Homogenization of thin structures and singular measures Andrey - PowerPoint PPT Presentation

Homogenization of thin structures and singular measures Andrey Piatnitski Narvik Universiti college, Norway and Lebedev Physical Institute RAS, Moscow, Russia Seix 11–16 June, 2006 . – p.1/51

Motiv a tion Dimension reduction. Shells, skeletons, rod structures → surfaces and segments structures; − Reduction of the number of parameters. Asymptotic problems with two small parameters (microscopic length scale of the medium and structure thickness) − → problems with only one parameter; Porous media with rough geometry. . – p.2/51

Let µ ( x ) be a positive finite Borel measure on a standard n -dimensional torus T n ≡ R n / Z n or in R n . We identify µ with the corresponding periodic measure in R n . Without loss of generality, we may assume that � dµ ( x ) = 1 . T n . – p.3/51

To clarify the idea of introducing Sobolev spaces with measure, consider a simple example. Let µ be a positive finite Borel measure in a smooth bounded domain G . Consider the variational problem � � � a ( x ) ∇ ϕ ( x ) · ∇ ϕ ( x ) + ϕ 2 ( x ) − 2 f ( x ) ϕ ( x ) inf dµ ( x ) , ϕ ∈ C ∞ 0 ( G ) G where a ( x ) ia a continuous positive definite matrix in G and f ( x ) is a continuous function in G . Our goal is to introduce a Sobolev space with measure µ in such a way that the mini- mum is attained and a minimizer is found as a solution to the corresponding Euler equation. . – p.4/51

Sobolev sp a es Definition 1. We say that a function u ∈ L 2 ( T n , µ ) belongs to the space H 1 ( T n , µ ) if there exists a vector-function z ∈ ( L 2 ( T n , µ )) n and a sequence ϕ k ∈ C ∞ ( T n ) such that in L 2 ( T n , µ ) as k → ∞ , ϕ k − → u in ( L 2 ( T n , µ )) n as k → ∞ . ∇ ϕ k − → z The function z ( x ) is called the gradient or µ -gradient of u ( x ) and is denoted by ∇ µ u . Similarly, we can define the spaces H 1 ( R n , µ ) , H 1 loc ( R n , µ ) and also the space H 1 ( G, µ ) for an arbitrary domain G ⊂ R n and a (locally) finite Borel measure µ on G . . – p.5/51

Example. Segment Generally speaking, the gradient of a function of class H 1 ( T n , µ ) is not unique. In particular, the zero function may have a nontrivial gradient. We illustrate this with Example 1 . In the square [ − 1 / 2 , 1 / 2] 2 , we consider the segment {− 1 / 4 ≤ x 1 ≤ 1 / 4 , x 2 = 0 } and introduce dµ = 2 χ ( x 1 ) dx 1 × δ ( x 2 ) , (1) where χ ( t ) is the characteristic function of the segment [ − 1 4 , 1 4 ] and δ ( t ) is the Dirac mass at zero. . – p.6/51

Example ( ont.) Let ψ ( x ) ∈ C ∞ coincide with a function of the form θ ( x 1 ) x 2 0 in a small neighborhood of the segment. Then ψ = 0 in L 2 ( T 2 , µ ) . Choosing ϕ k ( x ) = ψ ( x ) for all k in the definition of µ -gradient, we find z ( x ) = ∇ µ ψ ( x ) = (0 , θ ( x 1 )) . Thus, any vector-valued function of the form (0 , θ ( x 1 )) with smooth θ ( s ) serves as the µ -gradient of zero. In fact, this assertion is valid for any θ ( s ) in L 2 . . – p.7/51

