[PPT] - On homogenization of periodic hyperbolic systems Yulia Meshkova PowerPoint Presentation

SLIDE 1

On homogenization

f periodic hyperbolic systems

Yulia Meshkova

y.meshkova@spbu.ru

Chebyshev Laboratory, St. Petersburg State University (Russia)

Quantissima in the Serenissima III Palazzo Pesaro-Papafava, Venice 21 August 2019

SLIDE 2

Introduction Class of operators Effective operator Survey Problem Results Method

Introduction to homogenization

The simplest periodic operator. In L2(Rd), A = −div g(x)∇, g(x + a) = g(x), x ∈ Rd, a ∈ Zd; g ∈ L∞; g(x) > 0 = ⇒A = A∗ 0.

Fig. 1–3: A periodic medium with different values of ε.
Fig. 4: The

effective medium.

Homogenization. Aε = −div g(x/ε)∇, ε > 0 is small.

Typical problem: Aεuε + uε = F ∈ L2(Rd). Question: uε ∼? as ε → 0. Answer: uε ∼ u0, where u0 solves −div g0∇u0 + u0 = F.

Example. If d = 1, g0 = g :=

1

0 g(x)−1 dx

−1 . If d > 1, g g0 g :=

(0,1)d g(x) dx (Voigt-Reuss bracketing).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

SLIDE 3

Introduction Class of operators Effective operator Survey Problem Results Method

Introduction to homogenization

The simplest periodic operator. In L2(Rd), A = −div g(x)∇, g(x + a) = g(x), x ∈ Rd, a ∈ Zd; g ∈ L∞; g(x) > 0 = ⇒A = A∗ 0.

Fig. 1–3: A periodic medium with different values of ε.
Fig. 4: The

effective medium.

Homogenization. Aε = −div g(x/ε)∇, ε > 0 is small.

Typical problem: Aεuε + uε = F ∈ L2(Rd). Question: uε ∼? as ε → 0. Answer: uε ∼ u0, where u0 solves −div g0∇u0 + u0 = F.

Example. If d = 1, g0 = g :=

1

0 g(x)−1 dx

−1 . If d > 1, g g0 g :=

(0,1)d g(x) dx (Voigt-Reuss bracketing).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

SLIDE 4

Introduction Class of operators Effective operator Survey Problem Results Method

Introduction to homogenization

The simplest periodic operator. In L2(Rd), A = −div g(x)∇, g(x + a) = g(x), x ∈ Rd, a ∈ Zd; g ∈ L∞; g(x) > 0 = ⇒A = A∗ 0.

Fig. 1–3: A periodic medium with different values of ε.
Fig. 4: The

effective medium.

Homogenization. Aε = −div g(x/ε)∇, ε > 0 is small.

Typical problem: Aεuε + uε = F ∈ L2(Rd). Question: uε ∼? as ε → 0. Answer: uε ∼ u0, where u0 solves −div g0∇u0 + u0 = F.

Example. If d = 1, g0 = g :=

1

0 g(x)−1 dx

−1 . If d > 1, g g0 g :=

(0,1)d g(x) dx (Voigt-Reuss bracketing).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

SLIDE 5

Introduction Class of operators Effective operator Survey Problem Results Method

Introduction to homogenization

The simplest periodic operator. In L2(Rd), A = −div g(x)∇, g(x + a) = g(x), x ∈ Rd, a ∈ Zd; g ∈ L∞; g(x) > 0 = ⇒A = A∗ 0.

Fig. 1–3: A periodic medium with different values of ε.
Fig. 4: The

effective medium.

Homogenization. Aε = −div g(x/ε)∇, ε > 0 is small.

Typical problem: Aεuε + uε = F ∈ L2(Rd). Question: uε ∼? as ε → 0. Answer: uε ∼ u0, where u0 solves −div g0∇u0 + u0 = F.

Example. If d = 1, g0 = g :=

1

0 g(x)−1 dx

−1 . If d > 1, g g0 g :=

(0,1)d g(x) dx (Voigt-Reuss bracketing).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

SLIDE 6

Introduction Class of operators Effective operator Survey Problem Results Method

Introduction to homogenization

The simplest periodic operator. In L2(Rd), A = −div g(x)∇, g(x + a) = g(x), x ∈ Rd, a ∈ Zd; g ∈ L∞; g(x) > 0 = ⇒A = A∗ 0.

Fig. 1–3: A periodic medium with different values of ε.
Fig. 4: The

effective medium.

Homogenization. Aε = −div g(x/ε)∇, ε > 0 is small.

Typical problem: Aεuε + uε = F ∈ L2(Rd). Question: uε ∼? as ε → 0. Answer: uε ∼ u0, where u0 solves −div g0∇u0 + u0 = F.

Example. If d = 1, g0 = g :=

1

0 g(x)−1 dx

−1 . If d > 1, g g0 g :=

(0,1)d g(x) dx (Voigt-Reuss bracketing).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

SLIDE 7

Introduction Class of operators Effective operator Survey Problem Results Method

Introduction to homogenization

The simplest periodic operator. In L2(Rd), A = −div g(x)∇, g(x + a) = g(x), x ∈ Rd, a ∈ Zd; g ∈ L∞; g(x) > 0 = ⇒A = A∗ 0.

Fig. 1–3: A periodic medium with different values of ε.
Fig. 4: The

effective medium.

Homogenization. Aε = −div g(x/ε)∇, ε > 0 is small.

Typical problem: Aεuε + uε = F ∈ L2(Rd). Question: uε ∼? as ε → 0. Answer: uε ∼ u0, where u0 solves −div g0∇u0 + u0 = F.

Example. If d = 1, g0 = g :=

1

0 g(x)−1 dx

−1 . If d > 1, g g0 g :=

(0,1)d g(x) dx (Voigt-Reuss bracketing).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

SLIDE 8

Introduction Class of operators Effective operator Survey Problem Results Method

Introduction to homogenization

The simplest periodic operator. In L2(Rd), A = −div g(x)∇, g(x + a) = g(x), x ∈ Rd, a ∈ Zd; g ∈ L∞; g(x) > 0 = ⇒A = A∗ 0.

Fig. 1–3: A periodic medium with different values of ε.
Fig. 4: The

effective medium.

Homogenization. Aε = −div g(x/ε)∇, ε > 0 is small.

Typical problem: Aεuε + uε = F ∈ L2(Rd). Question: uε ∼? as ε → 0. Answer: uε ∼ u0, where u0 solves −div g0∇u0 + u0 = F.

Example. If d = 1, g0 = g :=

1

0 g(x)−1 dx

−1 . If d > 1, g g0 g :=

(0,1)d g(x) dx (Voigt-Reuss bracketing).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

SLIDE 9

Introduction Class of operators Effective operator Survey Problem Results Method

Introduction to homogenization

The simplest periodic operator. In L2(Rd), A = −div g(x)∇, g(x + a) = g(x), x ∈ Rd, a ∈ Zd; g ∈ L∞; g(x) > 0 = ⇒A = A∗ 0.

Fig. 1–3: A periodic medium with different values of ε.
Fig. 4: The

effective medium.

Homogenization. Aε = −div g(x/ε)∇, ε > 0 is small.

Typical problem: Aεuε + uε = F ∈ L2(Rd). Question: uε ∼? as ε → 0. Answer: uε ∼ u0, where u0 solves −div g0∇u0 + u0 = F.

Example. If d = 1, g0 = g :=

1

0 g(x)−1 dx

−1 . If d > 1, g g0 g :=

(0,1)d g(x) dx (Voigt-Reuss bracketing).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

SLIDE 10

Introduction Class of operators Effective operator Survey Problem Results Method

Introduction to homogenization

The simplest periodic operator. In L2(Rd), A = −div g(x)∇, g(x + a) = g(x), x ∈ Rd, a ∈ Zd; g ∈ L∞; g(x) > 0 = ⇒A = A∗ 0.

Fig. 1–3: A periodic medium with different values of ε.
Fig. 4: The

effective medium.

Homogenization. Aε = −div g(x/ε)∇, ε > 0 is small.

Typical problem: Aεuε + uε = F ∈ L2(Rd). Question: uε ∼? as ε → 0. Answer: uε ∼ u0, where u0 solves −div g0∇u0 + u0 = F.

Example. If d = 1, g0 = g :=

1

0 g(x)−1 dx

−1 . If d > 1, g g0 g :=

(0,1)d g(x) dx (Voigt-Reuss bracketing).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

SLIDE 11

Introduction Class of operators Effective operator Survey Problem Results Method

Introduction to homogenization

The simplest periodic operator. In L2(Rd), A = −div g(x)∇, g(x + a) = g(x), x ∈ Rd, a ∈ Zd; g ∈ L∞; g(x) > 0 = ⇒A = A∗ 0.

