Faber-Krahn Type Inequalities in Inverse Scattering Theory David - PowerPoint PPT Presentation

Faber-Krahn Type Inequalities in Inverse Scattering Theory David Colton colton@math.udel.edu Department of Mathematical Sciences, University of Delaware, USA AIP , Vienna July 20-24, 2009 – p. 1/18

Scattering by a Perfect Conductor in R 2 \ D ∆ u + k 2 u = 0 in R 2 \ D u = u s + e ikx · d u i D u = 0 on ∂D ν � ∂u s u √ r � s ∂r − iku s lim = 0 r →∞ The scattered field u s has the asymptotic behavior u s ( x ) = e ikr � r − 3 / 2 � √ r u ∞ (ˆ x, d ) + O as r → ∞ , where ˆ x = x/ | x | , | d | = 1 , r = | x | and k > 0 is the wave x, d ) is called the far field pattern of the scattered field u s . number. u ∞ (ˆ AIP , Vienna July 20-24, 2009 – p. 2/18

Scattering by a Perfect Conductor We define the far field operator F : L 2 ( S ) → L 2 ( S ) by � ( Fg )(ˆ x ) := u ∞ (ˆ x, d ) g ( d ) ds ( d ) , S is the unit circle . S The far field equation is g ∈ L 2 ( S ) ( Fg )(ˆ x ) = Φ ∞ (ˆ x, z ) , x, z ) = e iπ/ 4 e − ik ˆ x · z √ Φ ∞ (ˆ where 8 πk is the far field pattern of the fundamental solution Φ( x, z ) := i 4 H (1) ( k | x − z | ) . 0 AIP , Vienna July 20-24, 2009 – p. 3/18

The Linear Sampling Method The linear sampling method for determining D from u ∞ (ˆ x, d ) attempts to use regularization methods to solve the far field equation z ∈ R 2 ( Fg )(ˆ x ) = Φ ∞ (ˆ x, z ) , then looks for points z where the norm of g is relatively large. This method can only be justified if k 2 is not a Dirichlet eigenvalue. AIP , Vienna July 20-24, 2009 – p. 4/18

Dirichlet Eigenvalues What happens when k 2 is equal to a Dirichlet eigenvalue? Example: Let D be the unit disk and set z = 0 . Then √ Φ ∞ = γ := e iπ/ 4 / 8 πk and Fg = γ can be solved explicitly to give g ( d ) := γH (1) 0 ( k ) 2 πJ 0 ( k ) . Let k 01 be the first zero of J 0 ( k ) . Then k 2 01 is the first Dirichlet eigenvalue λ 1 , for D and k 2 → λ 1 � g � = ∞ . lim In general the far field equation is not solvable but for z ∈ D it can be shown using a result of Tilo Arens that the regularized solution of Fg = Φ ∞ has relatively large norm for k 2 near λ 1 . AIP , Vienna July 20-24, 2009 – p. 5/18

Dirichlet Eigenvalues Let the scattering object D be the rectangle [ − 0 . 4 , 0 . 4] × [ − 0 . 5 , 0 . 5] . Norm of g Dirichlet b.c. 20 18 16 Faber-Krahn inequality 14 12 λ 1 ≥ πk 2 10 01 8 area D 6 4 2 0 0 2 4 6 8 10 12 14 16 18 20 k Graph of � g � versus k From the Faber-Krahn inequality, a lower bound for the area of D can be determined from a knowledge of the smallest eigenvalue λ 1 . AIP , Vienna July 20-24, 2009 – p. 6/18

Scattering by a Dielectric in R 2 ∆ u + k 2 n ( x ) u = 0 u = u s + u i in R 2 u i D � ∂u s √ r � ∂r − iku s lim = 0 ν u s r →∞ where u i ( x ) := e ikx · d . We assume that n − 1 has compact support D , n ( x ) > 0 for x ∈ D and n is in L ∞ ( D ) . u s again has the asymptotic behavior u s ( x ) = e ikr � r − 3 / 2 � √ r u ∞ (ˆ x, d ) + O AIP , Vienna July 20-24, 2009 – p. 7/18

Transmission Eigenvalues We again consider the far field operator � ( Fg )(ˆ x ) := u ∞ (ˆ x, d ) g ( d ) ds ( d ) S and the corresponding far field equation ( Fg )(ˆ x ) = Φ ∞ (ˆ x, z ) . Instead of � g � becoming large near a Dirichlet eigenvalue, � g � now becomes large when k is a transmission eigenvalue. The linear sampling method for determining D from u ∞ (ˆ x, d ) only works if k is not a transmission eigenvalue. AIP , Vienna July 20-24, 2009 – p. 8/18

Transmission Eigenvalues Definition: k > 0 is a transmission eigenvalue if there exists a nontrivial solution v ∈ L 2 ( D ) , w ∈ L 2 ( D ) , v − w ∈ H 2 ( D ) of the interior transmission problem ∆ w + k 2 n ( x ) w = 0 D in ∆ v + k 2 v = 0 D in w = v ∂D on ∂w ∂ν = ∂v ∂D on ∂ν Remark: Note that if n = 1 the interior transmission problem is degenerate. AIP , Vienna July 20-24, 2009 – p. 9/18