Gradients of zer o, Example of H 1 sp a e The gradients of zero form a closed subspace of ( L 2 ( T n , µ )) n , denote it Γ µ (0) . The set of the gradients of any H 1 ( T n , µ ) -function is the sum of its arbitrary gradient and Γ µ (0) . Example 2 (Segment). Consider the space H 1 ( T n , µ ) (or H 1 ( R n , µ ) ) for 1D Lebesgue measure µ on the segment I = { x ∈ R n : 0 ≤ x 1 ≤ a, x 2 = x 3 = · · · = x n = 0 } . Proposition 1. The space H 1 ( T n , µ ) consists of all Borel functions u ( x ) such that u ( s, 0 , 0 , . . . , 0) ∈ H 1 (0 , a ) . Moreover, ∇ µ u ( x ) = ( u ′ x 1 ( x 1 , 0) , ψ 2 ( x 1 ) , . . . , ψ n ( x 1 )) , where � x 1 ≡ d � u ′ dsu ( s, 0 , 0 , . . . , 0) , and ψ 2 , ψ 3 , . . . , ψ n are arbitrary � s = x 1 functions in L 2 (0 , a ) . . – p.8/51

Example. Ur hin Example 3 (”Urchin”). Consider the segments I 1 , I 2 , I N starting at the origin and directed along vectors v 1 , v 2 , . . . , v N . Let µ 1 , µ 2 , . . . , µ N be the standard 1 D Lebesgue measures on the segments I 1 , . . . , I N respectively, and let λ 1 , . . . , λ N be arbitrary positive numbers. We set N � µ = λ j µ j . j =1 � � A function u ( x ) belongs H 1 ( T n , µ ) if and only if u I j ∈ H 1 ( I j ) , and the values of the restricted functions at the origin coin- cide for all segments (recall that an H 1 -function of a single variable is continuous). . – p.9/51

Example. Reinf or ed shell Example 4 (Reinforced shells). Let Π 0 = { x ∈ T n : x 1 = 0 } . We set µ ( x ) = δ ( x 1 ) × dx ′ + dx, x ′ = ( x 2 , . . . , x n ) . d ˜ A function u ( x ) ∈ H 1 ( T 2 , ˜ µ ) if and only if u ∈ H 1 ( T n ) and the � � Π 0 ∈ H 1 ( T n − 1 ) . trace u ( x ) Remark 1 . If the co-dimension of a plane Π ⊂ R n is greater than one, then the trace of a H 1 ( R n ) -function on Π is not well-defined. Therefore, µ = dµ + dx , then H 1 ( T n , ˜ if µ is the Lebesgue measure on Π and d ˜ µ ) is isomorphic to the direct sum of the spaces H 1 ( R n ) and H 1 ( R n , µ ) . We denote H ( R n , µ ) = { ( u, z ) : u ∈ H 1 ( R n , µ ) , z = ∇ µ u ) } . . – p.10/51

Conver gen e in v ariable sp a es Suppose that Radon measures µ k weakly converges, as k → ∞ , to µ in R n . Definition 2. We say that g k ∈ L 2 ( R n , µ k ) weakly converges in L 2 ( R n , µ k ) to g ∈ L 2 ( R n , µ ) as k → ∞ if - � g k � L 2 ( R n ,µ ) ≤ C ; � � - lim g k ( x ) ϕ ( x ) dµ k ( x ) = g ( x ) ϕ ( x ) dµ ( x ) k →∞ R n R n for all ϕ ∈ C ∞ 0 ( R n ) . . – p.11/51

Conver gen e in v ariable sp a es Definition 3. A sequence { g k } converges strongly to g ( x ) ∈ L 2 ( R n , µ k ) if it weakly converges and � � lim g k ( x ) h k ( x ) dµ k ( x ) = g ( x ) h ( x ) dµ ( x ) k →∞ R n R n for any sequence { h k ( x ) } weakly converging to h ( x ) ∈ L 2 ( R n , µ ) in L 2 ( R n , µ k ) . Lemma 1. Let { g k } weakly converge to g ( x ) in L 2 ( R n , µ k ) . Then { g k } converges strongly if and only if k →∞ � g k � L 2 ( R n ,µ k ) = � g � L 2 ( R n ,µ ) . lim . – p.12/51

Conver gen e in v ariable sp a es Lemma 2. Let { µ k } converge weakly to µ . Then any bounded sequence { g k ( x ) } , � g k � L 2 ( R n ,µ k ) ≤ C converges weakly along a subsequence in L 2 ( R n , µ k ) towards some function g ( x ) ∈ L 2 ( R n , µ ) . . – p.13/51

