Corners Scattering and Inverse Scattering Jingni Xiao Department of Mathematics Rutgers University Joint with Emilia Bl˚ asten, Fioralba Cakoni and Hongyu Liu IAS, HKUST May 22, 2019 1 / 27
Introduction Introduction 1 Corner of Media Scatters 2 Applications 3 Shape Determination Approximation by Herglotz Wave Functions Sketch of the Proof 4 Corner of Sources Scatter - EM case 5 Concluding remarks 6 2 / 27
Introduction Scattering u in ( x ) D u sc ( x ) 3 / 27
Introduction Scattering u in ( x ) D u sc ( x ) Inverse scattering Invisibility 3 / 27
Introduction Scattering u in ( x ) D u sc ( x ) Inverse scattering Invisibility Question: ∃ { [ D ; u in ] } , s.t. u sc ≡ 0 in D c (or equiv., u ∞ ≡ 0 ) ? 3 / 27
Introduction A resulting problem u in ( x ) D ∆ u in + k 2 u in = 0 u sc ( x ) = 0 Interior Transmission Eigenvalue Problem ∇ · a ∇ u + k 2 cu = 0 , ∆ v + k 2 v = 0 , in D, u = v, a∂ ν u = ∂ ν v, on ∂D. 4 / 27
Introduction A resulting problem u in ( x ) D ∆ u in + k 2 u in = 0 u sc ( x ) = 0 Interior Transmission Eigenvalue Problem ∇ · a ∇ u + k 2 cu = 0 , ∆ v + k 2 v = 0 , in D, u = v, a∂ ν u = ∂ ν v, on ∂D. Equivalence? 4 / 27
Introduction Some known result Certain shapes: invisible under certain probing waves: Spherically stratified media: [Colton-Monk ’88] 5 / 27
Introduction Some known result Certain shapes: invisible under certain probing waves: Spherically stratified media: [Colton-Monk ’88] Media with corners/edges: ∆ + k 2 q : [Bl˚ asten-P¨ aiv¨ arinta-Sylvester ’14, P¨ aiv¨ arinta-Salo-Vesalainen ’17, Elschner-Hu ’15&’18, Hu-Salo-Vesalainen ’16, Bl˚ asten-Vesalainen ’18, etc.] Source problem: [Bl˚ asten ’18] Maxwell’s equations: [Liu-X. ’17 (Right corner)] 5 / 27
Introduction Some known result Certain shapes: invisible under certain probing waves: Spherically stratified media: [Colton-Monk ’88] Media with corners/edges: ∆ + k 2 q : [Bl˚ asten-P¨ aiv¨ arinta-Sylvester ’14, P¨ aiv¨ arinta-Salo-Vesalainen ’17, Elschner-Hu ’15&’18, Hu-Salo-Vesalainen ’16, Bl˚ asten-Vesalainen ’18, etc.] Source problem: [Bl˚ asten ’18] Maxwell’s equations: [Liu-X. ’17 (Right corner)] Cloaking: invisible for all probing fields Anisotropic and singular medium [Greenleaf-Lassas-Uhlmann ’03, etc.] 5 / 27
Corner of Media Scatters Introduction 1 Corner of Media Scatters 2 Applications 3 Shape Determination Approximation by Herglotz Wave Functions Sketch of the Proof 4 Corner of Sources Scatter - EM case 5 Concluding remarks 6 6 / 27
Corner of Media Scatters The scattering problem Consider the scattering problem ∇ · a ∇ u + k 2 cu = 0 in R n , ∆ u in + k 2 u in = 0 in R n , x · ∇ u sc − iku sc = o ( | x | n − 1 2 ) , ˆ | x | → ∞ . where a, c ∈ L ∞ ( R n ) , a = ( a ij ) symm. and positive definite, and a − I and c − 1 compactly supported, 7 / 27
Corner of Media Scatters The scattering problem Consider the scattering problem ∇ · a ∇ u + k 2 cu = 0 in R n , ∆ u in + k 2 u in = 0 in R n , x · ∇ u sc − iku sc = o ( | x | n − 1 2 ) , ˆ | x | → ∞ . where a, c ∈ L ∞ ( R n ) , a = ( a ij ) symm. and positive definite, and a − I and c − 1 compactly supported, Corner scattering: Convex corner(s) at the support of a − I or/and that of c − 1 ; Around the corner, a is locally W 3 , 1+ ε and scalar, and/or c is locally W 1 , 1+ ε . 7 / 27
Corner of Media Scatters Corner Scatters Theorem 1 ([Cakoni-X ’19 ]) (Roughly) Corners of c with a jump across the corner always scatter any incident wave, when a − 1 vanishes to the second order at the same corner. 8 / 27
Corner of Media Scatters Corner Scatters Theorem 1 ([Cakoni-X ’19 ]) (Roughly) Corners of c with a jump across the corner always scatter any incident wave, when a − 1 vanishes to the second order at the same corner. Figure: dotted: supp ( c − 1) ; colored: supp ( a − I ) . ( ρ ( x ) − 1) γ − 1 / 2 ( x ) = ρ 0 + O ( | x − x 0 | σ ) , ( γ ( x ) − 1) γ − 1 / 2 ( x ) = O ( | x − x 0 | 2+ σ ) , where ρ 0 = const � = 0 , ( γ, ρ ) | C ε ( x 0 ) = ( a, c ) | C ε ( x 0 ) in W 3 , 1+ ε × W 1 , 1+ ε . 8 / 27
Corner of Media Scatters Corner Scatters Theorem 1 ([Cakoni-X ’19 ]) (Roughly) Corners of c with a jump across the corner always scatter any incident wave, when a − 1 vanishes to the second order at the same corner. Figure: dotted: supp ( c − 1) ; colored: supp ( a − I ) . ( ρ ( x ) − 1) γ − 1 / 2 ( x ) = ρ 0 + O ( | x − x 0 | σ ) , ( γ ( x ) − 1) γ − 1 / 2 ( x ) = O ( | x − x 0 | 2+ σ ) , where ρ 0 = const � = 0 , ( γ, ρ ) | C ε ( x 0 ) = ( a, c ) | C ε ( x 0 ) in W 3 , 1+ ε × W 1 , 1+ ε . Generalization of the previous results concerning the operator ∆ + k 2 q . 8 / 27