Fig. 1–3: A periodic medium with different values of ε.
Fig. 4: The

effective medium.

Homogenization. Aε = −div g(x/ε)∇, ε > 0 is small.

Typical problem: Aεuε + uε = F ∈ L2(Rd). Question: uε ∼? as ε → 0. Answer: uε ∼ u0, where u0 solves −div g0∇u0 + u0 = F.

Example. If d = 1, g0 = g :=

1

0 g(x)−1 dx

−1 . If d > 1, g g0 g :=

(0,1)d g(x) dx (Voigt-Reuss bracketing).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

SLIDE 12

Introduction Class of operators Effective operator Survey Problem Results Method

Introduction to homogenization

The simplest periodic operator. In L2(Rd), A = −div g(x)∇, g(x + a) = g(x), x ∈ Rd, a ∈ Zd; g ∈ L∞; g(x) > 0 = ⇒A = A∗ 0.

Fig. 1–3: A periodic medium with different values of ε.
Fig. 4: The

effective medium.

Homogenization. Aε = −div g(x/ε)∇, ε > 0 is small.

Typical problem: Aεuε + uε = F ∈ L2(Rd). Question: uε ∼? as ε → 0. Answer: uε ∼ u0, where u0 solves −div g0∇u0 + u0 = F.

Example. If d = 1, g0 = g :=

1

0 g(x)−1 dx

−1 . If d > 1, g g0 g :=

(0,1)d g(x) dx (Voigt-Reuss bracketing).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

SLIDE 13

Introduction Class of operators Effective operator Survey Problem Results Method

Introduction to homogenization

The simplest periodic operator. In L2(Rd), A = −div g(x)∇, g(x + a) = g(x), x ∈ Rd, a ∈ Zd; g ∈ L∞; g(x) > 0 = ⇒A = A∗ 0.

Fig. 1–3: A periodic medium with different values of ε.
Fig. 4: The

effective medium.

Homogenization. Aε = −div g(x/ε)∇, ε > 0 is small.

Typical problem: Aεuε + uε = F ∈ L2(Rd). Question: uε ∼? as ε → 0. Answer: uε ∼ u0, where u0 solves −div g0∇u0 + u0 = F.

Example. If d = 1, g0 = g :=

1

0 g(x)−1 dx

−1 . If d > 1, g g0 g :=

(0,1)d g(x) dx (Voigt-Reuss bracketing).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

SLIDE 14

Introduction Class of operators Effective operator Survey Problem Results Method

Introduction to homogenization

The simplest periodic operator. In L2(Rd), A = −div g(x)∇, g(x + a) = g(x), x ∈ Rd, a ∈ Zd; g ∈ L∞; g(x) > 0 = ⇒A = A∗ 0.

Fig. 1–3: A periodic medium with different values of ε.
Fig. 4: The

effective medium.

Homogenization. Aε = −div g(x/ε)∇, ε > 0 is small.

Typical problem: Aεuε + uε = F ∈ L2(Rd). Question: uε ∼? as ε → 0. Answer: uε ∼ u0, where u0 solves −div g0∇u0 + u0 = F.

Example. If d = 1, g0 = g :=

1

0 g(x)−1 dx

−1 . If d > 1, g g0 g :=

(0,1)d g(x) dx (Voigt-Reuss bracketing).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

SLIDE 15

Introduction Class of operators Effective operator Survey Problem Results Method

Introduction to homogenization

The simplest periodic operator. In L2(Rd), A = −div g(x)∇, g(x + a) = g(x), x ∈ Rd, a ∈ Zd; g ∈ L∞; g(x) > 0 = ⇒A = A∗ 0.

Fig. 1–3: A periodic medium with different values of ε.
Fig. 4: The

effective medium.

Homogenization. Aε = −div g(x/ε)∇, ε > 0 is small.

Typical problem: Aεuε + uε = F ∈ L2(Rd). Question: uε ∼? as ε → 0. Answer: uε ∼ u0, where u0 solves −div g0∇u0 + u0 = F.

Example. If d = 1, g0 = g :=

1

0 g(x)−1 dx

−1 . If d > 1, g g0 g :=

(0,1)d g(x) dx (Voigt-Reuss bracketing).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

SLIDE 16

Introduction Class of operators Effective operator Survey Problem Results Method

Introduction to homogenization

The simplest periodic operator. In L2(Rd), A = −div g(x)∇, g(x + a) = g(x), x ∈ Rd, a ∈ Zd; g ∈ L∞; g(x) > 0 = ⇒A = A∗ 0.

Fig. 1–3: A periodic medium with different values of ε.
Fig. 4: The

effective medium.

Homogenization. Aε = −div g(x/ε)∇, ε > 0 is small.

Typical problem: Aεuε + uε = F ∈ L2(Rd). Question: uε ∼? as ε → 0. Answer: uε ∼ u0, where u0 solves −div g0∇u0 + u0 = F.

Example. If d = 1, g0 = g :=

1

0 g(x)−1 dx

−1 . If d > 1, g g0 g :=

(0,1)d g(x) dx (Voigt-Reuss bracketing).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

SLIDE 17

Introduction Class of operators Effective operator Survey Problem Results Method

Introduction to homogenization

The simplest periodic operator. In L2(Rd), A = −div g(x)∇, g(x + a) = g(x), x ∈ Rd, a ∈ Zd; g ∈ L∞; g(x) > 0 = ⇒A = A∗ 0.

Fig. 1–3: A periodic medium with different values of ε.
Fig. 4: The

effective medium.

Homogenization. Aε = −div g(x/ε)∇, ε > 0 is small.

Typical problem: Aεuε + uε = F ∈ L2(Rd). Question: uε ∼? as ε → 0. Answer: uε ∼ u0, where u0 solves −div g0∇u0 + u0 = F.

Example. If d = 1, g0 = g :=

1

0 g(x)−1 dx

−1 . If d > 1, g g0 g :=

(0,1)d g(x) dx (Voigt-Reuss bracketing).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

SLIDE 18

Introduction Class of operators Effective operator Survey Problem Results Method

Introduction to homogenization

The simplest periodic operator. In L2(Rd), A = −div g(x)∇, g(x + a) = g(x), x ∈ Rd, a ∈ Zd; g ∈ L∞; g(x) > 0 = ⇒A = A∗ 0.

Fig. 1–3: A periodic medium with different values of ε.
Fig. 4: The

effective medium.

Homogenization. Aε = −div g(x/ε)∇, ε > 0 is small.

Typical problem: Aεuε + uε = F ∈ L2(Rd). Question: uε ∼? as ε → 0. Answer: uε ∼ u0, where u0 solves −div g0∇u0 + u0 = F.

Example. If d = 1, g0 = g :=

1

0 g(x)−1 dx

−1 . If d > 1, g g0 g :=

(0,1)d g(x) dx (Voigt-Reuss bracketing).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

SLIDE 19

Introduction Class of operators Effective operator Survey Problem Results Method

Introduction to homogenization

The simplest periodic operator. In L2(Rd), A = −div g(x)∇, g(x + a) = g(x), x ∈ Rd, a ∈ Zd; g ∈ L∞; g(x) > 0 = ⇒A = A∗ 0.

Fig. 1–3: A periodic medium with different values of ε.
Fig. 4: The

effective medium.

Homogenization. Aε = −div g(x/ε)∇, ε > 0 is small.

Typical problem: Aεuε + uε = F ∈ L2(Rd). Question: uε ∼? as ε → 0. Answer: uε ∼ u0, where u0 solves −div g0∇u0 + u0 = F.

Example. If d = 1, g0 = g :=

1

0 g(x)−1 dx

−1 . If d > 1, g g0 g :=

(0,1)d g(x) dx (Voigt-Reuss bracketing).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

SLIDE 20

Introduction Class of operators Effective operator Survey Problem Results Method

Introduction to homogenization

The simplest periodic operator. In L2(Rd), A = −div g(x)∇, g(x + a) = g(x), x ∈ Rd, a ∈ Zd; g ∈ L∞; g(x) > 0 = ⇒A = A∗ 0.

Fig. 1–3: A periodic medium with different values of ε.
Fig. 4: The

effective medium.

Homogenization. Aε = −div g(x/ε)∇, ε > 0 is small.

Typical problem: Aεuε + uε = F ∈ L2(Rd). Question: uε ∼? as ε → 0. Answer: uε ∼ u0, where u0 solves −div g0∇u0 + u0 = F.

