ZHACM C OLLOQUIUM — Z URICH — 2.11.2016 Sobolev spaces on non-Lipschitz sets with application to BIEs on fractal screens Andrea Moiola D EPARTMENT OF M ATHEMATICS AND S TATISTICS , U NIVERSITY OF R EADING with S.N. Chandler-Wilde (Reading) and D.P . Hewett (UCL)
Motivation: acoustic scattering by fractal screen Γ bounded open subset of { x ∈ R n + 1 : x n + 1 = 0 } ∼ = R n , n = 1 , 2 (∆ + k 2 ) u = 0 D := R n + 1 \ { Γ × { 0 }} x 3 u i = � N x 2 j = 1 a j e i k d j · x | d j | = 1 , k > 0 u = − u i or Γ ∂ n = − ∂ u i ∂ u ∂ n x 1 u satisfies Sommerfeld radiation condition (SRC) at infinity � r − ( n − 1 ) / 2 � (i.e. ∂ r u − iku = o uniformly as r = | x | → ∞ ). Classical problem when Γ is Lipschitz. What happens for arbitrary (e.g. fractal) Γ ? 2
Fractal antennas (Figures from http://www.antenna-theory.com/antennas/fractal.php ) Fractal antennas are a popular topic in engineering: Wideband/multiband, compact, cheap, metamaterials, cloaking. . . Not analysed by mathematicians. 3
Example: Dirichlet scattering Given g D ∈ H 1 / 2 (Γ) , we can write 3 different BVPs: Find u ∈ C 2 ( D ) ∩ { u ∈ L 2 loc ( D ) , ∇ u ∈ L 2 loc ( D ) n } s.t. ∆ u + k 2 u = 0 in D γ ± ( u ) | Γ = g D P if { u ∈ H 1 / 2 ( R n ) SRC supp u ⊂ ∂ Γ } = { 0 } � P P’ [ u ] = γ + u − γ − u = 0 if D (Γ) dense in { u ∈ H − 1 / 2 ( R n ) supp u ⊂ Γ } � P ′ P” H 1 / 2 ( R n ) [ ∂ u /∂ x n ] ∈ D (Γ) Jump [ u ] is supported in Γ , while γ ± is restricted to Γ only. H − 1 / 2 P’ is equivalent to BIE ( − S k φ = g D , S k coercive in D (Γ) ). P” is uniquely solvable. ⇒ P” holds if Γ is C 0 . P ⇐ ⇒ P’ ⇐ Swap ± 1 / 2 for Neumann problem. (Chandler–Wilde, Hewett 2013) 4
Questions ◮ For which Γ and s are conditions above satisfied? { u ∈ H s ( R n ) : supp u ⊂ ∂ Γ } = { 0 } , D (Γ) dense in { u ∈ H s ( R n ) : supp u ⊂ Γ } ◮ Given Γ , which other � Γ � = Γ give the same scattered fields ∀ u i ? ◮ When is Γ “inaudible”, i.e. u s = 0 for all g D ? Can a screen with zero mass scatter waves? ◮ Does it matter whether Γ is open or closed? ◮ Where do Galerkin (BEM) solutions converge to? To try to answer these questions we need to learn more about Sobolev spaces on non-Lipschitz sets. Many results available (Maz’ya, Triebel, Polking, Adams, Hedberg,. . . ) but not entirely clear/satisfactory/useful for us. 5
Part II Definitions and duality
Basic definitions I: Sobolev spaces on R n W k := { u ∈ L 2 ( R n ) : ∂ α u ∈ L 2 ( R n ) , ∀| α | ≤ k } , For k ∈ N 0 , � � � u � 2 R n | ∂ α u ( x ) | 2 d x . W k := | α |≤ k H s := { u ∈ S ∗ ( R n ) : ˆ u ∈ L 1 loc ( R n ) and � u � H s < ∞} , For s ∈ R , � R n ( 1 + | ξ | 2 ) s | � u ( ξ ) | 2 d ξ . � u � 2 H s := ◮ For k ∈ N 0 , H k = W k with equivalent norms. ◮ For t > s , H t ⊂ H s (continuous embedding, norm 1). ◮ ( H s ) ∗ = H − s , with duality pairing � � u , v � H − s × H s := R n ˆ u ( ξ )ˆ v ( ξ ) d ξ . ◮ H s ⊂ C ( R n ) for s > n / 2 (Sobolev embedding theorem). δ x 0 ∈ H s ⇐ ⇒ s < − n / 2 ( � δ x 0 , φ � = φ ( x 0 ) ). 6
