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

On homogenization of 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

Introduction Class of operators Effective operator Survey Problem Results Method Introduction to homogenization The simplest periodic operator. In L 2 ( R d ) , A = − div g ( x ) ∇ , ⇒ A = A ∗ � 0 . g ( x + a ) = g ( x ) , x ∈ R d , a ∈ Z d ; g ∈ L ∞ ; g ( x ) > 0 = Fig. 4: The Fig. 1–3: A periodic medium with different values of ε . effective medium. Homogenization. A ε = − div g ( x /ε ) ∇ , ε > 0 is small. Typical problem: A ε u ε + u ε = F ∈ L 2 ( R d ) . Question: u ε ∼ ? as ε → 0 . Answer: u ε ∼ u 0 , where u 0 solves − div g 0 ∇ u 0 + u 0 = F . �� 1 � − 1 Example. If d = 1 , g 0 = g := 0 g ( x ) − 1 d x . � If d > 1 , g � g 0 � g := (0 , 1) d g ( x ) d x (Voigt-Reuss bracketing). Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

Introduction Class of operators Effective operator Survey Problem Results Method Introduction to homogenization The simplest periodic operator. In L 2 ( R d ) , A = − div g ( x ) ∇ , ⇒ A = A ∗ � 0 . g ( x + a ) = g ( x ) , x ∈ R d , a ∈ Z d ; g ∈ L ∞ ; g ( x ) > 0 = Fig. 4: The Fig. 1–3: A periodic medium with different values of ε . effective medium. Homogenization. A ε = − div g ( x /ε ) ∇ , ε > 0 is small. Typical problem: A ε u ε + u ε = F ∈ L 2 ( R d ) . Question: u ε ∼ ? as ε → 0 . Answer: u ε ∼ u 0 , where u 0 solves − div g 0 ∇ u 0 + u 0 = F . �� 1 � − 1 Example. If d = 1 , g 0 = g := 0 g ( x ) − 1 d x . � If d > 1 , g � g 0 � g := (0 , 1) d g ( x ) d x (Voigt-Reuss bracketing). Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

Introduction Class of operators Effective operator Survey Problem Results Method Introduction to homogenization The simplest periodic operator. In L 2 ( R d ) , A = − div g ( x ) ∇ , ⇒ A = A ∗ � 0 . g ( x + a ) = g ( x ) , x ∈ R d , a ∈ Z d ; g ∈ L ∞ ; g ( x ) > 0 = Fig. 4: The Fig. 1–3: A periodic medium with different values of ε . effective medium. Homogenization. A ε = − div g ( x /ε ) ∇ , ε > 0 is small. Typical problem: A ε u ε + u ε = F ∈ L 2 ( R d ) . Question: u ε ∼ ? as ε → 0 . Answer: u ε ∼ u 0 , where u 0 solves − div g 0 ∇ u 0 + u 0 = F . �� 1 � − 1 Example. If d = 1 , g 0 = g := 0 g ( x ) − 1 d x . � If d > 1 , g � g 0 � g := (0 , 1) d g ( x ) d x (Voigt-Reuss bracketing). Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

Introduction Class of operators Effective operator Survey Problem Results Method Introduction to homogenization The simplest periodic operator. In L 2 ( R d ) , A = − div g ( x ) ∇ , ⇒ A = A ∗ � 0 . g ( x + a ) = g ( x ) , x ∈ R d , a ∈ Z d ; g ∈ L ∞ ; g ( x ) > 0 = Fig. 4: The Fig. 1–3: A periodic medium with different values of ε . effective medium. Homogenization. A ε = − div g ( x /ε ) ∇ , ε > 0 is small. Typical problem: A ε u ε + u ε = F ∈ L 2 ( R d ) . Question: u ε ∼ ? as ε → 0 . Answer: u ε ∼ u 0 , where u 0 solves − div g 0 ∇ u 0 + u 0 = F . �� 1 � − 1 Example. If d = 1 , g 0 = g := 0 g ( x ) − 1 d x . � If d > 1 , g � g 0 � g := (0 , 1) d g ( x ) d x (Voigt-Reuss bracketing). Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

Introduction Class of operators Effective operator Survey Problem Results Method Introduction to homogenization The simplest periodic operator. In L 2 ( R d ) , A = − div g ( x ) ∇ , ⇒ A = A ∗ � 0 . g ( x + a ) = g ( x ) , x ∈ R d , a ∈ Z d ; g ∈ L ∞ ; g ( x ) > 0 = Fig. 4: The Fig. 1–3: A periodic medium with different values of ε . effective medium. Homogenization. A ε = − div g ( x /ε ) ∇ , ε > 0 is small. Typical problem: A ε u ε + u ε = F ∈ L 2 ( R d ) . Question: u ε ∼ ? as ε → 0 . Answer: u ε ∼ u 0 , where u 0 solves − div g 0 ∇ u 0 + u 0 = F . �� 1 � − 1 Example. If d = 1 , g 0 = g := 0 g ( x ) − 1 d x . � If d > 1 , g � g 0 � g := (0 , 1) d g ( x ) d x (Voigt-Reuss bracketing). Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

Introduction Class of operators Effective operator Survey Problem Results Method Introduction to homogenization The simplest periodic operator. In L 2 ( R d ) , A = − div g ( x ) ∇ , ⇒ A = A ∗ � 0 . g ( x + a ) = g ( x ) , x ∈ R d , a ∈ Z d ; g ∈ L ∞ ; g ( x ) > 0 = Fig. 4: The Fig. 1–3: A periodic medium with different values of ε . effective medium. Homogenization. A ε = − div g ( x /ε ) ∇ , ε > 0 is small. Typical problem: A ε u ε + u ε = F ∈ L 2 ( R d ) . Question: u ε ∼ ? as ε → 0 . Answer: u ε ∼ u 0 , where u 0 solves − div g 0 ∇ u 0 + u 0 = F . �� 1 � − 1 Example. If d = 1 , g 0 = g := 0 g ( x ) − 1 d x . � If d > 1 , g � g 0 � g := (0 , 1) d g ( x ) d x (Voigt-Reuss bracketing). Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