Example. If d = 1, g0 = g :=

1

0 g(x)−1 dx

−1 . If d > 1, g g0 g :=

(0,1)d g(x) dx (Voigt-Reuss bracketing).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

SLIDE 21

Introduction Class of operators Effective operator Survey Problem Results Method

Introduction to homogenization

The simplest periodic operator. In L2(Rd), A = −div g(x)∇, g(x + a) = g(x), x ∈ Rd, a ∈ Zd; g ∈ L∞; g(x) > 0 = ⇒A = A∗ 0.

Fig. 1–3: A periodic medium with different values of ε.
Fig. 4: The

effective medium.

Homogenization. Aε = −div g(x/ε)∇, ε > 0 is small.

Typical problem: Aεuε + uε = F ∈ L2(Rd). Question: uε ∼? as ε → 0. Answer: uε ∼ u0, where u0 solves −div g0∇u0 + u0 = F.

Example. If d = 1, g0 = g :=

1

0 g(x)−1 dx

−1 . If d > 1, g g0 g :=

(0,1)d g(x) dx (Voigt-Reuss bracketing).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

SLIDE 22

Introduction Class of operators Effective operator Survey Problem Results Method

Classical results VS operator error estimates

Aε = −divg(x/ε)∇; A0 := −divg0∇ is the effective operator. Classical results [∼70]: uε − u0L2(Rd) C(F)ε. Recall, that uε = (Aε + I)−1F and u0 = (A0 + I)−1F. Operator error estimates [BSu’01]: uε − u0L2(Rd) CεFL2(Rd) ⇐ ⇒ (Aε + I)−1 − (A0 + I)−1L2(Rd)→L2(Rd) Cε.

M. Birman, T. Suslina, Threshold effects near the lower edge
f the spectrum for periodic differential operators
f mathematical physics, Oper. Theory Adv. Appl. 129 (2001).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 3 / 13

SLIDE 23

Introduction Class of operators Effective operator Survey Problem Results Method

Classical results VS operator error estimates

Aε = −divg(x/ε)∇; A0 := −divg0∇ is the effective operator. Classical results [∼70]: uε − u0L2(Rd) C(F)ε. Recall, that uε = (Aε + I)−1F and u0 = (A0 + I)−1F. Operator error estimates [BSu’01]: uε − u0L2(Rd) CεFL2(Rd) ⇐ ⇒ (Aε + I)−1 − (A0 + I)−1L2(Rd)→L2(Rd) Cε.

M. Birman, T. Suslina, Threshold effects near the lower edge
f the spectrum for periodic differential operators
f mathematical physics, Oper. Theory Adv. Appl. 129 (2001).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 3 / 13

SLIDE 24

Introduction Class of operators Effective operator Survey Problem Results Method

Classical results VS operator error estimates

Aε = −divg(x/ε)∇; A0 := −divg0∇ is the effective operator. Classical results [∼70]: uε − u0L2(Rd) C(F)ε. Recall, that uε = (Aε + I)−1F and u0 = (A0 + I)−1F. Operator error estimates [BSu’01]: uε − u0L2(Rd) CεFL2(Rd) ⇐ ⇒ (Aε + I)−1 − (A0 + I)−1L2(Rd)→L2(Rd) Cε.

M. Birman, T. Suslina, Threshold effects near the lower edge
f the spectrum for periodic differential operators
f mathematical physics, Oper. Theory Adv. Appl. 129 (2001).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 3 / 13

SLIDE 25

Introduction Class of operators Effective operator Survey Problem Results Method

Classical results VS operator error estimates

Aε = −divg(x/ε)∇; A0 := −divg0∇ is the effective operator. Classical results [∼70]: uε − u0L2(Rd) C(F)ε. Recall, that uε = (Aε + I)−1F and u0 = (A0 + I)−1F. Operator error estimates [BSu’01]: uε − u0L2(Rd) CεFL2(Rd) ⇐ ⇒ (Aε + I)−1 − (A0 + I)−1L2(Rd)→L2(Rd) Cε.

M. Birman, T. Suslina, Threshold effects near the lower edge
f the spectrum for periodic differential operators
f mathematical physics, Oper. Theory Adv. Appl. 129 (2001).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 3 / 13

SLIDE 26

Introduction Class of operators Effective operator Survey Problem Results Method

Classical results VS operator error estimates

Aε = −divg(x/ε)∇; A0 := −divg0∇ is the effective operator. Classical results [∼70]: uε − u0L2(Rd) C(F)ε. Recall, that uε = (Aε + I)−1F and u0 = (A0 + I)−1F. Operator error estimates [BSu’01]: uε − u0L2(Rd) CεFL2(Rd) ⇐ ⇒ (Aε + I)−1 − (A0 + I)−1L2(Rd)→L2(Rd) Cε.

M. Birman, T. Suslina, Threshold effects near the lower edge
f the spectrum for periodic differential operators
f mathematical physics, Oper. Theory Adv. Appl. 129 (2001).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 3 / 13

SLIDE 27

Introduction Class of operators Effective operator Survey Problem Results Method

Classical results VS operator error estimates

Aε = −divg(x/ε)∇; A0 := −divg0∇ is the effective operator. Classical results [∼70]: uε − u0L2(Rd) C(F)ε. Recall, that uε = (Aε + I)−1F and u0 = (A0 + I)−1F. Operator error estimates [BSu’01]: uε − u0L2(Rd) CεFL2(Rd) ⇐ ⇒ (Aε + I)−1 − (A0 + I)−1L2(Rd)→L2(Rd) Cε.

M. Birman, T. Suslina, Threshold effects near the lower edge
f the spectrum for periodic differential operators
f mathematical physics, Oper. Theory Adv. Appl. 129 (2001).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 3 / 13

SLIDE 28

Introduction Class of operators Effective operator Survey Problem Results Method

Spectral approach

Homogenization as a spectral threshold effect. Unitary scaling in L2(Rd): Aε ∼ ε−2A. Floquet-Bloch theory: A ∼

(−π;π)d ⊕A(k) dk.

Analytic perturbation theory w. r. t. |k|.

V. V. Zhikov, Spectral approach to asymptotic diffusion problems,

Differential Equations 25:1 (1989), 33–39.

C. Conca, M. Vanninathan, Homogenization of periodic structures via

Bloch decomposition, SIAM J. Appl. Math. 57:6 (1997), 1639–1659.

V. V. Zhikov, S. E. Pastukhova, Operator estimates in homogenization

theory, Russian Math. Surveys 71:3 (2016), 417–511.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 4 / 13

SLIDE 29

Introduction Class of operators Effective operator Survey Problem Results Method

Method. Analytic perturbation theory w. r. t. |k|

A(k) = (D + k)∗g(x)(D + k) in L2((−π; π)d) with periodic b. c. Unperturbed operator: A(0) = D∗g(x)D = −div g(x)∇, Ker A(0) = C. λ1(0) spec A(0) λ2(0) λ1(k) ∈ [0, δ] δ 3δ spec A(k) for small |k| λ2(k), · · · ∈ [3δ, ∞) Goes to the error term. Only λ1(k) and the corresponding eigenfunction φ1(k) are important for homogenization procedure!

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 5 / 13

SLIDE 30

Introduction Class of operators Effective operator Survey Problem Results Method

Method. Analytic perturbation theory w. r. t. |k|

A(k) = (D + k)∗g(x)(D + k) in L2((−π; π)d) with periodic b. c. Unperturbed operator: A(0) = D∗g(x)D = −div g(x)∇, Ker A(0) = C. λ1(0) spec A(0) λ2(0) λ1(k) ∈ [0, δ] δ 3δ spec A(k) for small |k| λ2(k), · · · ∈ [3δ, ∞) Goes to the error term. Only λ1(k) and the corresponding eigenfunction φ1(k) are important for homogenization procedure!

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 5 / 13

SLIDE 31

Introduction Class of operators Effective operator Survey Problem Results Method

Method. Analytic perturbation theory w. r. t. |k|

A(k) = (D + k)∗g(x)(D + k) in L2((−π; π)d) with periodic b. c. Unperturbed operator: A(0) = D∗g(x)D = −div g(x)∇, Ker A(0) = C. λ1(0) spec A(0) λ2(0) λ1(k) ∈ [0, δ] δ 3δ spec A(k) for small |k| λ2(k), · · · ∈ [3δ, ∞) Goes to the error term. Only λ1(k) and the corresponding eigenfunction φ1(k) are important for homogenization procedure!

