Inverse boundary problems for elliptic PDE and best approximation by analytic functions Juliette Leblond Sophia-Antipolis, France team APICS Joint work with L. Baratchart (team APICS), Y. Fischer (team Magique3D, INRIA Bordeaux)
Overview • Boundary value problems Dirichlet, Cauchy • Normed spaces of generalized analytic functions Hardy • Application: a physical free boundary problem plasma • Conclusion
Conductivity equation Let Ω ⊂ R 2 with smooth boundary Γ = ∂ Ω (H¨ older or Dini-smooth) Ω simply connected: Ω ≃ disk D , Γ ≃ circle T ≃ conformally or annular: Ω ≃ A , Γ ≃ T ∪ ̺ T (also in multiply connected domains) 0 < ̺ < 1 Conductivity coefficient σ Lipschitz smooth function in Ω (known) Consider solutions u to (u): div ( σ grad u ) = div ( σ ∇ u ) = 0 in Ω (u) distributional sense 0 < c ≤ σ ≤ C second order elliptic equation ∆ u + ∇ (log σ ) . ∇ u = 0
Boundary value problems - Cauchy (inverse) problem: | I | , | J | > 0 partial overdetermined boundary data Given measures u and σ ∂ n u on I ⊂ Γ of a solution u to (u), recover u , σ ∂ n u on J = Γ \ I (u): div ( σ ∇ u ) = 0 in Ω (or ∂ n u ) n outer unit normal σ given R ( I ), ψ I ∈ W − 1 , 2 pair of Dirichlet-Neumann data ( φ I , ψ I ) on I , φ I ∈ L 2 ( I )... compatibility... R - Dirichlet (direct) problem: Given measures of u on Γ, recover u in Ω (and σ∂ n u , on Γ) well-posed for Dirichlet data φ ∈ L 2 R (Γ)... (already for smooth data) L 2 boundary data � smooth conductivity σ , tradeoff practically: pointwise corrupted boundary measurements
Ω = A I = T , J = ̺ T J u, ∂ n u u, ∂ n u ? Ω = A u, u, ∂ n u ∂ n u ? I ̺ T T Ω = D , I ⊂ T , J = T \ I
σ -harmonic conjugation Generalized Cauchy-Riemann equations: for Ω = D u solution to (u): div ( σ ∇ u ) = 0 = ⇒ ∃ v such that in Ω: � ∂ x v = − σ∂ y u � 1 � whence div σ ∇ v = 0 ∂ y v = σ∂ x u Function v : σ -conjugated to u v unique up to additive constant If u solution to (u) and its σ -conjugated v have L 2 (Γ) trace, then Cauchy-Riemann equations hold up to boundary Γ: ∂ θ v = σ∂ n u ∂ θ tangential derivative for Ω = A : ∃ v if compatibility boundary condition
Generalized analytic functions In Ω ≃ D ⊂ R 2 ≃ C complex plane 1 1 ( ∂ x − i ∂ y ) , ¯ X = ( x , y ) ≃ z = x + iy , ∂ = ∂ z = ∂ = ∂ ¯ z = ( ∂ x + i ∂ y ) 2 2 u solution to (u): div ( σ ∇ u ) = 0 ∇ ≃ ¯ ∂ , div ≃ Re ∂ ⇔ f = u + i v satisfies conjugated Beltrami equation pseudoanalytic ¯ ∂ f = ν∂ f (f) for ν = 1 − σ 1 + σ ∈ W 1 , ∞ (Ω) , | ν | ≤ κ < 1 in Ω f solution to (f) ⇐ ⇒ u = Re f solution to (u) (f) conformally invariant ( R -linear, first order) ∂ g = ν∂ g , quasi-conformal map. � = C -linear Beltrami equation: ¯ f ( z , ¯ z ), u ( x , y ), v ( x , y ) in Ω ≃ A , compatibility condition needed for ⇐
