Analysis of the anomalous localized resonance Hyundae Lee(Inha University, Korea) Joint work with Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Graeme Milton. UCI, June 22, 2012 A conference on inverse problems in honor of Gunther Uhlmann Analysis of the anomalous localized resonance Hyundae Lee(Inha University, Korea) Joint work with Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Graeme Milton.
Outline • Introduction • Integral operators and its symmetry • Spectral analysis of ALR • ALR in annulus region Analysis of the anomalous localized resonance Hyundae Lee(Inha University, Korea) Joint work with Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Graeme Milton.
Surface plasmon Let � 1 in { ( x , y ) : y ≥ 0 } , ǫ = − 1 in { ( x , y ) : y < 0 } . Consider in R 2 . ∇ · ǫ ∇ u = 0 Then one solution is � e − y + ix in { ( x , y ) : y ≥ 0 } , u = e y + ix in { ( x , y ) : y < 0 } . Analysis of the anomalous localized resonance Hyundae Lee(Inha University, Korea) Joint work with Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Graeme Milton.
Let Ω be a smooth domain in R 2 and let D ⊂ Ω. The permittivity distribution in R 2 is given by in R 2 \ Ω , 1 ǫ δ = − 1 + i δ in Ω \ D , 1 in D . 1 1 − 1 + i δ f Analysis of the anomalous localized resonance Hyundae Lee(Inha University, Korea) Joint work with Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Graeme Milton.
Problem For a given function f compactly supported in R 2 satisfying � R 2 fdx = 0 , we consider the following equation: in R 2 , ∇ · ǫ δ ∇ V δ = f with decaying condition V δ ( x ) → 0 as | x | → ∞ . Since the equation degenerates as δ → 0, we can expect some singular behavior of the solution, depending on the source term f . Analysis of the anomalous localized resonance Hyundae Lee(Inha University, Korea) Joint work with Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Graeme Milton.
Milton-Nicorovici(2006) 4 4 2 2 0 0 � 2 � 2 � 4 � 4 � 5 � 2.5 0 2.5 5 7.5 10 � 5 � 2.5 0 2.5 5 7.5 10 Figure: Anomalous resonance, Milton et al (2006). • Energy concentration near interfaces, depending on the location of source. • Associated with the cloaking effect of polarizable dipole. • Generalized to a small inclusion with a specific boundary condition by Bouchitt´ e and B. Schweizer(2010). Analysis of the anomalous localized resonance Hyundae Lee(Inha University, Korea) Joint work with Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Graeme Milton.
Numerical simulation by Bruno-Linter(2007). • There is some cloaking effect even in the presence of a small dielectric inclusion, not perfect. • Blow-up may not depend on the location of the source in a layer of general shape. Analysis of the anomalous localized resonance Hyundae Lee(Inha University, Korea) Joint work with Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Graeme Milton.
A fundamental problem is to find a region Ω ∗ containing Ω such that if f is supported in Ω ∗ \ Ω, then � δ |∇ V δ | 2 dx → ∞ as δ → 0 . Ω \ D • Such a region Ω ∗ \ Ω is called the anomalous resonance region or cloaking Ω \ D δ |∇ V δ | 2 dx is a part of the absorbed energy. region. The quantity � • The blow-up of the energy may or may not occur depending on f . So the problem is not only finding the anomalous resonance region Ω ∗ \ Ω but also characterizing those source terms f which actually make the anomalous resonance happen. Analysis of the anomalous localized resonance Hyundae Lee(Inha University, Korea) Joint work with Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Graeme Milton.
Relation to cloaking Suppose f is a polarizable dipole at x 0 , i.e. , V δ ( x ) = U δ ( x ) + A δ · ∇ G ( x − x 0 ) , A δ = k ∇ U δ ( x 0 ) , for some given coefficient k . If ALR happens, then we should have A δ → 0 as δ → 0 . � Ω \ D δ |∇ V δ | 2 dx blows up, which is not physical. Otherwise Analysis of the anomalous localized resonance Hyundae Lee(Inha University, Korea) Joint work with Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Graeme Milton.
Let F be the Newtonian potential of f , i.e. , � x ∈ R 2 . F ( x ) = R 2 G ( x − y ) f ( y ) dy , Then F satisfies ∆ F = f in R 2 , and the solution V δ may be represented as V δ ( x ) = F ( x ) + S Γ i [ ϕ i ]( x ) + S Γ e [ ϕ e ]( x ) for some functions ϕ i ∈ L 2 0 (Γ i ) and ϕ e ∈ L 2 0 (Γ e ) ( L 2 0 is the collection of all square integrable functions with the integral zero). The transmission conditions along the interfaces Γ e and Γ i satisfied by V δ read ( − 1 + i δ ) ∂ V δ + = ∂ V δ � � on Γ i � � ∂ν ∂ν � � − ∂ V δ + = ( − 1 + i δ ) ∂ V δ � � on Γ e . � � ∂ν ∂ν � � − Analysis of the anomalous localized resonance Hyundae Lee(Inha University, Korea) Joint work with Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Graeme Milton.
