BMO − H1/2 condition then maximal L2-regularity holds (Auscher-Egert, Archiv Math. 2016). Fackler’s counter-example actually shows that A(.) ∈ W 1/2,p(0, τ; L(V, V ′)) for p > 2 does NOT imply maximal L2-regularity (Arendt-Dier-Fackler, Archiv. Math. 2017). Dier’s counter-example actually shows that A(.) ∈ W 1/2,p(0, τ; L(V, V ′)) for p < 2 does NOT imply maximal L2-regularity (at least for forms which do not satisfy the Kato square root property).

Problem: What about the case p = 2, i.e., A(.) ∈ H

1 2 (0, τ; L(V, V ′)) ?

Problem: What about the case p = 2, i.e., A(.) ∈ H

1 2 (0, τ; L(V, V ′)) ?

Theorem (Achache-Ou, 2017)

Suppose the forms a(t) satisfy the uniform Kato square root property and a little of regularity (e.g., Cǫ) and A(.) ∈ H

1 2 (0, τ; L(V, V ′)) (even just

piecewise). Then the maximal L2-regularity holds for any u0 ∈ V.

Problem: What about the case p = 2, i.e., A(.) ∈ H

1 2 (0, τ; L(V, V ′)) ?

Theorem (Achache-Ou, 2017)

Suppose the forms a(t) satisfy the uniform Kato square root property and a little of regularity (e.g., Cǫ) and A(.) ∈ H

1 2 (0, τ; L(V, V ′)) (even just

piecewise). Then the maximal L2-regularity holds for any u0 ∈ V.

• The uniform Kato square root property:

c1u2

V ≤ Rea(t, u, u) ≤ c2u2 V.

Problem: What about the case p = 2, i.e., A(.) ∈ H

1 2 (0, τ; L(V, V ′)) ?

Theorem (Achache-Ou, 2017)

Suppose the forms a(t) satisfy the uniform Kato square root property and a little of regularity (e.g., Cǫ) and A(.) ∈ H

1 2 (0, τ; L(V, V ′)) (even just

piecewise). Then the maximal L2-regularity holds for any u0 ∈ V.

• The uniform Kato square root property:

c1u2

V ≤ Rea(t, u, u) ≤ c2u2 V.

• The proof is different from the case A(.) ∈ C

1 2 +ε(0, τ; L(V, V ′)). It is based

• n certain L∞ for the solution of the Cauchy problem together with tools from

harmonic analysis such as square function estimates, functional calculus...

Problem: What about the case p = 2, i.e., A(.) ∈ H

1 2 (0, τ; L(V, V ′)) ?

Theorem (Achache-Ou, 2017)

Suppose the forms a(t) satisfy the uniform Kato square root property and a little of regularity (e.g., Cǫ) and A(.) ∈ H

1 2 (0, τ; L(V, V ′)) (even just

piecewise). Then the maximal L2-regularity holds for any u0 ∈ V.

• The uniform Kato square root property:

c1u2

V ≤ Rea(t, u, u) ≤ c2u2 V.

• The proof is different from the case A(.) ∈ C

1 2 +ε(0, τ; L(V, V ′)). It is based

• n certain L∞ for the solution of the Cauchy problem together with tools from

harmonic analysis such as square function estimates, functional calculus...

• The order of smoothness 1

2 can be improved into γ 2 if in addition

A(t) − A(s) : V → [H, V]γ This latter condition holds in some situation such as Robin boundary conditions or Schr¨

• dinger operators with time dependent potentials.

Examples:

Let Ω ⊂ Rd be a bounded Lipschitz domain, H = L2(Ω).

Examples:

Let Ω ⊂ Rd be a bounded Lipschitz domain, H = L2(Ω). A Linear Problem (1):    ∂tu(t) − d

k,l=1 ∂k(akl(t, x)∂lu) = f(t), t ∈ [0, T],

u(0) = u0 ∈ V + Dirichlet or Neumann on b.c. on ∂Ω.

Examples:

Let Ω ⊂ Rd be a bounded Lipschitz domain, H = L2(Ω). A Linear Problem (1):    ∂tu(t) − d

k,l=1 ∂k(akl(t, x)∂lu) = f(t), t ∈ [0, T],

u(0) = u0 ∈ V + Dirichlet or Neumann on b.c. on ∂Ω. Then if t → akl(t, x) is piecewise Cα for some α > 1/2 then the above problem has L2-maximal regularity.

Examples:

Let Ω ⊂ Rd be a bounded Lipschitz domain, H = L2(Ω). A Linear Problem (1):    ∂tu(t) − d

k,l=1 ∂k(akl(t, x)∂lu) = f(t), t ∈ [0, T],

u(0) = u0 ∈ V + Dirichlet or Neumann on b.c. on ∂Ω. Then if t → akl(t, x) is piecewise Cα for some α > 1/2 then the above problem has L2-maximal regularity. Proof: apply the above results to a(t, u, v) =

• k,l
• Ω

akl(t, x)∂lu∂kv dx and note that D(A(t)1/2) = W 1,2 (Ω) or W 1,2(Ω) depending on the b.c. This is the Kato square root property (cf. Auscher, Hofmann, Lacey, McIntosh and Tchamitchian 2002).

