Homogenization and uniform resolvent convergence for elliptic - PowerPoint PPT Presentation

Homogenization and uniform resolvent convergence for elliptic operators in a strip perforated along a curve Giuseppe CARDONE Dep. of Engineering, University of Sannio, Benevento, Italy joint works with D. Borisov, T. Durante G. Cardone (University of Sannio, Italy) Strip perforated along a curve 1 / 57

Formulation of the problem We consider an infinite planar straight strip Ω := { x : 0 < x 2 < d } , d > 0 perforated by small holes located closely one to another along an infinite, or finite and closed, curve. In Ω we consider a general second order elliptic operator subject to classical boundary conditions on the holes. If the perforation is non-periodic and satisfies rather weak assumptions, we describe possible homogenized problems G. Cardone (University of Sannio, Italy) Strip perforated along a curve 2 / 57

The curve γ Let γ be a curve - lying in Ω and separated from ∂ Ω by a fixed distance, - is C 3 -smooth, has no self-intersection, - is either an infinite or finite closed curve. Let us denote by: - s its arc length, s ∈ ( − s ∗ , + s ∗ ), where s ∗ is either finite or infinite - ρ = ρ ( s ) ∈ C 3 ( − s ∗ , + s ∗ ) the vector function describing the curve γ . G. Cardone (University of Sannio, Italy) Strip perforated along a curve 3 / 57

The holes Let us denote by: - ε be a small positive parameter, M ε ⊂ Z , - for k ∈ M ε , s ε k ∈ [ − s ∗ , + s ∗ ] set of points satisfying s ε k < s ε k +1 . - ω k , k ∈ Z , sequence of bounded domains in R 2 having C 2 -boundaries. - the domain θ ε defined by � θ ε := θ ε 0 ∪ θ ε θ ε ω ε 1 , i := k , i = 0 , 1 , k ∈ M ε i ω ε k := { x : ε − 1 η − 1 ( ε )( x − y ε y ε k := ρ ( s ε k ) ∈ ω k } , k ) , where M ε i ⊂ Z , M ε 0 ∩ M ε 1 = ∅ , M ε 0 ∪ M ε 1 = Z , - η = η ( ε ) is a some function and 0 < η ( ε ) � 1. G. Cardone (University of Sannio, Italy) Strip perforated along a curve 4 / 57

Operator Ω ε Ω ε := Ω \ θ ε perforated domain (a) Perforation along an infinite curve (b) Perforation along a closed curve Figure: Perforated domain Remarks: The sizes of the holes and the distance between them are described by means of two small parameters. The perforation is quite general and no periodicity is assumed:both the shapes and the distribution of the holes can be rather arbitrary. G. Cardone (University of Sannio, Italy) Strip perforated along a curve 5 / 57

Formulation of the problem H ε singularly perturbed operator: 2 2 � � ∂ ∂ ∂ − ∂ − A ij + A j A j + A 0 (1) ∂ x i ∂ x j ∂ x j ∂ x j i , j =1 j =1 in Ω ε subject to: Dirichlet condition on ∂ Ω ∪ ∂θ ε 0 Robin condition � ∂ � 2 2 � � ∂ ∂ ∂θ ε A ij ν ε A j ν ε ∂ N ε + a u = 0 on 1 , ∂ N ε := + j , i ∂ x j i , j =1 j =1 where - ν ε = ( ν ε 1 , ν ε 2 ) is the inward normal to ∂θ ε 1 , - a ∈ W 1 ∞ ( { x : | τ | < τ 0 } ). G. Cardone (University of Sannio, Italy) Strip perforated along a curve 6 / 57

Formulation of the problem - ( s , τ ) are the local coordinates introduced in the vicinity of γ , - τ is the distance to a point measured along the normal ν 0 to γ which is inward for Ω − - Ω − and Ω + are the partitions of Ω originated by γ . Remarks: On the boundary of the holes we impose Dirichlet or Neumann or Robin condition. Boundaries of different holes can be subject to different types of boundary conditions. Such mixtures of boundary conditions were not considered before. G. Cardone (University of Sannio, Italy) Strip perforated along a curve 7 / 57

Physical interpretation Waveguide theory: - Our operator describes a quantum particle in a waveguide - The waveguide is not isotropic: coefficients of the operator variable. - The perforation represents small defects distributed along a given line, - Conditions on the boundaries of the holes impose certain regime: Dirichlet condition describes a wall and the particle can not pass through such boundary. - The homogenization describes the effective behavior of our model once the perforation becomes finer. - The type of resolvent convergence characterizes in which sense the perturbed model is close to the effective one G. Cardone (University of Sannio, Italy) Strip perforated along a curve 8 / 57

Operator Ω ε - Sesquilinear form � � � � 2 2 � � ∂ u , ∂ v ∂ u a ε ( u , v ) := A ij + A j , v ∂ x j ∂ x i ∂ x j L 2 (Ω ε ) L 2 (Ω ε ) i , j =1 j =1 (2) � � 2 � ∂ v + u , A j + ( A 0 u , v ) L 2 (Ω ε ) ∂ x j L 2 (Ω ε ) j =1 in L 2 (Ω ε ) on the domain W 1 2 (Ω ε ). - H ε self-adjoint operator in L 2 (Ω ε ) associated with the sesquilinear form h ε ( u , v ) := a ε ( u , v ) + ( au , v ) L 2 ( ∂θ ε 1 ) in L 2 (Ω ε ) on ˚ W 1 2 (Ω ε , ∂ Ω ∪ ∂θ ε 0 ). - ˚ W 1 2 (Ω ε , ∂ Ω ∪ ∂θ ε 0 ) functions in W 1 2 (Ω ε ) with zero trace on ∂ Ω ∪ ∂θ ε 0 . G. Cardone (University of Sannio, Italy) Strip perforated along a curve 9 / 57