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 5 / 13

SLIDE 32

Introduction Class of operators Effective operator Survey Problem Results Method

The class of operators

Notation: Γ ⊂ Rd is a lattice, Ω is the cell of Γ. Example: Γ = Zd, Ω = (0, 1)d. Aε = b(D)∗g(x/ε)b(D) g is a Γ-periodic (m × m)-matrix-valued function with complex entries: g(x) > 0; g, g−1 ∈ L∞(Rd); b(D) = d

j=1 bjDj. Here bj are constant (m × n)-matrices with

complex entries. Suppose that m n and rank b(ξ) = n, 0 = ξ ∈ Rd. Then Aε is a strongly elliptic operator acting in L2(Rd; Cn). Example: The acoustics operator Aε = −divg(x/ε)∇.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 6 / 13

SLIDE 33

Introduction Class of operators Effective operator Survey Problem Results Method

The class of operators

Notation: Γ ⊂ Rd is a lattice, Ω is the cell of Γ. Example: Γ = Zd, Ω = (0, 1)d. Aε = b(D)∗g(x/ε)b(D) g is a Γ-periodic (m × m)-matrix-valued function with complex entries: g(x) > 0; g, g−1 ∈ L∞(Rd); b(D) = d

j=1 bjDj. Here bj are constant (m × n)-matrices with

complex entries. Suppose that m n and rank b(ξ) = n, 0 = ξ ∈ Rd. Then Aε is a strongly elliptic operator acting in L2(Rd; Cn). Example: The acoustics operator Aε = −divg(x/ε)∇.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 6 / 13

SLIDE 34

Introduction Class of operators Effective operator Survey Problem Results Method

The class of operators

Notation: Γ ⊂ Rd is a lattice, Ω is the cell of Γ. Example: Γ = Zd, Ω = (0, 1)d. Aε = b(D)∗g(x/ε)b(D) g is a Γ-periodic (m × m)-matrix-valued function with complex entries: g(x) > 0; g, g−1 ∈ L∞(Rd); b(D) = d

j=1 bjDj. Here bj are constant (m × n)-matrices with

complex entries. Suppose that m n and rank b(ξ) = n, 0 = ξ ∈ Rd. Then Aε is a strongly elliptic operator acting in L2(Rd; Cn). Example: The acoustics operator Aε = −divg(x/ε)∇.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 6 / 13

SLIDE 35

Introduction Class of operators Effective operator Survey Problem Results Method

The class of operators

Notation: Γ ⊂ Rd is a lattice, Ω is the cell of Γ. Example: Γ = Zd, Ω = (0, 1)d. Aε = b(D)∗g(x/ε)b(D) g is a Γ-periodic (m × m)-matrix-valued function with complex entries: g(x) > 0; g, g−1 ∈ L∞(Rd); b(D) = d

j=1 bjDj. Here bj are constant (m × n)-matrices with

complex entries. Suppose that m n and rank b(ξ) = n, 0 = ξ ∈ Rd. Then Aε is a strongly elliptic operator acting in L2(Rd; Cn). Example: The acoustics operator Aε = −divg(x/ε)∇.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 6 / 13

SLIDE 36

Introduction Class of operators Effective operator Survey Problem Results Method

The class of operators

Notation: Γ ⊂ Rd is a lattice, Ω is the cell of Γ. Example: Γ = Zd, Ω = (0, 1)d. Aε = b(D)∗g(x/ε)b(D) g is a Γ-periodic (m × m)-matrix-valued function with complex entries: g(x) > 0; g, g−1 ∈ L∞(Rd); b(D) = d

j=1 bjDj. Here bj are constant (m × n)-matrices with

complex entries. Suppose that m n and rank b(ξ) = n, 0 = ξ ∈ Rd. Then Aε is a strongly elliptic operator acting in L2(Rd; Cn). Example: The acoustics operator Aε = −divg(x/ε)∇.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 6 / 13

SLIDE 37

Introduction Class of operators Effective operator Survey Problem Results Method

The class of operators

Notation: Γ ⊂ Rd is a lattice, Ω is the cell of Γ. Example: Γ = Zd, Ω = (0, 1)d. Aε = b(D)∗g(x/ε)b(D) g is a Γ-periodic (m × m)-matrix-valued function with complex entries: g(x) > 0; g, g−1 ∈ L∞(Rd); b(D) = d

j=1 bjDj. Here bj are constant (m × n)-matrices with

complex entries. Suppose that m n and rank b(ξ) = n, 0 = ξ ∈ Rd. Then Aε is a strongly elliptic operator acting in L2(Rd; Cn). Example: The acoustics operator Aε = −divg(x/ε)∇.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 6 / 13

SLIDE 38

Introduction Class of operators Effective operator Survey Problem Results Method

The class of operators

Notation: Γ ⊂ Rd is a lattice, Ω is the cell of Γ. Example: Γ = Zd, Ω = (0, 1)d. Aε = b(D)∗g(x/ε)b(D) g is a Γ-periodic (m × m)-matrix-valued function with complex entries: g(x) > 0; g, g−1 ∈ L∞(Rd); b(D) = d

j=1 bjDj. Here bj are constant (m × n)-matrices with

complex entries. Suppose that m n and rank b(ξ) = n, 0 = ξ ∈ Rd. Then Aε is a strongly elliptic operator acting in L2(Rd; Cn). Example: The acoustics operator Aε = −divg(x/ε)∇.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 6 / 13

SLIDE 39

Introduction Class of operators Effective operator Survey Problem Results Method

The class of operators

Notation: Γ ⊂ Rd is a lattice, Ω is the cell of Γ. Example: Γ = Zd, Ω = (0, 1)d. Aε = b(D)∗g(x/ε)b(D) g is a Γ-periodic (m × m)-matrix-valued function with complex entries: g(x) > 0; g, g−1 ∈ L∞(Rd); b(D) = d

j=1 bjDj. Here bj are constant (m × n)-matrices with

complex entries. Suppose that m n and rank b(ξ) = n, 0 = ξ ∈ Rd. Then Aε is a strongly elliptic operator acting in L2(Rd; Cn). Example: The acoustics operator Aε = −divg(x/ε)∇.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 6 / 13

SLIDE 40

Introduction Class of operators Effective operator Survey Problem Results Method

The effective operator

A0 = b(D)∗g0b(D) Here g0 is the constant positive definite effective matrix.

Definition of the effective matrix

Let Λ be a Γ-periodic (n × m)-matrix-valued solution of the cell problem: b(D)∗g(x)(b(D)Λ(x) + 1m) = 0,

Ω

Λ(x) dx = 0. Then g0 = |Ω|−1

Ω

g(x)(b(D)Λ(x) + 1m) dx.

Remark. Λ ∈ H1(Ω); in general, Λ ∈ L∞.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 7 / 13

SLIDE 41

Introduction Class of operators Effective operator Survey Problem Results Method

The effective operator

A0 = b(D)∗g0b(D) Here g0 is the constant positive definite effective matrix.

Definition of the effective matrix

Let Λ be a Γ-periodic (n × m)-matrix-valued solution of the cell problem: b(D)∗g(x)(b(D)Λ(x) + 1m) = 0,

Ω

Λ(x) dx = 0. Then g0 = |Ω|−1

Ω

g(x)(b(D)Λ(x) + 1m) dx.

Remark. Λ ∈ H1(Ω); in general, Λ ∈ L∞.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 7 / 13

SLIDE 42

Introduction Class of operators Effective operator Survey Problem Results Method

The effective operator

A0 = b(D)∗g0b(D) Here g0 is the constant positive definite effective matrix.

Definition of the effective matrix

Let Λ be a Γ-periodic (n × m)-matrix-valued solution of the cell problem: b(D)∗g(x)(b(D)Λ(x) + 1m) = 0,

Ω

Λ(x) dx = 0. Then g0 = |Ω|−1

Ω

g(x)(b(D)Λ(x) + 1m) dx.

Remark. Λ ∈ H1(Ω); in general, Λ ∈ L∞.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 7 / 13

SLIDE 43

Introduction Class of operators Effective operator Survey Problem Results Method

The effective operator

A0 = b(D)∗g0b(D) Here g0 is the constant positive definite effective matrix.

Definition of the effective matrix

Let Λ be a Γ-periodic (n × m)-matrix-valued solution of the cell problem: b(D)∗g(x)(b(D)Λ(x) + 1m) = 0,

Ω

Λ(x) dx = 0. Then g0 = |Ω|−1

Ω

g(x)(b(D)Λ(x) + 1m) dx.

Remark. Λ ∈ H1(Ω); in general, Λ ∈ L∞.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 7 / 13

SLIDE 44

Introduction Class of operators Effective operator Survey Problem Results Method

The effective operator

A0 = b(D)∗g0b(D) Here g0 is the constant positive definite effective matrix.

Definition of the effective matrix

Let Λ be a Γ-periodic (n × m)-matrix-valued solution of the cell problem: b(D)∗g(x)(b(D)Λ(x) + 1m) = 0,

Ω

Λ(x) dx = 0. Then g0 = |Ω|−1

Ω

g(x)(b(D)Λ(x) + 1m) dx.