Basic definitions II: Sobolev sp. on subsets of R n Notation: Γ ⊂ R n open, F ⊂ R n closed, K ⊂ R n compact. H s � H s (Γ) := D (Γ) ( D (Γ) := C ∞ 0 (Γ) ⊂ C ∞ ( R n )) F := { u ∈ H s : supp u ⊂ F } = { u ∈ H s : u ( ϕ ) = 0 ∀ ϕ ∈ D ( F c ) } H s H s (Γ) := { u | Γ : u ∈ H s } H s (Γ) H s 0 (Γ) := D (Γ) | Γ (notation from McLean) “Global” and “local” spaces: Γ ⊂ H s ⊂ D ∗ ( R n ) , � H s (Γ) ⊂ H s H s 0 (Γ) ⊂ H s (Γ) ⊂ D ∗ (Γ) When Γ is Lipschitz it holds that ◮ � H s (Γ) = ( H − s (Γ)) ∗ ◮ � H s (Γ) = H s Γ ◮ H ± 1 / 2 = { 0 } ∂ Γ ◮ { H s (Γ) } s ∈ R and { � H s (Γ) } s ∈ R are interpolation scales 7
Duality Theorem (Chandler-Wilde and Hewett, 2013) Let Γ be any open subset of R n and let s ∈ R . Then ( H s (Γ)) ∗ = � H s (Γ)) ∗ = H − s (Γ) with equal norms and H − s (Γ) and ( � H s (Γ) = � U , w � H − s × H s for any U ∈ H − s , U | Γ = u . � u , w � H − s (Γ) × � Well-known for Lipschitz but not in general case. Main ideas of proof: ◮ H Hilbert, V ⊂ H closed ssp, H unitary realisation of H ∗ , then ( V a , H ) ⊥ = { ψ ∈ H , � ψ, φ � = 0 ∀ φ ∈ V } ⊥ is unitary realisation of V ∗ Γ c = { u ∈ H − s : u ( ψ ) = � u , ψ � = 0 ∀ ψ ∈ D (Γ) } = ( � H s (Γ)) a , H − s ◮ H − s Γ c ) ⊥ → H − s (Γ) ◮ Restriction operator | Γ is unitary isomorphism | Γ : ( H − s (from identification of H − s (Γ) with H − s / H − s Γ c ) ◮ Choose V = � H s (Γ) , H = H s , H = H − s 8
Sobolev space questions We will address the following questions: ◮ When does E ⊂ R n support non-zero u ∈ H s ? ◮ When is � H s (Γ) = H s Γ ? ◮ When is H s (Γ) = H s 0 (Γ) ? ◮ For which spaces is | Γ an isomorphism? ◮ When are H s (Γ) and � H s (Γ) interpolation scales? ◮ What’s the limit of a sequence of Galerkin solutions to a variational problem on prefractals? 9
Part III s -nullity
s -nullity Definition Given s ∈ R we say that a set E ⊂ R n is s -null if there are no non-zero elements of H s supported in E . (I.e. if H s F = { 0 } for every closed set F ⊂ E .) Other terminology exists: “ ( − s ) -polar” (Maz’ya, Littman), “set of uniqueness for H s ” (Maz’ya, Adams/Hedberg). 10
Relevance of s -nullity For the screen scattering problem: ◮ For a compact screen K to be audible we need H ± 1 / 2 � = { 0 } . K ◮ For the solution of the classical Dirichlet/Neumann BVP to be unique we need H ± 1 / 2 = { 0 } . ∂ Γ ◮ Two screens Γ 1 and Γ 2 give the same scattered field for all incident waves if and only if Γ 1 ⊖ Γ 2 is ± 1 / 2 -null. Γ 1 Γ 2 For general Sobolev space results: ◮ H s F 1 = H s F 2 ⇐ ⇒ F 1 ⊖ F 2 is s -null. ◮ � H s (Γ 1 ) = � H s (Γ 2 ) ⇐ ⇒ Γ 1 ⊖ Γ 2 is ( − s ) -null. ◮ If int (Γ) \ Γ is not ( − s ) -null then � H s (Γ) � H s Γ . ◮ We’ll see many more uses of nullity. . . 11
