Jacobians Giovanni Alessandrini Coupled physics inverse problems Introduction The problem and Jacobians of σ -harmonic mappings Qualitative results in 2D Quantitative estimates in 2D Giovanni Alessandrini 3D End Università degli Studi di Trieste Geometric Properties for Parabolic and Elliptic PDE’s 4th Italian-Japanese Workshop Palinuro May 2015, 25–29
Jacobians Introduction Giovanni Alessandrini Introduction Since the ’80s, a dominant theme in Inverse Problems has The problem been: Qualitative results in 2D To image the interior of an object from measurements taken Quantitative in its exterior. estimates in 2D Consider the (direct) elliptic Dirichlet problem of finding a 3D weak solution u to End � div ( σ ∇ u ) = 0 in Ω , u = ϕ on ∂ Ω , where Ω is a bounded connected open set in R n , n ≥ 2, and � � σ = σ ij ( x ) satisfies uniform ellipticity for every x , ξ ∈ R 2 , K − 1 | ξ | 2 σ ( x ) ξ · ξ ≥ , for every x , ξ ∈ R 2 . σ − 1 ( x ) ξ · ξ K − 1 | ξ | 2 ≥ ,
Jacobians Introduction Giovanni Alessandrini Introduction Since the ’80s, a dominant theme in Inverse Problems has The problem been: Qualitative results in 2D To image the interior of an object from measurements taken Quantitative in its exterior. estimates in 2D Consider the (direct) elliptic Dirichlet problem of finding a 3D weak solution u to End � div ( σ ∇ u ) = 0 in Ω , u = ϕ on ∂ Ω , where Ω is a bounded connected open set in R n , n ≥ 2, and � � σ = σ ij ( x ) satisfies uniform ellipticity for every x , ξ ∈ R 2 , K − 1 | ξ | 2 σ ( x ) ξ · ξ ≥ , for every x , ξ ∈ R 2 . σ − 1 ( x ) ξ · ξ K − 1 | ξ | 2 ≥ ,
Jacobians Introduction Giovanni Alessandrini Introduction The problem Qualitative results in 2D The Calderón’s inverse problem (EIT) is: Quantitative estimates in Find σ , given all pairs of Cauchy data 2D 3D ( u | ∂ Ω , σ ∇ u · ν | ∂ Ω ) . End . Main problems: � � • If σ = σ ij ( x ) , nonuniqueness (Tartar ’84). � � • If σ = γ ( x ) δ ij , instability (Mandache ’01).
Jacobians Introduction Giovanni Alessandrini Introduction The problem Qualitative results in 2D The Calderón’s inverse problem (EIT) is: Quantitative estimates in Find σ , given all pairs of Cauchy data 2D 3D ( u | ∂ Ω , σ ∇ u · ν | ∂ Ω ) . End . Main problems: � � • If σ = σ ij ( x ) , nonuniqueness (Tartar ’84). � � • If σ = γ ( x ) δ ij , instability (Mandache ’01).
Jacobians Introduction Giovanni Alessandrini Introduction The problem Qualitative results in 2D The Calderón’s inverse problem (EIT) is: Quantitative estimates in Find σ , given all pairs of Cauchy data 2D 3D ( u | ∂ Ω , σ ∇ u · ν | ∂ Ω ) . End . Main problems: � � • If σ = σ ij ( x ) , nonuniqueness (Tartar ’84). � � • If σ = γ ( x ) δ ij , instability (Mandache ’01).
Jacobians Introduction Giovanni Alessandrini Introduction The problem Qualitative results in 2D Coupled physics: to combine electrical measurements with Quantitative other physical modalities. estimates in 2D • EIT + Magnetic Resonance (MREIT): interior values of 3D | σ ∇ u | (Kim, Kwon, Seo, Yoon ’02). End • EIT + Ultrasonic waves (UMEIT): by focusing ultrasonic waves on a tiny spot near x ∈ Ω and by applying various boundary potentials ϕ i it is possible to detect the local energies H ij = σ ∇ u i · ∇ u j ( x ) (Ammari et al. ’08).
Jacobians Introduction Giovanni Alessandrini Introduction The problem Qualitative results in 2D Coupled physics: to combine electrical measurements with Quantitative other physical modalities. estimates in 2D • EIT + Magnetic Resonance (MREIT): interior values of 3D | σ ∇ u | (Kim, Kwon, Seo, Yoon ’02). End • EIT + Ultrasonic waves (UMEIT): by focusing ultrasonic waves on a tiny spot near x ∈ Ω and by applying various boundary potentials ϕ i it is possible to detect the local energies H ij = σ ∇ u i · ∇ u j ( x ) (Ammari et al. ’08).
Jacobians Introduction Giovanni Alessandrini Introduction The problem Qualitative results in 2D Coupled physics: to combine electrical measurements with Quantitative other physical modalities. estimates in 2D • EIT + Magnetic Resonance (MREIT): interior values of 3D | σ ∇ u | (Kim, Kwon, Seo, Yoon ’02). End • EIT + Ultrasonic waves (UMEIT): by focusing ultrasonic waves on a tiny spot near x ∈ Ω and by applying various boundary potentials ϕ i it is possible to detect the local energies H ij = σ ∇ u i · ∇ u j ( x ) (Ammari et al. ’08).