Remark. Λ ∈ H1(Ω); in general, Λ ∈ L∞.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 7 / 13

SLIDE 45

Introduction Class of operators Effective operator Survey Problem Results Method

Survey

Principal term of Approximation in approximation the energy norm Birman, Suslina (’01) Zhikov, Pastukhova (’05) (Aε + I)−1 Zhikov (’05) Birman, Suslina (’06) Suslina (’04) Zhikov, Pastukhova (’05) exp(−tAε) Zhikov, Pastukhova (’05) Suslina (’10) cos(tA1/2

ε

) Birman, Suslina (’08) — 1 A−1/2

ε

sin(tA1/2

ε

) Dorodnyi, Suslina (’16,’17) Meshkova (’17) Our main goal is to approximate A−1/2

ε

sin(tA1/2

ε

) with the corrector.

1

S. Brahim-Otsmane, G. A. Francfort, F. Murat, J. Math. Pures Appl. 71

(1992), 197–231.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 8 / 13

SLIDE 46

Introduction Class of operators Effective operator Survey Problem Results Method

Survey

Principal term of Approximation in approximation the energy norm Birman, Suslina (’01) Zhikov, Pastukhova (’05) (Aε + I)−1 Zhikov (’05) Birman, Suslina (’06) Suslina (’04) Zhikov, Pastukhova (’05) exp(−tAε) Zhikov, Pastukhova (’05) Suslina (’10) cos(tA1/2

ε

) Birman, Suslina (’08) — 1 A−1/2

ε

sin(tA1/2

ε

) Dorodnyi, Suslina (’16,’17) Meshkova (’17) Our main goal is to approximate A−1/2

ε

sin(tA1/2

ε

) with the corrector.

1

S. Brahim-Otsmane, G. A. Francfort, F. Murat, J. Math. Pures Appl. 71

(1992), 197–231.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 8 / 13

SLIDE 47

Introduction Class of operators Effective operator Survey Problem Results Method

Survey

Principal term of Approximation in approximation the energy norm Birman, Suslina (’01) Zhikov, Pastukhova (’05) (Aε + I)−1 Zhikov (’05) Birman, Suslina (’06) Suslina (’04) Zhikov, Pastukhova (’05) exp(−tAε) Zhikov, Pastukhova (’05) Suslina (’10) cos(tA1/2

ε

) Birman, Suslina (’08) — 1 A−1/2

ε

sin(tA1/2

ε

) Dorodnyi, Suslina (’16,’17) Meshkova (’17) Our main goal is to approximate A−1/2

ε

sin(tA1/2

ε

) with the corrector.

1

S. Brahim-Otsmane, G. A. Francfort, F. Murat, J. Math. Pures Appl. 71

(1992), 197–231.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 8 / 13

SLIDE 48

Introduction Class of operators Effective operator Survey Problem Results Method

Survey

Principal term of Approximation in approximation the energy norm Birman, Suslina (’01) Zhikov, Pastukhova (’05) (Aε + I)−1 Zhikov (’05) Birman, Suslina (’06) Suslina (’04) Zhikov, Pastukhova (’05) exp(−tAε) Zhikov, Pastukhova (’05) Suslina (’10) cos(tA1/2

ε

) Birman, Suslina (’08) — 1 A−1/2

ε

sin(tA1/2

ε

) Dorodnyi, Suslina (’16,’17) Meshkova (’17) Our main goal is to approximate A−1/2

ε

sin(tA1/2

ε

) with the corrector.

1

S. Brahim-Otsmane, G. A. Francfort, F. Murat, J. Math. Pures Appl. 71

(1992), 197–231.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 8 / 13

SLIDE 49

Introduction Class of operators Effective operator Survey Problem Results Method

Survey

Principal term of Approximation in approximation the energy norm Birman, Suslina (’01) Zhikov, Pastukhova (’05) (Aε + I)−1 Zhikov (’05) Birman, Suslina (’06) Suslina (’04) Zhikov, Pastukhova (’05) exp(−tAε) Zhikov, Pastukhova (’05) Suslina (’10) cos(tA1/2

ε

) Birman, Suslina (’08) — 1 A−1/2

ε

sin(tA1/2

ε

) Dorodnyi, Suslina (’16,’17) Meshkova (’17) Our main goal is to approximate A−1/2

ε

sin(tA1/2

ε

) with the corrector.

1

S. Brahim-Otsmane, G. A. Francfort, F. Murat, J. Math. Pures Appl. 71

(1992), 197–231.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 8 / 13

SLIDE 50

Introduction Class of operators Effective operator Survey Problem Results Method

Survey

Principal term of Approximation in approximation the energy norm Birman, Suslina (’01) Zhikov, Pastukhova (’05) (Aε + I)−1 Zhikov (’05) Birman, Suslina (’06) Suslina (’04) Zhikov, Pastukhova (’05) exp(−tAε) Zhikov, Pastukhova (’05) Suslina (’10) cos(tA1/2

ε

) Birman, Suslina (’08) — 1 A−1/2

ε

sin(tA1/2

ε

) Dorodnyi, Suslina (’16,’17) Meshkova (’17) Our main goal is to approximate A−1/2

ε

sin(tA1/2

ε

) with the corrector.

1

S. Brahim-Otsmane, G. A. Francfort, F. Murat, J. Math. Pures Appl. 71

(1992), 197–231.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 8 / 13

SLIDE 51

Introduction Class of operators Effective operator Survey Problem Results Method

Survey

Principal term of Approximation in approximation the energy norm Birman, Suslina (’01) Zhikov, Pastukhova (’05) (Aε + I)−1 Zhikov (’05) Birman, Suslina (’06) Suslina (’04) Zhikov, Pastukhova (’05) exp(−tAε) Zhikov, Pastukhova (’05) Suslina (’10) cos(tA1/2

ε

) Birman, Suslina (’08) — 1 A−1/2

ε

sin(tA1/2

ε

) Dorodnyi, Suslina (’16,’17) Meshkova (’17) Our main goal is to approximate A−1/2

ε

sin(tA1/2

ε

) with the corrector.

1

S. Brahim-Otsmane, G. A. Francfort, F. Murat, J. Math. Pures Appl. 71

(1992), 197–231.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 8 / 13

SLIDE 52

Introduction Class of operators Effective operator Survey Problem Results Method

Survey

Principal term of Approximation in approximation the energy norm Birman, Suslina (’01) Zhikov, Pastukhova (’05) (Aε + I)−1 Zhikov (’05) Birman, Suslina (’06) Suslina (’04) Zhikov, Pastukhova (’05) exp(−tAε) Zhikov, Pastukhova (’05) Suslina (’10) cos(tA1/2

ε

) Birman, Suslina (’08) — 1 A−1/2

ε

sin(tA1/2

ε

) Dorodnyi, Suslina (’16,’17) Meshkova (’17) Our main goal is to approximate A−1/2

ε

sin(tA1/2

ε

) with the corrector.

1

S. Brahim-Otsmane, G. A. Francfort, F. Murat, J. Math. Pures Appl. 71

(1992), 197–231.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 8 / 13

SLIDE 53

Introduction Class of operators Effective operator Survey Problem Results Method

Survey

Principal term of Approximation in approximation the energy norm Birman, Suslina (’01) Zhikov, Pastukhova (’05) (Aε + I)−1 Zhikov (’05) Birman, Suslina (’06) Suslina (’04) Zhikov, Pastukhova (’05) exp(−tAε) Zhikov, Pastukhova (’05) Suslina (’10) cos(tA1/2

ε

) Birman, Suslina (’08) — 1 A−1/2

ε

sin(tA1/2

ε

) Dorodnyi, Suslina (’16,’17) Meshkova (’17) Our main goal is to approximate A−1/2

ε

sin(tA1/2

ε

) with the corrector.

1

S. Brahim-Otsmane, G. A. Francfort, F. Murat, J. Math. Pures Appl. 71

(1992), 197–231.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 8 / 13

SLIDE 54

Introduction Class of operators Effective operator Survey Problem Results Method

Survey

Principal term of Approximation in approximation the energy norm Birman, Suslina (’01) Zhikov, Pastukhova (’05) (Aε + I)−1 Zhikov (’05) Birman, Suslina (’06) Suslina (’04) Zhikov, Pastukhova (’05) exp(−tAε) Zhikov, Pastukhova (’05) Suslina (’10) cos(tA1/2

ε

) Birman, Suslina (’08) — 1 A−1/2

ε

sin(tA1/2

ε

) Dorodnyi, Suslina (’16,’17) Meshkova (’17) Our main goal is to approximate A−1/2

ε

sin(tA1/2

ε

) with the corrector.