Harmonic and analytic functions Generalization of homogeneous situations σ = cst � σ = 1, ν = 0 Holomorphic / complex analytic functions ¯ ∂ F = 0 in D ⊂ C : Ω = D unit disc or Ω ≃ D conformally equivalent 1 1 ( ∂ x − i ∂ y ) , ¯ X = ( x , y ) ≃ z = x + iy , ∂ = ∂ z = ∂ = ∂ ¯ z = ( ∂ x + i ∂ y ) 2 2 Laplace operator ∆ = 4 ¯ ∂ ∂ = 4 ∂ ¯ ∂ = ∂ 2 x + ∂ 2 y F k z k = F k r k e ik θ , z = r e i θ ∈ D , r < 1 ˆ ˆ � � F ( z ) = k ≥ 0 k ≥ 0 ¯ ∂ F = 0 ( F holomorphic) ⇔ F = u + iv (Fourier series, coefficients ˆ F k ) with ∆ u = 0 and ∆ v = 0: harmonic u and conjugate function v satisfying Cauchy-Riemann equations in D : � ∂ x v = − ∂ y u ∂ y v = ∂ x u
Hardy spaces H 2 of analytic functions in D H 2 ( D ): solutions to ¯ ∂ F = 0 in D , � F � 2 < ∞ � 2 π | F ( re i θ ) | 2 d θ � F � 2 � | ˆ F k | 2 2 = ess sup 2 π = 0 < r < 1 0 k ≥ 0 Hilbert space ⊂ L 2 ( D ) Parseval p = 2, also Ω = A and Banach H p � L 2 boundary values on T : tr H 2 ( D ) ⊂ L 2 ( T ) traces, non tg lim L 2 ( T ) = tr H 2 ( D ) ⊕ tr H 2 , 0 ( C \ D ) ⊥ decomposition, projection P + � equivalent boundary L 2 ( T ) norm: � F � 2 = � tr F � L 2 ( T ) � Cauchy-Riemann equation in D , up to boundary T : F = u + iv , ∂ θ v = ∂ n u , ∂ n v = − ∂ θ u tr v = H tr u also Cauchy integral formula, Poisson kernel, Hilbert-Riesz operator + further properties [Duren, Garnett] � results for σ = 1, ν = 0, Laplace equations (dimension 2 or 3)
Generalized Hardy space H 2 ν Hilbert space H 2 ν = H 2 Ω = D or A ν (Ω): also Ω ≃ D or A conformally, and Banach H p ν , 1 < p < ∞ ¯ - solutions f to (f) ∂ f = ν∂ f in Ω - bounded in Hardy norm in Ω � f � 2 < ∞ (sup of L 2 norms on circles in Ω) ν shares many properties of H 2 = H 2 H 2 0 [Baratchart-L.-Rigat-Russ, 2010], [Fischer, 2011], [F.-L.-Partington-Sincich, 2011], [BFL, 2012]
Properties of H 2 ν Generalize those of H 2 Ω = D or A also H p ν [BFL,F] ¯ f ∈ H 2 Theorem [BLRR] ν (Ω) ∂ f = ν∂ f , � f � 2 < ∞ - f admits a non tangential limit tr f ∈ L 2 (Γ) on Γ - tr f = 0 a.e. on I ⊂ Γ, | I | > 0 implies that f ≡ 0 if f �≡ 0, then log | tr f | ∈ L 1 (Γ), and f admits isolated zeroes (+ Blaschke condition) - � tr f � L 2 (Γ) is equivalent to � f � 2 on H 2 ν (Ω) Hardy norm tr H 2 ν (Ω) is closed in L 2 (Γ) - Closedness of traces: - Re tr f = 0 a.e. on Γ implies that f ≡ 0 in Ω (up to constant) whenever normalization on Γ � f ∈ H 2 , 0 ν (Ω) = { f ∈ H 2 ν (Ω) , Im tr f = 0 } T Γ = T or T ∪ ̺ T + maximum principle in modulus
Properties of tr H 2 ν ( D ) Corollary [BLRR] Dirichlet in H 2 ν ( D ), density R ( T ), ∃ ! f ∈ H 2 , 0 - ∀ φ ∈ L 2 ν ( D ) such that Re tr f = φ moreover , � tr f � L 2 ( T ) ≤ c ν � φ � L 2 ( T ) - conjugation operator H ν bounded on L 2 R ( T ) Hilbert-Riesz transform, L 2 ( T ) H ν Re tr f = φ �− → Im tr f = H ν φ f ∈ H 2 , 0 ⇒ tr f = ( I + i H ν ) φ , φ ∈ L 2 ν ( D ) ⇐ R ( T ) let I ⊂ T , J = T \ I such that | J | > 0 - density: then, restrictions to I of functions in tr H 2 ν ( D ) dense in L 2 ( I ) also in H p ν , 1 < p < ∞