A Linear Problem (2):

A Linear Problem (2):    ∂tu(t) − ∆u(t) = f(t), t ∈ [0, T], u(0) = u0

∂u ∂n + β(t, .)u = 0 on ∂Ω.

A Linear Problem (2):    ∂tu(t) − ∆u(t) = f(t), t ∈ [0, T], u(0) = u0

∂u ∂n + β(t, .)u = 0 on ∂Ω.

Define a(t, u, v) =

• Ω

∇u.∇vdx +

• ∂Ω

β(t, .)uvdσ, u, v ∈ W 1,2(Ω).

A Linear Problem (2):    ∂tu(t) − ∆u(t) = f(t), t ∈ [0, T], u(0) = u0

∂u ∂n + β(t, .)u = 0 on ∂Ω.

Define a(t, u, v) =

• Ω

∇u.∇vdx +

• ∂Ω

β(t, .)uvdσ, u, v ∈ W 1,2(Ω). If β is Cα w.r.t. t for some α > 1/4 and u0 ∈ W 1,2(Ω), then L2-maximal regularity holds.

A Linear Problem (2):    ∂tu(t) − ∆u(t) = f(t), t ∈ [0, T], u(0) = u0

∂u ∂n + β(t, .)u = 0 on ∂Ω.

Define a(t, u, v) =

• Ω

∇u.∇vdx +

• ∂Ω

β(t, .)uvdσ, u, v ∈ W 1,2(Ω). If β is Cα w.r.t. t for some α > 1/4 and u0 ∈ W 1,2(Ω), then L2-maximal regularity holds. Indeed, |a(t; u, v) − a(s; u, v)| = |

• ∂Ω

[β(t, .) − β(s, .)]Tr(u)Tr(v) dσ| ≤ C|t − s|α uH1/2+ε(Ω)vH1/2+ε(Ω)

• ,

(the trace operator is bounded from H1/2+ε(Ω) into L2(∂Ω) for ε > 0).

A Non-linear Problem:

A Non-linear Problem: (NLCP)    ∂tu(t) − m(t, u(t))∆u(t) = f(t), t ∈ [0, T], u(0) = u0 ∈ W 1,2(Ω)

∂u ∂n + β(t, .)u = 0 on ∂Ω.

The function m : [0, T] × R → [δ, 1

δ] is continuous.

A Non-linear Problem: (NLCP)    ∂tu(t) − m(t, u(t))∆u(t) = f(t), t ∈ [0, T], u(0) = u0 ∈ W 1,2(Ω)

∂u ∂n + β(t, .)u = 0 on ∂Ω.

The function m : [0, T] × R → [δ, 1

δ] is continuous.

Theorem

Suppose β is C1/4+ε (w.r.t. t). Let f ∈ L2(0, T, L2(Ω)) and u0 ∈ W 1,2(Ω). Then there exists a solution u ∈ W 1,2(0, T, L2(Ω)) ∩ L2(0, T, W 1,2(Ω)) of (NLCP).

A Non-linear Problem: (NLCP)    ∂tu(t) − m(t, u(t))∆u(t) = f(t), t ∈ [0, T], u(0) = u0 ∈ W 1,2(Ω)

∂u ∂n + β(t, .)u = 0 on ∂Ω.

The function m : [0, T] × R → [δ, 1

δ] is continuous.

Theorem

Suppose β is C1/4+ε (w.r.t. t). Let f ∈ L2(0, T, L2(Ω)) and u0 ∈ W 1,2(Ω). Then there exists a solution u ∈ W 1,2(0, T, L2(Ω)) ∩ L2(0, T, W 1,2(Ω)) of (NLCP). Proof:

A Non-linear Problem: (NLCP)    ∂tu(t) − m(t, u(t))∆u(t) = f(t), t ∈ [0, T], u(0) = u0 ∈ W 1,2(Ω)

∂u ∂n + β(t, .)u = 0 on ∂Ω.

The function m : [0, T] × R → [δ, 1

δ] is continuous.

Theorem

Suppose β is C1/4+ε (w.r.t. t). Let f ∈ L2(0, T, L2(Ω)) and u0 ∈ W 1,2(Ω). Then there exists a solution u ∈ W 1,2(0, T, L2(Ω)) ∩ L2(0, T, W 1,2(Ω)) of (NLCP). Proof: For v ∈ L2(0, T, H), define: g ∈ H → Bv(t)g = m(t, v(t))g.

A Non-linear Problem: (NLCP)    ∂tu(t) − m(t, u(t))∆u(t) = f(t), t ∈ [0, T], u(0) = u0 ∈ W 1,2(Ω)

∂u ∂n + β(t, .)u = 0 on ∂Ω.