Introduction Class of operators Effective operator Survey Problem Results Method Introduction to homogenization The simplest periodic operator. In L 2 ( R d ) , A = − div g ( x ) ∇ , ⇒ A = A ∗ � 0 . g ( x + a ) = g ( x ) , x ∈ R d , a ∈ Z d ; g ∈ L ∞ ; g ( x ) > 0 = Fig. 4: The Fig. 1–3: A periodic medium with different values of ε . effective medium. Homogenization. A ε = − div g ( x /ε ) ∇ , ε > 0 is small. Typical problem: A ε u ε + u ε = F ∈ L 2 ( R d ) . Question: u ε ∼ ? as ε → 0 . Answer: u ε ∼ u 0 , where u 0 solves − div g 0 ∇ u 0 + u 0 = F . �� 1 � − 1 Example. If d = 1 , g 0 = g := 0 g ( x ) − 1 d x . � If d > 1 , g � g 0 � g := (0 , 1) d g ( x ) d x (Voigt-Reuss bracketing). Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

Introduction Class of operators Effective operator Survey Problem Results Method Introduction to homogenization The simplest periodic operator. In L 2 ( R d ) , A = − div g ( x ) ∇ , ⇒ A = A ∗ � 0 . g ( x + a ) = g ( x ) , x ∈ R d , a ∈ Z d ; g ∈ L ∞ ; g ( x ) > 0 = Fig. 4: The Fig. 1–3: A periodic medium with different values of ε . effective medium. Homogenization. A ε = − div g ( x /ε ) ∇ , ε > 0 is small. Typical problem: A ε u ε + u ε = F ∈ L 2 ( R d ) . Question: u ε ∼ ? as ε → 0 . Answer: u ε ∼ u 0 , where u 0 solves − div g 0 ∇ u 0 + u 0 = F . �� 1 � − 1 Example. If d = 1 , g 0 = g := 0 g ( x ) − 1 d x . � If d > 1 , g � g 0 � g := (0 , 1) d g ( x ) d x (Voigt-Reuss bracketing). Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

Introduction Class of operators Effective operator Survey Problem Results Method Introduction to homogenization The simplest periodic operator. In L 2 ( R d ) , A = − div g ( x ) ∇ , ⇒ A = A ∗ � 0 . g ( x + a ) = g ( x ) , x ∈ R d , a ∈ Z d ; g ∈ L ∞ ; g ( x ) > 0 = Fig. 4: The Fig. 1–3: A periodic medium with different values of ε . effective medium. Homogenization. A ε = − div g ( x /ε ) ∇ , ε > 0 is small. Typical problem: A ε u ε + u ε = F ∈ L 2 ( R d ) . Question: u ε ∼ ? as ε → 0 . Answer: u ε ∼ u 0 , where u 0 solves − div g 0 ∇ u 0 + u 0 = F . �� 1 � − 1 Example. If d = 1 , g 0 = g := 0 g ( x ) − 1 d x . � If d > 1 , g � g 0 � g := (0 , 1) d g ( x ) d x (Voigt-Reuss bracketing). Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

Introduction Class of operators Effective operator Survey Problem Results Method Introduction to homogenization The simplest periodic operator. In L 2 ( R d ) , A = − div g ( x ) ∇ , ⇒ A = A ∗ � 0 . g ( x + a ) = g ( x ) , x ∈ R d , a ∈ Z d ; g ∈ L ∞ ; g ( x ) > 0 = Fig. 4: The Fig. 1–3: A periodic medium with different values of ε . effective medium. Homogenization. A ε = − div g ( x /ε ) ∇ , ε > 0 is small. Typical problem: A ε u ε + u ε = F ∈ L 2 ( R d ) . Question: u ε ∼ ? as ε → 0 . Answer: u ε ∼ u 0 , where u 0 solves − div g 0 ∇ u 0 + u 0 = F . �� 1 � − 1 Example. If d = 1 , g 0 = g := 0 g ( x ) − 1 d x . � If d > 1 , g � g 0 � g := (0 , 1) d g ( x ) d x (Voigt-Reuss bracketing). Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13

Introduction Class of operators Effective operator Survey Problem Results Method Introduction to homogenization The simplest periodic operator. In L 2 ( R d ) , A = − div g ( x ) ∇ , ⇒ A = A ∗ � 0 . g ( x + a ) = g ( x ) , x ∈ R d , a ∈ Z d ; g ∈ L ∞ ; g ( x ) > 0 = Fig. 4: The Fig. 1–3: A periodic medium with different values of ε . effective medium. Homogenization. A ε = − div g ( x /ε ) ∇ , ε > 0 is small. Typical problem: A ε u ε + u ε = F ∈ L 2 ( R d ) . Question: u ε ∼ ? as ε → 0 . Answer: u ε ∼ u 0 , where u 0 solves − div g 0 ∇ u 0 + u 0 = F . �� 1 � − 1 Example. If d = 1 , g 0 = g := 0 g ( x ) − 1 d x . � If d > 1 , g � g 0 � g := (0 , 1) d g ( x ) d x (Voigt-Reuss bracketing). Yulia Meshkova (SPbU) On homogenization of periodic hyperbolic systems 2 / 13