Other situations - Generalization to Ω = A annulus A = D \ ̺ D or multiply connected smooth domains [BFL, F] Dirichlet in H 2 ν ( A ) for data in L 2 R ( A ) ⊖ S ν ( A ) = solutions to (f): ¯ H 2 ∂ f = ν∂ f in A with � f � 2 < ∞ S = { φ ∈ L 2 R ( ∂ A ) s.t. φ | T = C , φ | ̺ T = − C , C ∈ R } Density of restrictions on I ⊆ T of tr H 2 ν ( A ) in L 2 ( I ) (or I ⊆ ̺ T ) - Conformal invariance of (f): Ω ≃ D or Ω ≃ A older smooth ν ∈ W 1 , r (Ω), r > 2 - For H¨ (and σ ) in H p ν (Ω) with ∞ > p > r / ( r − 1)
For related conductivity PDE u solution to (u) in Ω: Ω ≃ D or A div ( σ ∇ u ) = 0 ⇐ u = Re f with f solution to (f) in Ω if Ω ≃ D , ⇔ Dirichlet boundary value problems: from prescribed boundary data φ ∈ L 2 R (Γ) Ω ≃ A : φ ⊥ S recover u in Ω solution to (u) such that tr u = φ on Γ From Dirichlet theorem in H 2 , 0 ν (Ω): ∃ ! u in L p R (Ω) solution to (u) such that tr u = φ � tr f = φ + i σ∂ n u = φ + i H ν φ , � u � 2 = � tr u � L 2 (Γ) = � φ � L 2 (Γ) Γ Also, unique continuation properties bounded conjugation operator � stability properties for (u)... Dirichlet-Neumann map: Λ φ = ∂ θ H ν φ
For related conductivity PDE Cauchy inverse problems, I ⊂ T Ω = D or A Given φ I and ψ I in L 2 R ( I ) recover u solution to (u) in Ω such that tr u = φ I , σ∂ n u = ψ I on I � ψ I ∈ L 2 ( I ) Let Φ = φ I + i I Density results: [tr H 2 ν ] | I dense in L 2 ( I ) Runge property (compatible boundary data) ∃ f k ∈ tr H 2 ν , � Φ − f k � L 2 ( I ) → 0 ( k → ∞ ) either Φ ∈ tr H 2 ν | I already and � Φ − f k � L 2 ( T ) → 0 However ∈ tr H 2 or Φ / and � f k � L 2 ( J ) → ∞ ν | I
For related conductivity PDE � Cauchy problem ill-posed for non compatible data φ I , ψ I on I I ψ I ∈ L 2 ( I ) \ (tr H 2 � Φ = φ I + i ν ) | I : ∃ u k = Re f k solution to (u) in Ω � tr u k − φ I � L 2 ( I ) − → 0 but � tr u k � L 2 ( J ) − → ∞ � ∂ θ H ν u k − ψ I � L 2( I ) → 0 � Look for tr u ≃ φ I , σ∂ n u ≃ ψ I on I with tr u bounded on J ... � Bounded extremal problems (BEP) in tr H 2 ν best constrained approximation
Best constrained approximation in H 2 ν Regularization: bounded extremal problems (BEP) Let I ⊂ Γ, | I | , | J | > 0, ε > 0 Ω = D , Γ = T , J = T \ I � � f ∈ tr H 2 | I ⊂ L 2 ( I ) . B = ν , � Re f � L 2 ( J ) ≤ ε Theorem [BFL, FLPS] (BEP) well-posed ν = 0: [BLP] ∀ function Φ ∈ L 2 ( I ), ∃ unique f ∗ ∈ B such that � Φ − f ∗ � L 2 ( I ) = min f ∈B � Φ − f � L 2 ( I ) Moreover, if Φ / ∈ B , then � Re f ∗ � L 2 ( J ) = ε Proof : bounded conjugation, density result also in Ω ≃ A , with I ⊂ T , J = ( T \ I ) ∪ ̺ T also in H p ν , for L p ( I ) data, or with other norm constraints
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