The function m : [0, T] × R → [δ, 1

δ] is continuous.

Theorem

Suppose β is C1/4+ε (w.r.t. t). Let f ∈ L2(0, T, L2(Ω)) and u0 ∈ W 1,2(Ω). Then there exists a solution u ∈ W 1,2(0, T, L2(Ω)) ∩ L2(0, T, W 1,2(Ω)) of (NLCP). Proof: For v ∈ L2(0, T, H), define: g ∈ H → Bv(t)g = m(t, v(t))g. By maximal regularity there exists a solution u ∈ W 1,2(0, T, H) ∩ L2(0, T, W 1,2(Ω)) of

• ∂tu(t) + Bv(t)A(t)u(t) = f(t),

u(0) = u0 ∈ W 1,2(Ω).

A Non-linear Problem: (NLCP)    ∂tu(t) − m(t, u(t))∆u(t) = f(t), t ∈ [0, T], u(0) = u0 ∈ W 1,2(Ω)

∂u ∂n + β(t, .)u = 0 on ∂Ω.

The function m : [0, T] × R → [δ, 1

δ] is continuous.

Theorem

Suppose β is C1/4+ε (w.r.t. t). Let f ∈ L2(0, T, L2(Ω)) and u0 ∈ W 1,2(Ω). Then there exists a solution u ∈ W 1,2(0, T, L2(Ω)) ∩ L2(0, T, W 1,2(Ω)) of (NLCP). Proof: For v ∈ L2(0, T, H), define: g ∈ H → Bv(t)g = m(t, v(t))g. By maximal regularity there exists a solution u ∈ W 1,2(0, T, H) ∩ L2(0, T, W 1,2(Ω)) of

• ∂tu(t) + Bv(t)A(t)u(t) = f(t),

u(0) = u0 ∈ W 1,2(Ω). Consider S : L2(0, T, H) → L2(0, T, H), Sv = u.

A Non-linear Problem: (NLCP)    ∂tu(t) − m(t, u(t))∆u(t) = f(t), t ∈ [0, T], u(0) = u0 ∈ W 1,2(Ω)

∂u ∂n + β(t, .)u = 0 on ∂Ω.

The function m : [0, T] × R → [δ, 1

δ] is continuous.

Theorem

Suppose β is C1/4+ε (w.r.t. t). Let f ∈ L2(0, T, L2(Ω)) and u0 ∈ W 1,2(Ω). Then there exists a solution u ∈ W 1,2(0, T, L2(Ω)) ∩ L2(0, T, W 1,2(Ω)) of (NLCP). Proof: For v ∈ L2(0, T, H), define: g ∈ H → Bv(t)g = m(t, v(t))g. By maximal regularity there exists a solution u ∈ W 1,2(0, T, H) ∩ L2(0, T, W 1,2(Ω)) of

• ∂tu(t) + Bv(t)A(t)u(t) = f(t),

u(0) = u0 ∈ W 1,2(Ω). Consider S : L2(0, T, H) → L2(0, T, H), Sv = u. Maximal Regularity (apriori estimate) implies continuity of S.

A Non-linear Problem: (NLCP)    ∂tu(t) − m(t, u(t))∆u(t) = f(t), t ∈ [0, T], u(0) = u0 ∈ W 1,2(Ω)

∂u ∂n + β(t, .)u = 0 on ∂Ω.

The function m : [0, T] × R → [δ, 1

δ] is continuous.

Theorem

Suppose β is C1/4+ε (w.r.t. t). Let f ∈ L2(0, T, L2(Ω)) and u0 ∈ W 1,2(Ω). Then there exists a solution u ∈ W 1,2(0, T, L2(Ω)) ∩ L2(0, T, W 1,2(Ω)) of (NLCP). Proof: For v ∈ L2(0, T, H), define: g ∈ H → Bv(t)g = m(t, v(t))g. By maximal regularity there exists a solution u ∈ W 1,2(0, T, H) ∩ L2(0, T, W 1,2(Ω)) of

• ∂tu(t) + Bv(t)A(t)u(t) = f(t),

u(0) = u0 ∈ W 1,2(Ω). Consider S : L2(0, T, H) → L2(0, T, H), Sv = u. Maximal Regularity (apriori estimate) implies continuity of S. By Aubin-Lions lemma we can apply Schauder’s fixed point theorem to obtain u ∈ W 1,2(0, T, L2(Ω)) ∩ L2(0, T, W 1,2(Ω)) such that Su = u.

Two remaining problems:

Two remaining problems:

• What about the theorem with condition H1/2 if the uniform Kato square root

property is not satisfied (or not known) ?

• For the particular case of divergence form elliptic operators

A(t) = −

• k,j

∂k(akj(t, .)∂j) can one relax the required regularity t → akj(t, x) is C1/2+ε (or H1/2 ) and

• btain maximal regularity ?

Thank you for your attention !!