Formulation of the problem Aim: to study the resolvent convergence and the spectrum’s behavior of the operator H ε as ε → +0 , i.e. the asymptotic behavior of the resolvent of such operator as ε tends to zero G. Cardone (University of Sannio, Italy) Strip perforated along a curve 10 / 57

Formulation of the problem - Effective operator H 0 D : operator (1) in L 2 (Ω) subject to the Dirichlet condition on γ and ∂ Ω. D ( u , v ) := a ( u , v ) in L 2 (Ω) on ˚ - Associated form: h 0 W 1 2 (Ω , ∂ Ω ∪ γ ), - a : form (2), where Ω ε is replaced by Ω. D ) = ˚ - Domain of operator H 0 D : D ( H 0 W 1 2 (Ω , ∂ Ω ∪ γ ) ∩ W 2 2 (Ω \ γ ). G. Cardone (University of Sannio, Italy) Strip perforated along a curve 11 / 57

Assumptions on holes (A1) There exist 0 < R 1 < R 2 , b > 1, L > 0 and x k ∈ ω k , k ∈ M ε , such that B R 1 ( x k ) ⊂ ω k ⊂ B R 2 (0) , k ∈ M ε , | ∂ω k | � L for each B bR 2 ε ( y ε k ) ∩ B bR 2 ε ( y ε i , k ∈ M ε , i ) = ∅ i � = k , for each and for all sufficiently small ε . G. Cardone (University of Sannio, Italy) Strip perforated along a curve 12 / 57

Assumptions on holes Remarks: the sizes of holes are of the same order and there is a minimal distance between them. no periodicity for the perforation is assumed. since M ε is arbitrary, number of holes can be infinite or finite in the latter case, by an appropriate choice of M ε , the distances between the holes can be even not small, but finite. G. Cardone (University of Sannio, Italy) Strip perforated along a curve 13 / 57

Assumptions on holes (A2) For b and R 2 in (A1) and k ∈ M ε there exists a generalized solution X k : B b ∗ R 2 (0) \ ω k �→ R 2 , b ∗ := ( b + 1) / 2, of div X k = 0 in B b ∗ R 2 (0) \ ω k , X k · ν = − 1 on ∂ω k , (3) X k · ν = ϕ k on ∂ B b ∗ R 2 (0) , - belonging to L ∞ ( B b ∗ R 2 (0) \ ω k ) - bounded uniformly in k ∈ M ε in L ∞ ( B b ∗ R 2 (0) \ ω k ). ν is the outward normal to ∂ B b ∗ R 2 (0) and to ∂ω k ϕ k ∈ L ∞ ( ∂ B b ∗ R 2 (0)) satisfying � ϕ k ds = | ∂ω k | . (4) ∂ B b ∗ R 2 (0) G. Cardone (University of Sannio, Italy) Strip perforated along a curve 14 / 57

Assumptions on holes Remarks: Assumption (A2) is a restriction for the geometry of boundaries ∂ω k . Problem (3) can be rewritten to the Neumann problem for the Laplace equation by letting X k = ∇ V k . Then identity (4) is the solvability condition and this is the only restriction for ϕ k we suppose. Problem (3) is solvable for each fixed k and its solution belongs to L ∞ ( B b ∗ R 2 (0) \ ω k ). we assume that the norm � X k � L ∞ ( B b ∗ R 2 (0) \ ω k ) is bounded uniformly in k . G. Cardone (University of Sannio, Italy) Strip perforated along a curve 15 / 57

Main Result Theorem Let us assume ε ln η ( ε ) → 0 , ε → +0 , (5) suppose (A1), (A2), and (A3) There exists a constant R 3 > bR 2 such that � B R 3 ε ( y ε ω ε k ⊂ B R 3 ε ( y ε ∀ k ∈ M ε { x : | τ | < ε bR 2 } ⊂ k ) , k ) 0 . k ∈ M ε 0 Then the estimate 2 � � � ( H ε − i ) − 1 − ( H 0 1 1 D − i ) − 1 � L 2 (Ω) → W 1 2 + 1 2 (Ω ε ) � C ε | ln η ( ε ) | (6) holds true, where C is a positive constant independent of ε . G. Cardone (University of Sannio, Italy) Strip perforated along a curve 16 / 57

Main Result Assumptions: (5): the sizes of the holes are not too small (A3): the holes with the Dirichlet condition are, roughly speaking, distributed “uniformly” Results: homogenized operator is subject to the Dirichlet condition on γ norm resolvent condition in the sense of the operator norm � · � L 2 (Ω) → W 1 2 (Ω ε ) G. Cardone (University of Sannio, Italy) Strip perforated along a curve 17 / 57

Main Result Remark: Relation (5) admits the situation when the sizes of the holes are much smaller than the distances between them for example, η ( ε ) = ε α , α = const > 0, nevertheless the homogenized operator is still subject to the Dirichlet condition on γ . G. Cardone (University of Sannio, Italy) Strip perforated along a curve 18 / 57