1

S. Brahim-Otsmane, G. A. Francfort, F. Murat, J. Math. Pures Appl. 71

(1992), 197–231.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 8 / 13

SLIDE 55

Introduction Class of operators Effective operator Survey Problem Results Method

Homogenization problem for hyperbolic systems

∂2

t uε(x, t) = −b(D)∗g(x/ε)b(D) Aε

uε(x, t),

uε(x, 0) = ϕ(x), ∂tuε(x, 0) = ψ(x). Here ϕ ∈ H1(Rd; Cn) ψ ∈ L2(Rd; Cn). Then uε(·, t) = cos(tA1/2

ε

)ϕ + A−1/2

ε

sin(tA1/2

ε

)ψ.

Theorem 1 [Birman, Suslina, A&A 20:6 (2008)]

cos(tA1/2

ε

) − cos(t(A0)1/2)H2(Rd)→L2(Rd) Cε(1 + |t|), (1) ε > 0, t ∈ R. By the identity A−1/2

ε

sin(tA1/2

ε

) = t

0 cos(

tA1/2

ε

) d t, A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2)H2(Rd)→L2(Rd) Cε(1 + |t|)2.

Theorem 2 [Dorodnyi, Suslina, FAA 50:4 (2016)]

Estimate (1) is sharp with respect to the type of the operator norm.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 9 / 13

SLIDE 56

Introduction Class of operators Effective operator Survey Problem Results Method

Homogenization problem for hyperbolic systems

∂2

t uε(x, t) = −b(D)∗g(x/ε)b(D) Aε

uε(x, t),

uε(x, 0) = ϕ(x), ∂tuε(x, 0) = ψ(x). Here ϕ ∈ H1(Rd; Cn) ψ ∈ L2(Rd; Cn). Then uε(·, t) = cos(tA1/2

ε

)ϕ + A−1/2

ε

sin(tA1/2

ε

)ψ.

Theorem 1 [Birman, Suslina, A&A 20:6 (2008)]

cos(tA1/2

ε

) − cos(t(A0)1/2)H2(Rd)→L2(Rd) Cε(1 + |t|), (1) ε > 0, t ∈ R. By the identity A−1/2

ε

sin(tA1/2

ε

) = t

0 cos(

tA1/2

ε

) d t, A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2)H2(Rd)→L2(Rd) Cε(1 + |t|)2.

Theorem 2 [Dorodnyi, Suslina, FAA 50:4 (2016)]

Estimate (1) is sharp with respect to the type of the operator norm.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 9 / 13

SLIDE 57

Introduction Class of operators Effective operator Survey Problem Results Method

Homogenization problem for hyperbolic systems

∂2

t uε(x, t) = −b(D)∗g(x/ε)b(D) Aε

uε(x, t),

uε(x, 0) = ϕ(x), ∂tuε(x, 0) = ψ(x). Here ϕ ∈ H1(Rd; Cn) ψ ∈ L2(Rd; Cn). Then uε(·, t) = cos(tA1/2

ε

)ϕ + A−1/2

ε

sin(tA1/2

ε

)ψ.

Theorem 1 [Birman, Suslina, A&A 20:6 (2008)]

cos(tA1/2

ε

) − cos(t(A0)1/2)H2(Rd)→L2(Rd) Cε(1 + |t|), (1) ε > 0, t ∈ R. By the identity A−1/2

ε

sin(tA1/2

ε

) = t

0 cos(

tA1/2

ε

) d t, A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2)H2(Rd)→L2(Rd) Cε(1 + |t|)2.

Theorem 2 [Dorodnyi, Suslina, FAA 50:4 (2016)]

Estimate (1) is sharp with respect to the type of the operator norm.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 9 / 13

SLIDE 58

Introduction Class of operators Effective operator Survey Problem Results Method

Homogenization problem for hyperbolic systems

∂2

t uε(x, t) = −b(D)∗g(x/ε)b(D) Aε

uε(x, t),

uε(x, 0) = ϕ(x), ∂tuε(x, 0) = ψ(x). Here ϕ ∈ H1(Rd; Cn) ψ ∈ L2(Rd; Cn). Then uε(·, t) = cos(tA1/2

ε

)ϕ + A−1/2

ε

sin(tA1/2

ε

)ψ.

Theorem 1 [Birman, Suslina, A&A 20:6 (2008)]

cos(tA1/2

ε

) − cos(t(A0)1/2)H2(Rd)→L2(Rd) Cε(1 + |t|), (1) ε > 0, t ∈ R. By the identity A−1/2

ε

sin(tA1/2

ε

) = t

0 cos(

tA1/2

ε

) d t, A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2)H2(Rd)→L2(Rd) Cε(1 + |t|)2.

Theorem 2 [Dorodnyi, Suslina, FAA 50:4 (2016)]

Estimate (1) is sharp with respect to the type of the operator norm.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 9 / 13

SLIDE 59

Introduction Class of operators Effective operator Survey Problem Results Method

Homogenization problem for hyperbolic systems

∂2

t uε(x, t) = −b(D)∗g(x/ε)b(D) Aε

uε(x, t),

uε(x, 0) = ϕ(x), ∂tuε(x, 0) = ψ(x). Here ϕ ∈ H1(Rd; Cn) ψ ∈ L2(Rd; Cn). Then uε(·, t) = cos(tA1/2

ε

)ϕ + A−1/2

ε

sin(tA1/2

ε

)ψ.

Theorem 1 [Birman, Suslina, A&A 20:6 (2008)]

cos(tA1/2

ε

) − cos(t(A0)1/2)H2(Rd)→L2(Rd) Cε(1 + |t|), (1) ε > 0, t ∈ R. By the identity A−1/2

ε

sin(tA1/2

ε

) = t

0 cos(

tA1/2

ε

) d t, A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2)H2(Rd)→L2(Rd) Cε(1 + |t|)2.

Theorem 2 [Dorodnyi, Suslina, FAA 50:4 (2016)]

Estimate (1) is sharp with respect to the type of the operator norm.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 9 / 13

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Introduction Class of operators Effective operator Survey Problem Results Method

Homogenization problem for hyperbolic systems

∂2

t uε(x, t) = −b(D)∗g(x/ε)b(D) Aε

uε(x, t),

uε(x, 0) = ϕ(x), ∂tuε(x, 0) = ψ(x). Here ϕ ∈ H1(Rd; Cn) ψ ∈ L2(Rd; Cn). Then uε(·, t) = cos(tA1/2

ε

)ϕ + A−1/2

ε

sin(tA1/2

ε

)ψ.

Theorem 1 [Birman, Suslina, A&A 20:6 (2008)]

cos(tA1/2

ε

) − cos(t(A0)1/2)H2(Rd)→L2(Rd) Cε(1 + |t|), (1) ε > 0, t ∈ R. By the identity A−1/2

ε

sin(tA1/2

ε

) = t

0 cos(

tA1/2

ε

) d t, A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2)H2(Rd)→L2(Rd) Cε(1 + |t|)2.

Theorem 2 [Dorodnyi, Suslina, FAA 50:4 (2016)]

Estimate (1) is sharp with respect to the type of the operator norm.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 9 / 13

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Introduction Class of operators Effective operator Survey Problem Results Method

Homogenization problem for hyperbolic systems

∂2

t uε(x, t) = −b(D)∗g(x/ε)b(D) Aε

uε(x, t),

uε(x, 0) = ϕ(x), ∂tuε(x, 0) = ψ(x). Here ϕ ∈ H1(Rd; Cn) ψ ∈ L2(Rd; Cn). Then uε(·, t) = cos(tA1/2

ε

)ϕ + A−1/2

ε

sin(tA1/2

ε

)ψ.

Theorem 1 [Birman, Suslina, A&A 20:6 (2008)]

cos(tA1/2

ε

) − cos(t(A0)1/2)H2(Rd)→L2(Rd) Cε(1 + |t|), (1) ε > 0, t ∈ R. By the identity A−1/2

ε

sin(tA1/2

ε

) = t

0 cos(

tA1/2

ε

) d t, A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2)H2(Rd)→L2(Rd) Cε(1 + |t|)2.

Theorem 2 [Dorodnyi, Suslina, FAA 50:4 (2016)]

Estimate (1) is sharp with respect to the type of the operator norm.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 9 / 13

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Introduction Class of operators Effective operator Survey Problem Results Method

Homogenization problem for hyperbolic systems

∂2

t uε(x, t) = −b(D)∗g(x/ε)b(D) Aε

uε(x, t),

uε(x, 0) = ϕ(x), ∂tuε(x, 0) = ψ(x). Here ϕ ∈ H1(Rd; Cn) ψ ∈ L2(Rd; Cn). Then uε(·, t) = cos(tA1/2

ε

)ϕ + A−1/2

ε

sin(tA1/2

ε

)ψ.