Jacobians The problem Giovanni Alessandrini Introduction The problem Qualitative results in 2D � � Monard and Bal ’12, ’13: reconstruction of σ from H ij , Quantitative provided U = ( u 1 , . . . , u n ) is a σ –harmonic mapping (i.e.: a estimates in 2D n –tuple of solutions) such that 3D End det DU > 0 , in Ω . Question: Can we find Φ = ( ϕ 1 , . . . , ϕ n ) , independent of σ , such that det DU > 0 everywhere?
Jacobians The problem Giovanni Alessandrini Introduction The problem Qualitative results in 2D � � Monard and Bal ’12, ’13: reconstruction of σ from H ij , Quantitative provided U = ( u 1 , . . . , u n ) is a σ –harmonic mapping (i.e.: a estimates in 2D n –tuple of solutions) such that 3D End det DU > 0 , in Ω . Question: Can we find Φ = ( ϕ 1 , . . . , ϕ n ) , independent of σ , such that det DU > 0 everywhere?
Jacobians The problem Giovanni Alessandrini Introduction The problem Qualitative results in 2D � � Monard and Bal ’12, ’13: reconstruction of σ from H ij , Quantitative provided U = ( u 1 , . . . , u n ) is a σ –harmonic mapping (i.e.: a estimates in 2D n –tuple of solutions) such that 3D End det DU > 0 , in Ω . Question: Can we find Φ = ( ϕ 1 , . . . , ϕ n ) , independent of σ , such that det DU > 0 everywhere?
Jacobians n = 2. The Classical Results Giovanni Alessandrini Let Ω ⊂ R 2 be a Jordan domain and let Introduction The problem Φ = ( ϕ 1 , ϕ 2 ) : ∂ Ω → ∂ G , Qualitative results in 2D Quantitative be a homeomorphism. Consider estimates in 2D � ∆ U = 0 , in Ω , 3D End U = Φ , on ∂ Ω . Theorem ( H. Kneser ’26) If G is convex, then U is a homeomorphism of Ω onto G. Posed as a problem by Radó (’26), rediscovered by Choquet (’45). Theorem (H. Lewy ’36) If U : Ω → R 2 is a harmonic homeomorphism, then it is a diffeomorphism.
Jacobians n = 2. The Classical Results Giovanni Alessandrini Let Ω ⊂ R 2 be a Jordan domain and let Introduction The problem Φ = ( ϕ 1 , ϕ 2 ) : ∂ Ω → ∂ G , Qualitative results in 2D Quantitative be a homeomorphism. Consider estimates in 2D � ∆ U = 0 , in Ω , 3D End U = Φ , on ∂ Ω . Theorem ( H. Kneser ’26) If G is convex, then U is a homeomorphism of Ω onto G. Posed as a problem by Radó (’26), rediscovered by Choquet (’45). Theorem (H. Lewy ’36) If U : Ω → R 2 is a harmonic homeomorphism, then it is a diffeomorphism.
Jacobians n = 2. The Classical Results Giovanni Alessandrini Let Ω ⊂ R 2 be a Jordan domain and let Introduction The problem Φ = ( ϕ 1 , ϕ 2 ) : ∂ Ω → ∂ G , Qualitative results in 2D Quantitative be a homeomorphism. Consider estimates in 2D � ∆ U = 0 , in Ω , 3D End U = Φ , on ∂ Ω . Theorem ( H. Kneser ’26) If G is convex, then U is a homeomorphism of Ω onto G. Posed as a problem by Radó (’26), rediscovered by Choquet (’45). Theorem (H. Lewy ’36) If U : Ω → R 2 is a harmonic homeomorphism, then it is a diffeomorphism.
Jacobians n = 2. Variable coefficients. Giovanni Alessandrini Introduction � div ( σ ∇ U ) = 0 , The problem in Ω , Qualitative U = Φ , on ∂ Ω . results in 2D Quantitative Let estimates in 2D Φ : ∂ Ω → ∂ G , 3D be a homeomorphism, and let G be convex. End Theorem (Bauman-Marini-Nesi ’01) Assume Ω , G be C 1 ,α –smooth, σ ∈ C α and Φ a C 1 ,α diffeomorphism. � div ( σ ∇ U ) = 0 , in Ω , U = Φ , on ∂ Ω . then U : Ω → G is a diffeomorphism.
Jacobians n = 2. Variable, nonsmooth, Giovanni Alessandrini coefficients. Introduction The problem Qualitative results in 2D Quantitative Theorem (A.-Nesi ’01) estimates in 2D If we only assume σ ∈ L ∞ , then U is a homeomorphism of 3D Ω onto G. End Theorem (A., Nesi ’01) If U : Ω → R 2 is a σ − harmonic homeomorphism, then | det DU | > 0 a.e. . In fact, | det DU | is a Muckenhoupt weight (A., Nesi ’09) .
Jacobians n = 2. Variable, nonsmooth, Giovanni Alessandrini coefficients. Introduction The problem Qualitative results in 2D Quantitative Theorem (A.-Nesi ’01) estimates in 2D If we only assume σ ∈ L ∞ , then U is a homeomorphism of 3D Ω onto G. End Theorem (A., Nesi ’01) If U : Ω → R 2 is a σ − harmonic homeomorphism, then | det DU | > 0 a.e. . In fact, | det DU | is a Muckenhoupt weight (A., Nesi ’09) .
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