Transmission Eigenvalues Theorem: Let 0 < n ∈ L ∞ ( D ) and either n − 1 ≥ δ > 0 or 1 − n ≥ δ > 0 in D . Then the set of transmission eigenvalues is a discrete set. Colton-Kirsch-P¨ aiv¨ arinta (1989), Rynne-Sleeman (1991). Theorem: Let 0 < n ∈ L ∞ ( D ) and either n − 1 ≥ δ > 0 or 1 − n ≥ δ > 0 in D . Then there exist an infinite number of transmission eigenvalues. P¨ aiv¨ arinta and Sylvester (2008) Cakoni, Gintides, Haddar (to appear) Theorem: Let k be a transmission eigenvalue. If sup D | n ( x ) − 1 | → 0 then k → ∞ . Cakoni-Colton-Monk (2007) AIP , Vienna July 20-24, 2009 – p. 10/18

Numerical Examples Transmission eigenvalues can be determined from the far field data, i.e. values of k for wich the regularized solution of Fg = Φ ∞ becomes relatively large: 1.5 9 1 8 7 0.5 6 Norm of g 5 0 4 3 −0.5 2 1 −1 0 0 0.5 1 1.5 2 2.5 Wave number k −1.5 −1.5 −1 −0.5 0 0.5 1 1.5 n = 16 The scatterer D AIP , Vienna July 20-24, 2009 – p. 11/18

Transmission Eigenvalues We now obtain an analog of the Faber-Krahn inequality for the first transmission eigenvalue k 1 . We again denote the first Dirichlet eigenvalue for D by λ 1 . Theorem: Suppose n − 1 ≥ δ > 0 for D . Then λ 1 k 2 1 > sup D n Colton-P¨ aiv¨ arinta-Sylvester (2007) Open problem: If 1 − n ≥ δ > 0 all that be said is k 2 1 > λ 1 . In our previous example where D is an L shaped obstacle and n = 16 we have that k 1 ≈ 1 . 09 . Since λ 1 ≈ 9 . 65 the above inequality now give the lower bound n > 8 . 1 . Upper bounds for n and improved lower bounds have recently, been obtained by Cakoni and Gintides . AIP , Vienna July 20-24, 2009 – p. 12/18

Transmission Eigenvalues The case when there are regions in D where n = 1 (i.e. cavities) is more D o D delicate. o D Let λ 1 be again the first eigenvalue of n = 1 in D 0 − ∆ in D . n − 1 ≥ δ > 0 in D \ D 0 Theorem: Suppose n − 1 ≥ δ > 0 for x ∈ D \ D 0 and n = 1 in D 0 . Then λ 1 k 2 1 > sup D \ D 0 n. Cakoni-Colton-Haddar (to appear) AIP , Vienna July 20-24, 2009 – p. 13/18

Transmission Eigenvalues Theorem: Let k be a transmission eigenvalue. Then if either a) sup D \ D 0 ( n ( x ) − 1) = 0 or b) area ( D \ D 0 ) → 0 then k → ∞ . Cakoni-Cay¨ oren-Colton (2008) 1 1 1 0.5 0.5 0.5 0 0 0 −0.5 −0.5 −0.5 −1 −1 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 (a) (b) (c) (a) is dielectric with n = 16 , (b) and (c) show medium with a circular void of radius 0 . 1 AIP , Vienna July 20-24, 2009 – p. 14/18

Transmission Eigenvalues 80 X: 4.17 70 Y: 74.78 60 50 X: 1.73 Y: 41.91 ||g(.)|| 40 X: 3.44 X: 1.86 30 X: 1.83 Y: 26.95 Y: 26.3 Y: 24.89 X: 3.71 Y: 20.42 20 10 0 1.5 2 2.5 3 3.5 4 4.5 k Black dashed line corresponds to square without cavity (a), black solid line corresponds to the square with the cavity (b), and gray solid line corresponds to the square with the cavity (c). AIP , Vienna July 20 -24, 2009 – p. 15/18

Near Field Data in R 2 \ { x 0 } ∆ u + k 2 n ( x ) u = 0 Λ δΩ for x ∈ R 2 u = Φ( · , x 0 ) + u s D O � ∂u s √ r � D ∂r − iku s lim = 0 Ω r →∞ Here the initial field is a point source located at x 0 ∈ Λ and the Cauchy data of the total field is measured on ∂ Ω , n − 1 has support D and there may be cavities D 0 ⊂ D where n = 1 . AIP , Vienna July 20 -24, 2009 – p. 16/18

Near Field Data Define v ∈ H 1 (Ω) : ∆ v + k 2 v = 0 � � H (Ω) := The reciprocity gap operator R : H (Ω) → L 2 (Λ) is defined by ∂ν − v∂u ( · , x 0 ) � u ( · , x 0 ) ∂v � � ( R v )( x 0 ) := ds ∂ν Ω Theorem: R : H (Ω) → L 2 (Λ) is injective if k is not a transmission eigenvalue for D . Cakoni-Cay¨ oren-Colton (2008) AIP , Vienna July 20 -24, 2009 – p. 17/18

Near Field Data Instead of the whole space H (Ω) we can consider the dense set of H consisting of all Herglotz wave functions � e ikx · d g ( d ) ds ( d ) , g ∈ L 2 ( S ) . v g ( x ) := S Assuming that z ∈ D and defining the near field operator N : L 2 ( S ) → L 2 (Λ) by N g := R v g we expect that the regularized solution of ( N g )( x 0 ) = ( R Φ( · z ))( x 0 ) , x 0 ∈ Λ will be large if k is a transmission eigenvalue. Remark: Other dense sets of H (Ω) can be used yielding different near field operators. AIP , Vienna July 20 -24, 2009 – p. 18/18