s -nullity: basic results ◮ A subset of an s -null set is s -null. ◮ If E is s -null and t > s then E is t -null. ◮ If E is s -null then has empty interior. ◮ If s > n / 2 then E is s -null ⇐ ⇒ int ( E ) = ∅ . ◮ For s < − n / 2 there are no non-empty s -null sets. Non-trivial results: ◮ The union of finitely many s -null closed sets is s -null. ◮ The union of countably many s -null Borel sets is s -null if s ≤ 0 . Union of non-closed s -null sets for s > 0 is not s -null: counterexample is E 1 = Q n , E 2 = R n \ Q n , s > n / 2 . 12
Nullity threshold For every E ⊂ R n with int ( E ) = ∅ there exists s E ∈ [ − n / 2 , n / 2 ] such that E is s -null for s > s E and not s -null for s < s E . We call s E the nullity threshold of E . E is not s -null E is s -null can support H s distributions cannot support H s distributions s E − n / 2 n / 2 s 0 Q1: Given E ⊂ R n , can we determine s E ? Q2: Given s ∈ [ − n / 2 , n / 2 ] , can we find some E ⊂ R n for which s E = s ? Q3: When is E s E -null? (i.e. is the maximum regularity attained?) We study separately sets with zero and positive Lebesgue measure. 13
Zero Lebesgue measure ⇒ s K ∈ [ − n / 2 , 0 ] Let K ⊂ R n be non-empty and compact. Then: ◮ H 0 K = L 2 ( K ) = { 0 } ⇐ ⇒ m ( K ) = 0 . ◮ If m ( K ) = 0 then H s K = { 0 } for s ≥ 0 (i.e. s K ≤ 0 ). ( H s ◮ If K is countable then s K = − n / 2 K = { 0 } ⇔ s ≥ − n / 2 ). Theorem If m ( K ) = 0 , then s K = dim H K − n . 2 ( dim H = Hausdorff dimension, m = Lebesgue measure) � � d > 0 : H ( d − n ) / 2 = { 0 } dim H K = inf K This does not tell us if K is s K -null; examples of both cases are possible. Sharpens previous results by Littman (1967) and Triebel (1997). 14
Examples Let Γ ⊂ R n be non-empty and open. ◮ If Γ is C 0 then s ∂ Γ ∈ [ − 1 / 2 , 0 ] . ◮ If Γ is C 0 ,α for some 0 < α < 1 then s ∂ Γ ∈ [ − 1 / 2 , − α/ 2 ] (sharp). ◮ If Γ is Lipschitz then s ∂ Γ = − 1 / 2 (and H − 1 / 2 = { 0 } ). ∂ Γ ◮ If K is boundary of Koch snowflake, s K = log 2 log 3 − 1 ≈ − 0 . 37 . For 0 < α < 1 / 2 let C α ⊂ [ 0 , 1 ] be the Cantor set with l j = α j , j ∈ N 0 : l 2 = α 2 l 3 = α 3 l 0 = 1 l 1 = α α := C α × C α ⊂ R 2 denote the associated “Cantor dust”: Let C 2 � � � � α = − n 1 + log 2 − n ∈ 2 , 0 s C n 2 log α Choose α = 2 − n / ( 2 s + n ) to have s C n α = s . Can also define “thin” Cantor dusts which have s K = − n / 2 15
Capacity Our proofs rely on the following equivalence, which follows from results by Grusin 1962, Littman 1967, Adams and Hedberg 1996 and Maz’ya 2011: Theorem For s > 0 , K compact, H − s = { 0 } ⇐ ⇒ cap s ( K ) = 0 , where K cap s ( K ) := inf {� u � 2 H s : u ∈ C ∞ 0 ( R n ) and u ≥ 1 on K } . This allows us to apply well-known results relating cap s ( E ) to dim H ( E ) (see e.g. Adams and Hedberg 1996). Requires relating different set capacities. 16
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