Corner of Media Scatters Corner Scatters Other cases: Corners of c − 1 with a jump across the corner ( ρ 0 � = 0 ). If a − 1 vanishes to the first order at the same corner, i.e., ( γ ( x ) − 1) γ − 1 / 2 ( x ) = O ( | x − x 0 | 1+ σ ) , then the corner scatters all incident fields u in for which N T u in = N T ∇ u in . 9 / 27
Corner of Media Scatters Corner Scatters Other cases: Corners of c − 1 with a jump across the corner ( ρ 0 � = 0 ). If a − 1 vanishes to the first order at the same corner, i.e., ( γ ( x ) − 1) γ − 1 / 2 ( x ) = O ( | x − x 0 | 1+ σ ) , then the corner scatters all incident fields u in for which N T u in = N T ∇ u in . If a − 1 vanishes at the corner, i.e., ( γ ( x ) − 1) γ − 1 / 2 ( x ) = O ( | x − x 0 | σ ) , then the corner scatters all incident fields u in satisfying u in ( x 0 ) � = 0 and ∇ u in ( x 0 ) = 0 . 9 / 27
Corner of Media Scatters Corner Scatters Other cases: Corners of c − 1 with a jump across the corner ( ρ 0 � = 0 ). If a − 1 vanishes to the first order at the same corner, i.e., ( γ ( x ) − 1) γ − 1 / 2 ( x ) = O ( | x − x 0 | 1+ σ ) , then the corner scatters all incident fields u in for which N T u in = N T ∇ u in . If a − 1 vanishes at the corner, i.e., ( γ ( x ) − 1) γ − 1 / 2 ( x ) = O ( | x − x 0 | σ ) , then the corner scatters all incident fields u in satisfying u in ( x 0 ) � = 0 and ∇ u in ( x 0 ) = 0 . Corners of a − 1 with a jump across the corner, i.e., ( γ ( x ) − 1) γ − 1 / 2 ( x ) = γ 0 + O ( | x − x 0 | σ ) with γ 0 = const � = 0 . If the corner is of aperture κπ , κ ∈ (0 , 1) , then it scatters all incident fields u in for which N T ∇ u in � = κl − 1 with some l ∈ N + . e.g.: ∇ u in ( x 0 ) � = 0 . 9 / 27
Applications Introduction 1 Corner of Media Scatters 2 Applications 3 Shape Determination Approximation by Herglotz Wave Functions Sketch of the Proof 4 Corner of Sources Scatter - EM case 5 Concluding remarks 6 10 / 27
Applications Shape Determination Shape Determination Recall: forward scattering: ∇ · a ∇ u + k 2 cu = 0 in R n , where a − I and c − 1 are compactly supported and u = u in + u sc with u in the incident field and u sc the scattered field. u sc ( x ) | B e → u ∞ (ˆ R ← x ) 11 / 27
Applications Shape Determination Shape Determination Recall: forward scattering: ∇ · a ∇ u + k 2 cu = 0 in R n , where a − I and c − 1 are compactly supported and u = u in + u sc with u in the incident field and u sc the scattered field. u sc ( x ) | B e → u ∞ (ˆ R ← x ) The inverse problem: To uniquely determine the convex hull of c − 1 or/and a − 1 from the far-field measurement u ∞ or from the near-field (scattered) measurement u sc | ∂B 2 R . 11 / 27
Applications Shape Determination Unique determination from a single measurement Assumptions (roughly): 1 The convex hull D of supp ( c − 1) is a bounded polygon, supp ( a − I ) � D . 2 Local regularity of c and a ( W 1 , 1+ ε and W 3 , 1+ ε ) around corners of D . 3 c − 1 has a jump while a − 1 vanishes to the second order at each corner. 12 / 27
Applications Shape Determination Unique determination from a single measurement Assumptions (roughly): 1 The convex hull D of supp ( c − 1) is a bounded polygon, supp ( a − I ) � D . 2 Local regularity of c and a ( W 1 , 1+ ε and W 3 , 1+ ε ) around corners of D . 3 c − 1 has a jump while a − 1 vanishes to the second order at each corner. Theorem 2 (Cakoni-X ’19) D can be uniquely determined from the far-field (or scattered) data u ∞ corresponding to a single incident field. Figure: dotted: supp ( c − 1) ; colored: supp ( a − I ) . 12 / 27
Applications Approximation by Herglotz Wave Functions ITEP and Herglotz functions Interior transmission eigenvalue problem: ∇ · a ∇ u + k 2 cu = 0 , ∆ v + k 2 v = 0 , in D, u = v, a∂ ν u = ∂ ν v, on ∂D. Herglotz wave functions: � S n − 1 g ( d ) e ikx · d ds d , g ∈ L 2 ( S n − 1 ) . v g ( x ) = Fact ([Cakoni-Colton-Haddar ’16 (Book), etc.]): the set of Herglotz wave functions is dense in { v ∈ H 1 (Ω) : ∆ v + k 2 v = 0 in Ω } . 13 / 27
Applications Approximation by Herglotz Wave Functions Blow-up of the kernel To approximate v (eigenfunctions) by v g . Question: Do the Herglotz kernels g ε keep bounded when v g ε → v ? 14 / 27
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