Theorem 1 [Birman, Suslina, A&A 20:6 (2008)]

cos(tA1/2

ε

) − cos(t(A0)1/2)H2(Rd)→L2(Rd) Cε(1 + |t|), (1) ε > 0, t ∈ R. By the identity A−1/2

ε

sin(tA1/2

ε

) = t

0 cos(

tA1/2

ε

) d t, A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2)H2(Rd)→L2(Rd) Cε(1 + |t|)2.

Theorem 2 [Dorodnyi, Suslina, FAA 50:4 (2016)]

Estimate (1) is sharp with respect to the type of the operator norm.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 9 / 13

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Introduction Class of operators Effective operator Survey Problem Results Method

Main results

Steklov smoothing operator: (Sεu)(x) = |Ω|−1

Ω u(x − εz) dz.

Theorem 3 [Yu. Meshkova, arXiv:1705.02531 (2017)]

A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2)H1(Rd)→L2(Rd) C1ε(1 + |t|), ε > 0, t ∈ R. (2) Denote K(t; ε) = Λεb(D)Sε(A0)−1/2 sin(t(A0)1/2). (For d 4, we can remove Sε from the corrector.) Then A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2) − εK(t; ε)H2(Rd)→H1(Rd) C2ε(1 + |t|). Sharpness of estimate (2) with respect to the type of the norm and refinement under the additional assumptions are proven in

M. Dorodnyi, T. Suslina, arXiv:1708.00859 (2017).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 10 / 13

SLIDE 64

Introduction Class of operators Effective operator Survey Problem Results Method

Main results

Steklov smoothing operator: (Sεu)(x) = |Ω|−1

Ω u(x − εz) dz.

Theorem 3 [Yu. Meshkova, arXiv:1705.02531 (2017)]

A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2)H1(Rd)→L2(Rd) C1ε(1 + |t|), ε > 0, t ∈ R. (2) Denote K(t; ε) = Λεb(D)Sε(A0)−1/2 sin(t(A0)1/2). (For d 4, we can remove Sε from the corrector.) Then A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2) − εK(t; ε)H2(Rd)→H1(Rd) C2ε(1 + |t|). Sharpness of estimate (2) with respect to the type of the norm and refinement under the additional assumptions are proven in

M. Dorodnyi, T. Suslina, arXiv:1708.00859 (2017).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 10 / 13

SLIDE 65

Introduction Class of operators Effective operator Survey Problem Results Method

Main results

Steklov smoothing operator: (Sεu)(x) = |Ω|−1

Ω u(x − εz) dz.

Theorem 3 [Yu. Meshkova, arXiv:1705.02531 (2017)]

A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2)H1(Rd)→L2(Rd) C1ε(1 + |t|), ε > 0, t ∈ R. (2) Denote K(t; ε) = Λεb(D)Sε(A0)−1/2 sin(t(A0)1/2). (For d 4, we can remove Sε from the corrector.) Then A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2) − εK(t; ε)H2(Rd)→H1(Rd) C2ε(1 + |t|). Sharpness of estimate (2) with respect to the type of the norm and refinement under the additional assumptions are proven in

M. Dorodnyi, T. Suslina, arXiv:1708.00859 (2017).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 10 / 13

SLIDE 66

Introduction Class of operators Effective operator Survey Problem Results Method

Main results

Steklov smoothing operator: (Sεu)(x) = |Ω|−1

Ω u(x − εz) dz.

Theorem 3 [Yu. Meshkova, arXiv:1705.02531 (2017)]

A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2)H1(Rd)→L2(Rd) C1ε(1 + |t|), ε > 0, t ∈ R. (2) Denote K(t; ε) = Λεb(D)Sε(A0)−1/2 sin(t(A0)1/2). (For d 4, we can remove Sε from the corrector.) Then A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2) − εK(t; ε)H2(Rd)→H1(Rd) C2ε(1 + |t|). Sharpness of estimate (2) with respect to the type of the norm and refinement under the additional assumptions are proven in

M. Dorodnyi, T. Suslina, arXiv:1708.00859 (2017).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 10 / 13

SLIDE 67

Introduction Class of operators Effective operator Survey Problem Results Method

Main results

Steklov smoothing operator: (Sεu)(x) = |Ω|−1

Ω u(x − εz) dz.

Theorem 3 [Yu. Meshkova, arXiv:1705.02531 (2017)]

A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2)H1(Rd)→L2(Rd) C1ε(1 + |t|), ε > 0, t ∈ R. (2) Denote K(t; ε) = Λεb(D)Sε(A0)−1/2 sin(t(A0)1/2). (For d 4, we can remove Sε from the corrector.) Then A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2) − εK(t; ε)H2(Rd)→H1(Rd) C2ε(1 + |t|). Sharpness of estimate (2) with respect to the type of the norm and refinement under the additional assumptions are proven in

M. Dorodnyi, T. Suslina, arXiv:1708.00859 (2017).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 10 / 13

SLIDE 68

Introduction Class of operators Effective operator Survey Problem Results Method

Main results

Steklov smoothing operator: (Sεu)(x) = |Ω|−1

Ω u(x − εz) dz.

Theorem 3 [Yu. Meshkova, arXiv:1705.02531 (2017)]

A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2)H1(Rd)→L2(Rd) C1ε(1 + |t|), ε > 0, t ∈ R. (2) Denote K(t; ε) = Λεb(D)Sε(A0)−1/2 sin(t(A0)1/2). (For d 4, we can remove Sε from the corrector.) Then A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2) − εK(t; ε)H2(Rd)→H1(Rd) C2ε(1 + |t|). Sharpness of estimate (2) with respect to the type of the norm and refinement under the additional assumptions are proven in

M. Dorodnyi, T. Suslina, arXiv:1708.00859 (2017).

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 10 / 13

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Introduction Class of operators Effective operator Survey Problem Results Method

Method. Scaling transformation

The method is a further development of the Birman-Suslina approach. A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2)H1(Rd)→L2(Rd) =

A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2)

(−∆ + I)−1/2
L2(Rd)→L2(Rd)

Cε(1 + |t|). (2) Scaling transformation: (Tεu)(x) = εd/2u(εx). Obviously, (−∆ + I)−1/2 = εT ∗

ε (−∆ + ε2I)−1/2Tε

and Aε = ε−2T ∗

ε ATε, where A = b(D)∗g(x)b(D). So,

A−1/2

ε

sin(tA1/2

ε

) = εT ∗

ε A−1/2 sin(ε−1tA1/2)Tε.

Thus, (2) ⇐ ⇒

A−1/2 sin(ε−1tA1/2) − (A0)−1/2 sin(ε−1t(A0)1/2)
× ε(−∆ + ε2I)−1/2
L2(Rd)→L2(Rd) C(1 + |t|).

(3)

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 11 / 13

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Introduction Class of operators Effective operator Survey Problem Results Method

Method. Scaling transformation

The method is a further development of the Birman-Suslina approach. A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2)H1(Rd)→L2(Rd) =

A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2)

(−∆ + I)−1/2
L2(Rd)→L2(Rd)

Cε(1 + |t|). (2) Scaling transformation: (Tεu)(x) = εd/2u(εx). Obviously, (−∆ + I)−1/2 = εT ∗

ε (−∆ + ε2I)−1/2Tε

and Aε = ε−2T ∗

ε ATε, where A = b(D)∗g(x)b(D). So,

A−1/2

ε

sin(tA1/2

ε

) = εT ∗

ε A−1/2 sin(ε−1tA1/2)Tε.

Thus, (2) ⇐ ⇒

A−1/2 sin(ε−1tA1/2) − (A0)−1/2 sin(ε−1t(A0)1/2)
× ε(−∆ + ε2I)−1/2
L2(Rd)→L2(Rd) C(1 + |t|).

(3)

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 11 / 13

SLIDE 71

Introduction Class of operators Effective operator Survey Problem Results Method

Method. Scaling transformation

The method is a further development of the Birman-Suslina approach. A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2)H1(Rd)→L2(Rd) =

A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2)

(−∆ + I)−1/2
L2(Rd)→L2(Rd)

Cε(1 + |t|). (2) Scaling transformation: (Tεu)(x) = εd/2u(εx). Obviously, (−∆ + I)−1/2 = εT ∗

ε (−∆ + ε2I)−1/2Tε

and Aε = ε−2T ∗

ε ATε, where A = b(D)∗g(x)b(D). So,

A−1/2

ε

sin(tA1/2

ε

) = εT ∗

ε A−1/2 sin(ε−1tA1/2)Tε.

