Approximations of the Neumann Laplacian in nonuniformly collapsing - PowerPoint PPT Presentation

Approximations of the Neumann Laplacian in nonuniformly collapsing strips C´ esar R. de Oliveira UFSCar QMath13 – Atlanta C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 1 / 25

Sources 1 Sources 2 Collapsing regions 3 Effective operator 4 Uniformly collapsing approximations 5 Examples C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 2 / 25

Sources Sources R. Bedoya, C. R. de Oliveira & A. A. Verri: Complex Γ-convergence and magnetic Dirichlet Laplacian in bounded thin tubes. J. Spectr. Theory 4 (2014) 621–642 C. R. de Oliveira & A. A. Verri: On the Neumann Laplacian in nonuniformly collapsing strips. Preprint. L. Friedlander & M. Solomyak: On the spectrum of the Dirichlet Laplacian in a narrow infinite strip. Amer. Math. Soc. Transl. 225 (2008) 103–116 J. K. Hale & G. Raugel: Reaction-diffusion equation in thin domains. J. Math. pures et appl. 71 (1992) 33–95 C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 3 / 25

Collapsing regions 1 Sources 2 Collapsing regions 3 Effective operator 4 Uniformly collapsing approximations 5 Examples C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 4 / 25

Collapsing regions Initial g( x ) - ∆ Λ 1 a C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 5 / 25

Collapsing regions Initial ε g( x ) nonuniformly collapsing - Δ Λ ε a C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 6 / 25

Collapsing regions (Un)Bounded region • We consider “thick regions” given by functions g : [ a, ∞ ) → (0 , ∞ ) with g ( x ) → ∞ as x → ∞ . Is there an effective operator S = S ( g ) as ε → 0 ? •• A more delicate question: Is there a family of uniformly collapsing regions Q ε whose effective operator coincides with S ? • • • Conditions on g : (c1) C 2 function and strictly increasing for large values of x ; (c2) j ( x ) := g ′ ( x ) 2 g ( x ) and j ′ ( x ) are bounded. C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 7 / 25

Collapsing regions (Un)Bounded region • We consider “thick regions” given by functions g : [ a, ∞ ) → (0 , ∞ ) with g ( x ) → ∞ as x → ∞ . Is there an effective operator S = S ( g ) as ε → 0 ? •• A more delicate question: Is there a family of uniformly collapsing regions Q ε whose effective operator coincides with S ? • • • Conditions on g : (c1) C 2 function and strictly increasing for large values of x ; (c2) j ( x ) := g ′ ( x ) 2 g ( x ) and j ′ ( x ) are bounded. C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 7 / 25

Collapsing regions (Un)Bounded region • We consider “thick regions” given by functions g : [ a, ∞ ) → (0 , ∞ ) with g ( x ) → ∞ as x → ∞ . Is there an effective operator S = S ( g ) as ε → 0 ? •• A more delicate question: Is there a family of uniformly collapsing regions Q ε whose effective operator coincides with S ? • • • Conditions on g : (c1) C 2 function and strictly increasing for large values of x ; (c2) j ( x ) := g ′ ( x ) 2 g ( x ) and j ′ ( x ) are bounded. C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 7 / 25

Collapsing regions (Un)Bounded region • We consider “thick regions” given by functions g : [ a, ∞ ) → (0 , ∞ ) with g ( x ) → ∞ as x → ∞ . Is there an effective operator S = S ( g ) as ε → 0 ? •• A more delicate question: Is there a family of uniformly collapsing regions Q ε whose effective operator coincides with S ? • • • Conditions on g : (c1) C 2 function and strictly increasing for large values of x ; (c2) j ( x ) := g ′ ( x ) 2 g ( x ) and j ′ ( x ) are bounded. C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 7 / 25

Collapsing regions (Un)Bounded region • We consider “thick regions” given by functions g : [ a, ∞ ) → (0 , ∞ ) with g ( x ) → ∞ as x → ∞ . Is there an effective operator S = S ( g ) as ε → 0 ? •• A more delicate question: Is there a family of uniformly collapsing regions Q ε whose effective operator coincides with S ? • • • Conditions on g : (c1) C 2 function and strictly increasing for large values of x ; (c2) j ( x ) := g ′ ( x ) 2 g ( x ) and j ′ ( x ) are bounded. C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 7 / 25

Collapsing regions (Un)Bounded region The region of interest is ( x, y ) ∈ R 2 | 0 < y < εg ( x ) , x ∈ [ a, ∞ ) � � Λ ε := , and the quadratic form (Neumann Laplacian) � |∇ v | 2 d x, dom m ε = H 1 (Λ ε ) . m ε ( v ) = Λ ε After changes of variables, m ε ( v ) is cast as �� 2 � + | ϕ y | 2 � ϕ ′ − g ′ 2 g ϕ − y ϕ y g ′ � � � � n ε ( ϕ ) := d x d y, � � ε 2 g 2 g � Q where Q := [ a, ∞ ) × (0 , 1) is a fixed region. Note that, as ε → 0, 2 � � � � ϕ ′ − � g ′ 2 g ϕ d x d y, if ϕ y = 0 , � � n ε ( ϕ ) − → n ( ϕ ) := Q � ∞ , if ϕ y � = 0 . Let S ε and S be the operators associated with n ε and n , respectively. C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 8 / 25