Potential and solenoid al fields Definition 4. The space L pot 2 ( R n , µ ) is the closure of the linear set {∇ ϕ : ϕ ∈ C ∞ 0 ( R n ) } in the ( L 2 ( R n , µ )) n -norm. Definition 5. The space L pot 2 ( R n , µ ) of solenoidal vector-valued functions is the orthogonal complement to the space L pot 2 ( R n , µ ) in ( L 2 ( R n , µ )) n . . – p.14/51

Smoothing a periodi measure � Let K ( x ) ≥ 0 be a C ∞ 0 function such that R n K ( x ) dx = 1 and K ( − x ) = K ( x ) . For a Radon measure µ ( x ) in R n or on T n we set � � x − y � dµ δ ( x ) = ρ δ ( x ) dx, ρ δ ( x ) = δ − n K dµ ( y ) . δ R n The measures µ δ locally weakly converge in R n to µ . . – p.15/51

Smoothing We also introduce � � y � ϕ δ ( x ) = δ − n K ϕ ( x − y ) dy. δ R n Then � � ϕ δ ( x ) dµ ( x ) = ϕ ( x ) dµ δ ( x ) R n R n . – p.16/51

Smoothing opera tor a greed with the measure Lemma 3. For every v ∈ L 2 ( R n , µ ) there is v δ ∈ L 2 ( R n , µ ) such that � � v δ ( x ) ϕ ( x ) dµ δ ( x ) = v ( x ) ϕ δ ( x ) dµ ( x ) R n R n for all ϕ ∈ C 0 ( R n ) . The family v δ ( x ) strongly converges to v ( x ) in L 2 ( R n , µ δ ) as δ → 0 . . – p.17/51

Diver gen e opera tor Definition 6. Let g ∈ L 2 ( R n , µ ) and v ∈ ( L 2 ( R n , µ )) n . We say that g ( x ) = div µ v ( x ) if � � g ( x ) ϕ ( x ) dµ ( x ) = − v ( x ) · ∇ ϕ ( x ) dµ ( x ) R n R n for any ϕ ∈ C ∞ 0 ( R n ) . . – p.18/51

Ellipti equa tions Let a ( x ) = { a ij ( x ) } be a symmetric n × n -matrix, Λ | ξ | 2 ≤ a ij ( x ) ξ i ξ j ≤ Λ − 1 | ξ | 2 , ξ ∈ R n Λ > 0 , µ -a.e. in R n . Suppose that f ∈ L 2 ( R n , µ ) and λ > 0 . Definition 7. We say that a pair ( u, ∇ µ u ) with u ∈ H 1 ( R n , µ ) , satisfies the equation − div µ ( a ( x ) ∇ µ u ( x )) + λu ( x ) = f ( x ) (2) in L 2 ( R n , µ ) , if for any v ∈ H 1 ( R n , µ ) and any of its gradient ∇ µ v it holds: � � � a ( x ) ∇ µ u ( x ) ·∇ µ v ( x ) dµ ( x )+ λ u ( x ) v ( x ) dµ ( x ) = f ( x ) v ( x ) dµ ( x R n R n R n . – p.19/51

Ellipti equa tions A function u ∈ H 1 ( R n , µ ) is called a solution if the last identity holds for some of its gradients. Lemma 4. The above equation has a unique solution ( u, ∇ µ u ) , u ∈ H 1 ( R n , µ ) . Moreover, the choice of the µ -gradient of u is uniquely determined by the condition a ( x ) ∇ µ u ( x ) ∈ (Γ µ (0)) ⊥ . In the special case a ( x ) = Id the integral identity reads � � � ∇ µ u ( x ) ·∇ µ v ( x ) dµ ( x )+ λ u ( x ) v ( x ) dµ ( x ) = f ( x ) v ( x ) dµ ( x ) . R n R n R n The expression div µ ∇ µ u is called the µ -Laplacian of u . . – p.20/51

T angential gradient A gradient ∇ µ u of a function u ∈ H 1 ( R n , µ ) is tangential if it is orthogonal to Γ µ (0) . Thus tangential gradient of u is the orthogonal projection of an arbitrary µ -gradient of u on (Γ µ (0)) ⊥ . . – p.21/51