Thus, (2) ⇐ ⇒

A−1/2 sin(ε−1tA1/2) − (A0)−1/2 sin(ε−1t(A0)1/2)
× ε(−∆ + ε2I)−1/2
L2(Rd)→L2(Rd) C(1 + |t|).

(3)

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 11 / 13

SLIDE 72

Introduction Class of operators Effective operator Survey Problem Results Method

Method. Scaling transformation

The method is a further development of the Birman-Suslina approach. A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2)H1(Rd)→L2(Rd) =

A−1/2

ε

sin(tA1/2

ε

) − (A0)−1/2 sin(t(A0)1/2)

(−∆ + I)−1/2
L2(Rd)→L2(Rd)

Cε(1 + |t|). (2) Scaling transformation: (Tεu)(x) = εd/2u(εx). Obviously, (−∆ + I)−1/2 = εT ∗

ε (−∆ + ε2I)−1/2Tε

and Aε = ε−2T ∗

ε ATε, where A = b(D)∗g(x)b(D). So,

A−1/2

ε

sin(tA1/2

ε

) = εT ∗

ε A−1/2 sin(ε−1tA1/2)Tε.

Thus, (2) ⇐ ⇒

A−1/2 sin(ε−1tA1/2) − (A0)−1/2 sin(ε−1t(A0)1/2)
× ε(−∆ + ε2I)−1/2
L2(Rd)→L2(Rd) C(1 + |t|).

(3)

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 11 / 13

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Introduction Class of operators Effective operator Survey Problem Results Method

Method. The Floquet-Bloch theory

Recall that (2) ⇐ ⇒

A−1/2 sin(ε−1tA1/2) − (A0)−1/2 sin(ε−1t(A0)1/2)
× ε(−∆ + ε2I)−1/2
L2(Rd)→L2(Rd) C(1 + |t|).

(3) The Floquet-Bloch theory: A ∼

Ω ⊕A(k) dk.
Ω is the Brillouin zone of the dual lattice;

k ∈ Ω is quasimomentum. In L2(Ω; Cn), A(k) = b(D + k)∗g(x)b(D + k) with the periodic b. c. (3) ⇐ ⇒

A(k)−1/2 sin(ε−1tA(k)1/2) − A0(k)−1/2 sin(ε−1tA0(k)1/2)
× ε((D + k)2 + ε2I)−1/2
L2(Ω)→L2(Ω) C(1 + |t|),
a. e. k ∈

Ω. (4) But A(k) c∗|k|2I and it sufficies to prove (4) for |k| κ0.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 12 / 13

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Introduction Class of operators Effective operator Survey Problem Results Method

Method. The Floquet-Bloch theory

Recall that (2) ⇐ ⇒

A−1/2 sin(ε−1tA1/2) − (A0)−1/2 sin(ε−1t(A0)1/2)
× ε(−∆ + ε2I)−1/2
L2(Rd)→L2(Rd) C(1 + |t|).

(3) The Floquet-Bloch theory: A ∼

Ω ⊕A(k) dk.
Ω is the Brillouin zone of the dual lattice;

k ∈ Ω is quasimomentum. In L2(Ω; Cn), A(k) = b(D + k)∗g(x)b(D + k) with the periodic b. c. (3) ⇐ ⇒

A(k)−1/2 sin(ε−1tA(k)1/2) − A0(k)−1/2 sin(ε−1tA0(k)1/2)
× ε((D + k)2 + ε2I)−1/2
L2(Ω)→L2(Ω) C(1 + |t|),
a. e. k ∈

Ω. (4) But A(k) c∗|k|2I and it sufficies to prove (4) for |k| κ0.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 12 / 13

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Introduction Class of operators Effective operator Survey Problem Results Method

Method. The Floquet-Bloch theory

Recall that (2) ⇐ ⇒

A−1/2 sin(ε−1tA1/2) − (A0)−1/2 sin(ε−1t(A0)1/2)
× ε(−∆ + ε2I)−1/2
L2(Rd)→L2(Rd) C(1 + |t|).

(3) The Floquet-Bloch theory: A ∼

Ω ⊕A(k) dk.
Ω is the Brillouin zone of the dual lattice;

k ∈ Ω is quasimomentum. In L2(Ω; Cn), A(k) = b(D + k)∗g(x)b(D + k) with the periodic b. c. (3) ⇐ ⇒

A(k)−1/2 sin(ε−1tA(k)1/2) − A0(k)−1/2 sin(ε−1tA0(k)1/2)
× ε((D + k)2 + ε2I)−1/2
L2(Ω)→L2(Ω) C(1 + |t|),
a. e. k ∈

Ω. (4) But A(k) c∗|k|2I and it sufficies to prove (4) for |k| κ0.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 12 / 13

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Introduction Class of operators Effective operator Survey Problem Results Method

Method. The Floquet-Bloch theory

Recall that (2) ⇐ ⇒

A−1/2 sin(ε−1tA1/2) − (A0)−1/2 sin(ε−1t(A0)1/2)
× ε(−∆ + ε2I)−1/2
L2(Rd)→L2(Rd) C(1 + |t|).

(3) The Floquet-Bloch theory: A ∼

Ω ⊕A(k) dk.
Ω is the Brillouin zone of the dual lattice;

k ∈ Ω is quasimomentum. In L2(Ω; Cn), A(k) = b(D + k)∗g(x)b(D + k) with the periodic b. c. (3) ⇐ ⇒

A(k)−1/2 sin(ε−1tA(k)1/2) − A0(k)−1/2 sin(ε−1tA0(k)1/2)
× ε((D + k)2 + ε2I)−1/2
L2(Ω)→L2(Ω) C(1 + |t|),
a. e. k ∈

Ω. (4) But A(k) c∗|k|2I and it sufficies to prove (4) for |k| κ0.

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 12 / 13

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Introduction Class of operators Effective operator Survey Problem Results Method

Method. Analytic perturbation theory w. r. t. |k|

Recall that A(k) = b(D + k)∗g(x)b(D + k) in L2(Ω; Cn) with periodic b. c. Unperturbed operator: A(0) = b(D)∗g(x)b(D), Ker A(0) = Cn. λ1(0) = · · · = λn(0) spec A(0) λn+1(0) λ1(k), . . . , λn(k) ∈ [0, δ] δ 3δ spec A(k) for small |k| λn+1(k), · · · ∈ [3δ, ∞) Goes to the error term. Homogenization is the spectral threshold effect at the bottom of the spectrum.

Thank you for your attention!

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 13 / 13

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Introduction Class of operators Effective operator Survey Problem Results Method

Method. Analytic perturbation theory w. r. t. |k|

Recall that A(k) = b(D + k)∗g(x)b(D + k) in L2(Ω; Cn) with periodic b. c. Unperturbed operator: A(0) = b(D)∗g(x)b(D), Ker A(0) = Cn. λ1(0) = · · · = λn(0) spec A(0) λn+1(0) λ1(k), . . . , λn(k) ∈ [0, δ] δ 3δ spec A(k) for small |k| λn+1(k), · · · ∈ [3δ, ∞) Goes to the error term. Homogenization is the spectral threshold effect at the bottom of the spectrum.

Thank you for your attention!

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 13 / 13

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Introduction Class of operators Effective operator Survey Problem Results Method

Method. Analytic perturbation theory w. r. t. |k|

Recall that A(k) = b(D + k)∗g(x)b(D + k) in L2(Ω; Cn) with periodic b. c. Unperturbed operator: A(0) = b(D)∗g(x)b(D), Ker A(0) = Cn. λ1(0) = · · · = λn(0) spec A(0) λn+1(0) λ1(k), . . . , λn(k) ∈ [0, δ] δ 3δ spec A(k) for small |k| λn+1(k), · · · ∈ [3δ, ∞) Goes to the error term. Homogenization is the spectral threshold effect at the bottom of the spectrum.

Thank you for your attention!

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 13 / 13

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Introduction Class of operators Effective operator Survey Problem Results Method

Method. Analytic perturbation theory w. r. t. |k|

Recall that A(k) = b(D + k)∗g(x)b(D + k) in L2(Ω; Cn) with periodic b. c. Unperturbed operator: A(0) = b(D)∗g(x)b(D), Ker A(0) = Cn. λ1(0) = · · · = λn(0) spec A(0) λn+1(0) λ1(k), . . . , λn(k) ∈ [0, δ] δ 3δ spec A(k) for small |k| λn+1(k), · · · ∈ [3δ, ∞) Goes to the error term. Homogenization is the spectral threshold effect at the bottom of the spectrum.

Thank you for your attention!

Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 13 / 13