Collapsing regions (Un)Bounded region The region of interest is ( x, y ) ∈ R 2 | 0 < y < εg ( x ) , x ∈ [ a, ∞ ) � � Λ ε := , and the quadratic form (Neumann Laplacian) � |∇ v | 2 d x, dom m ε = H 1 (Λ ε ) . m ε ( v ) = Λ ε After changes of variables, m ε ( v ) is cast as �� 2 � + | ϕ y | 2 � ϕ ′ − g ′ 2 g ϕ − y ϕ y g ′ � � � � n ε ( ϕ ) := d x d y, � � ε 2 g 2 g � Q where Q := [ a, ∞ ) × (0 , 1) is a fixed region. Note that, as ε → 0, 2 � � � � ϕ ′ − � g ′ 2 g ϕ d x d y, if ϕ y = 0 , � � n ε ( ϕ ) − → n ( ϕ ) := Q � ∞ , if ϕ y � = 0 . Let S ε and S be the operators associated with n ε and n , respectively. C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 8 / 25

Collapsing regions (Un)Bounded region The region of interest is ( x, y ) ∈ R 2 | 0 < y < εg ( x ) , x ∈ [ a, ∞ ) � � Λ ε := , and the quadratic form (Neumann Laplacian) � |∇ v | 2 d x, dom m ε = H 1 (Λ ε ) . m ε ( v ) = Λ ε After changes of variables, m ε ( v ) is cast as �� 2 � + | ϕ y | 2 � ϕ ′ − g ′ 2 g ϕ − y ϕ y g ′ � � � � n ε ( ϕ ) := d x d y, � � ε 2 g 2 g � Q where Q := [ a, ∞ ) × (0 , 1) is a fixed region. Note that, as ε → 0, 2 � � � � ϕ ′ − � g ′ 2 g ϕ d x d y, if ϕ y = 0 , � � n ε ( ϕ ) − → n ( ϕ ) := Q � ∞ , if ϕ y � = 0 . Let S ε and S be the operators associated with n ε and n , respectively. C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 8 / 25

Collapsing regions (Un)Bounded region The region of interest is ( x, y ) ∈ R 2 | 0 < y < εg ( x ) , x ∈ [ a, ∞ ) � � Λ ε := , and the quadratic form (Neumann Laplacian) � |∇ v | 2 d x, dom m ε = H 1 (Λ ε ) . m ε ( v ) = Λ ε After changes of variables, m ε ( v ) is cast as �� 2 � + | ϕ y | 2 � ϕ ′ − g ′ 2 g ϕ − y ϕ y g ′ � � � � n ε ( ϕ ) := d x d y, � � ε 2 g 2 g � Q where Q := [ a, ∞ ) × (0 , 1) is a fixed region. Note that, as ε → 0, 2 � � � � ϕ ′ − � g ′ 2 g ϕ d x d y, if ϕ y = 0 , � � n ε ( ϕ ) − → n ( ϕ ) := Q � ∞ , if ϕ y � = 0 . Let S ε and S be the operators associated with n ε and n , respectively. C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 8 / 25

Collapsing regions (Un)Bounded region � ϕ ( x, y ) = w ( x )1 | w ∈ L 2 ([ a, ∞ )) � Let L := . Theorem (1) (by Kato-Robinson Theorem) For all f ∈ L 2 ( Q ) one has, as ε → 0 , ε f − ( S − 1 ⊕ 0) f � S − 1 � − � � → 0 , where 0 is the null operator on L ⊥ . C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 9 / 25

Collapsing regions (Un)Bounded region � ϕ ( x, y ) = w ( x )1 | w ∈ L 2 ([ a, ∞ )) � Let L := . Theorem (1) (by Kato-Robinson Theorem) For all f ∈ L 2 ( Q ) one has, as ε → 0 , ε f − ( S − 1 ⊕ 0) f � S − 1 � − � � → 0 , where 0 is the null operator on L ⊥ . C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 9 / 25

Effective operator 1 Sources 2 Collapsing regions 3 Effective operator 4 Uniformly collapsing approximations 5 Examples C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 10 / 25

Effective operator (Un)Bounded region The goal now is to characterize S : for this we need (c2), i.e., bounded j = g ′ 2 g and j ′ . Theorem (2) For g as above, we have ( Sw )( x ) := − w ′′ ( x ) + ̺ ( x ) w ( x ) , with ̺ ( x ) := j 2 ( x ) + j ′ ( x ) and a Robin condition at the end point a , that is, dom S = { w ∈ H 2 ([ a, ∞ )) | j ( a ) w ( a ) = w ′ ( a ) } . C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 11 / 25