Using the jump formula for the normal derivative of the single layer potentials, the pair of potentials ( ϕ i , ϕ e ) is the solution to − ∂ ∂ F z δ I − K ∗ ∂ν i S Γ e Γ i � ϕ i � ∂ν i = . ∂ ϕ e − ∂ F z δ I + K ∗ ∂ν e S Γ i Γ e ∂ν e on L 2 0 (Γ i ) × L 2 0 (Γ e ), where we set i δ z δ = 2(2 − i δ ) . Note that the operator can be viewed as a compact perturbation of the operator z δ I − K ∗ � � 0 Γ i . z δ I + K ∗ 0 Γ e Analysis of the anomalous localized resonance Hyundae Lee(Inha University, Korea) Joint work with Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Graeme Milton.
• We now recall Kellogg’s result on the spectrums of K ∗ Γ i and K ∗ Γ e . The eigenvalues of K ∗ Γ i and K ∗ Γ e lie in the interval ] − 1 2 , 1 2 ]. • Observe that z δ → 0 as δ → 0 and that there are sequences of eigenvalues of K ∗ Γ i and K ∗ Γ e approaching to 0 since K ∗ Γ i and K ∗ Γ e are compact. So 0 is the essential singularity of the operator valued meromorphic function λ ∈ C �→ ( λ I + K ∗ Γ e ) − 1 . This causes a serious difficulty in dealing with (11). • We emphasize that K ∗ Γ e is not self-adjoint in general. In fact, K ∗ Γ e is self-adjoint only when Γ e is a circle or a sphere. Analysis of the anomalous localized resonance Hyundae Lee(Inha University, Korea) Joint work with Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Graeme Milton.
Properties of K ∗ Let H = L 2 (Γ i ) × L 2 (Γ e ). Let the Neumann-Poincar´ e-type operator K ∗ : H → H be defined by − ∂ −K ∗ ∂ν i S Γ e Γ i K ∗ := . ∂ K ∗ ∂ν e S Γ i Γ e Then the integral equation can be written as ( z δ I + K ∗ )Φ δ = g and the L 2 -adjoint of K ∗ , K , is given by � � −K Γ i D Γ e K = . −D Γ i K Γ e We may check that the spectrum of K ∗ lies in the interval [ − 1 / 2 , 1 / 2]. Analysis of the anomalous localized resonance Hyundae Lee(Inha University, Korea) Joint work with Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Graeme Milton.
Let S be given by � S Γ i � S Γ e S = . S Γ i S Γ e • The operator − S is self-adjoint and − S ≥ 0 on H . • The Calder´ on’s identity is generalized. SK ∗ = KS , i.e. , SK ∗ is self-adjoint. • K ∗ ∈ C 2 ( H ), Schatten-von Neumann class of compact operators. Analysis of the anomalous localized resonance Hyundae Lee(Inha University, Korea) Joint work with Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Graeme Milton.
We recall the result of Khavinson et al (2007) Let M ∈ C p ( H ). If there exists a strictly positive bounded operator R such that R 2 M is self adjoint, then there is a bounded self-adjoint operator A ∈ C p ( H ) such that AR = RM . Theorem There exists a bounded self-adjoint operator A ∈ C 2 ( H ) such that √ √ − SK ∗ . − S = A Analysis of the anomalous localized resonance Hyundae Lee(Inha University, Korea) Joint work with Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Graeme Milton.
Limiting properties of the solution • ALR occurs if and only if � � 2 � � δ |∇ V δ | 2 dx ≈ δ � ∇ ( S Γ i [ ϕ δ i ] + S Γ e [ ϕ δ e ]) dx → ∞ as δ → ∞ . � � � Ω \ D Ω \ D • One can use √ √ − SK ∗ − S = A to obtain � 2 dx = − 1 � � 2 � Φ δ , S Φ δ � + � K ∗ Φ δ , S Φ δ � � ∇ ( S Γ i [ ϕ δ i ] + S Γ e [ ϕ δ e ]) � � � Ω \ D √ √ √ √ = 1 2 � − S Φ δ , − S Φ δ � − � A − S Φ δ , − S Φ δ � . Analysis of the anomalous localized resonance Hyundae Lee(Inha University, Korea) Joint work with Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Graeme Milton.
Since A is self-adjoint, we have an orthogonal decomposition H = Ker A ⊕ ( Ker A ) ⊥ , and ( Ker A ) ⊥ = Range A . Let P and Q = I − P be the orthogonal projections from H onto Ker A and ( Ker A ) ⊥ , respectively. Let λ 1 , λ 2 , . . . with | λ 1 | ≥ | λ 2 | ≥ . . . be the nonzero eigenvalues of A and Ψ n be the corresponding (normalized) eigenfunctions. Since A ∈ C 2 ( H ), we have ∞ � λ 2 n < ∞ , n =1 and ∞ � A Φ = λ n � Φ , Ψ n � Ψ n , Φ ∈ H n =1 Analysis of the anomalous localized resonance Hyundae Lee(Inha University, Korea) Joint work with Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Graeme Milton.
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