Probl` emes inverses ` a la fronti` ere pour l’´ equation de Beltrami dans des domaines plans, approximation dans des classes de Hardy g´ en´ eralis´ ees Juliette Leblond projet APICS joint work with L. Baratchart, S. Rigat, E. Russ INRIA, Sophia-Antipolis, LATP-CMI, Univ. Provence, U. Paul C´ ezanne, Marseille
Conductivity equation Let Ω ⊂ R 2 smooth and σ ∈ C (¯ Ω), 0 < c ≤ σ ≤ C div ( σ ∇ u ) = 0 in Ω (1) • Cauchy problems: | I | , | ∂ Ω \ I | > 0 tr u and ∂ n u prescribed on I ⊂ ∂ Ω recover u in Ω and Cauchy data on J = ∂ Ω \ I • Dirichlet problem: tr u prescribed on ∂ Ω recover u in Ω and ∂ n u on ∂ Ω
A motivation... Recover shape of plasma boundary in a tokamak Tore Supra (CEA-IRFM Cadarache)
... A motivation... Maxwell equations, cylindrical coordinates ( x , y , φ ) of magnetic induction, axial symmetry (indep. of φ )
... A motivation � in poloidal section (annular domain) ( x , y ) ∈ Ω ⊂ R 2 poloidal magnetic induction [Bl] : � 0 � B x � � conductivity σ = 1 − 1 B = = σ ∇ u , B y 1 0 x for poloidal magnetic flux u : div ( σ ∇ u ) = 0 in Ω given u and B ≈ σ∂ n u on I ⊂ ∂ Ω look for u and ∂ n u on ∂ Ω \ I ? level line of u (plasma boundary)?
Conjugated ( R -linear) Beltrami equation u solution to (1): div ( σ ∇ u ) = 0 iff u = Re f where f = f ( z , ¯ z ) satisfies first order elliptic equation ¯ ∂ f = ν∂ f in Ω (2) with respect to complex variable z = x + iy and [AP] ν = 1 − σ 1 + σ ν ∈ C (¯ Ω) real-valued, | ν | ≤ κ < 1 in Ω C -linear Beltrami equation: ¯ ∂ g = ν∂ g quasi-conformal map. [Ahlf., Ast.]
Generalized σ -harmonic conjugation we have f = u + i v where v σ -harmonic conjugated function Hilbert-Riesz transform � 1 � div σ ∇ v = 0 in Ω unique up to additive constant generalized Cauchy-Riemann equations in Ω: � 0 � ∂ x v = − σ∂ y u � − 1 ∇ v = σ ∇ u : 1 0 ∂ y v = σ∂ x u
Proof ∂ = ∂ z = 1 z = 1 2( ∂ x − i ∂ y ) , ¯ ∂ = ∂ ¯ 2( ∂ x + i ∂ y ) .... generalization of ( σ constant) ∆ u = 0 ( u harmonic) ⇔ ¯ ∂ f = 0 ( f analytic) in Ω
Smooth solutions to Dirichlet problem Thm [Campanato] 1 < p < ∞ ∀ φ ∈ W 1 − 1 / p , p ( ∂ Ω), there exists f ∈ W 1 , p (Ω) solution to (2) in Ω R such that Re tr f = φ on ∂ Ω � unique if normalization condition Im tr f d θ = 0 (3) ∂ Ω further � f � W 1 , p (Ω) ≤ C � ϕ � W 1 − 1 / p , p ( ∂ Ω) u = Re f ∈ W 1 , p (Ω), u = φ on ∂ Ω unique solution to (1) (in W 1 , 2 (Ω) Lax-Milgram - also for σ ∈ L ∞ (Ω) - in W 2 , p (Ω) [ADN]; for σ ∈ VMO(Ω) [AQ]) allows to solve boundary approximation problems but with Sobolev norms and smooth boundary data
With L p ( ∂ Ω) boundary data? Ω = D unit disk, L p ( T ) data simply connected Ω smooth σ , ν ∈ W 1 , ∞ ( D ) Generalized Hardy spaces H p ν = H p ν ( D ) of solutions: functions f on D satisfying T r circle radius r � f � H p ν = ess sup � f � L p ( T r ) < + ∞ 0 < r < 1 solutions to (2) in D as distributions � 2 π ( � f � p 1 | f ( re i θ ) | p d θ ) L p ( T r ) = 2 π 0 H p ν ⊂ L p ( D ) real Banach space
Harmonic and analytic functions σ ≡ 1 (cst), ∆ u = 0 in D ( ν = 0) classical Hardy spaces H p = H p 0 ( D ) of analytic functions ¯ ∂ f = 0 and � f � H p < + ∞ f = u + i v, conjugated function v : ∆ v = 0 in D Hilbert-Riesz transform Cauchy-Riemann equations: � ∂ x v = − ∂ y u in D � ∂ n v = − ∂ θ u on T ∂ y v = ∂ x u ∂ θ v = ∂ n u
Hardy spaces H p • Properties of H p Banach spaces (below...) • Poisson-Cauchy-Green representation formulas, analytic projection • Hilbert H 2 , Fourier basis: H 2 = { f n z n , | f n | 2 } , tr H 2 : z = e i θ ∈ T � � n ≥ 0 n ≥ 0 • allow to state and solve above issues as best approximation problems on L p ( I ) or L p ( T ) [BL]
Properties of H p ν ... Generalize those of H p • Fatou: � tr f � L p ( T ) ≤ � f � H p ν ≤ c ν � tr f � L p ( T ) � 2 π p � � � f ( re i θ ) − tr f ( e i θ ) lim d θ = 0 � � � r → 1 0 • tr H p ν closed subspace of L p ( T ) If f ∈ H p ν : • log | tr f | ∈ L 1 ( T ) (does not vanish on positive measure subsets) unless f ≡ 0 in D • If f �≡ 0, then its zeros α j are isolated in D ∞ � (1 − | α j | ) < + ∞ (with multiplicity) j =1
... Properties of H p ν Let H p , 0 ⊂ H p ν of f such that (3) holds ν • If f ∈ H p , 0 is such that Re (tr f ) = 0 a.e. on T , ν then f ≡ 0 in D • If f ∈ W 1 , p ( D ) solution of (2), then f ∈ H p ν with � f � H p ν ≤ C ν, p � f � W 1 , p ( D ) + orthogonal space and duality
Density results Thm I ⊂ T measurable subset, | T \ I | > 0 • the space of restrictions to I of functions in tr H p ν is dense in L p ( I ) • tr H p ν weakly closed in L p ( T ) • let ( f k ) k ≥ 1 ∈ H p ν whose trace on I converges to φ in L p ( I ): either φ is already the trace on I of an H p ν function or � tr f k � L p ( T \ I ) → + ∞ � bounded approximation problems (BEP) if I = Int I � = T (in particular, I is open), the space of restrictions to I of traces on T of solutions to (CB) in W 1 , p ( D ) is dense in W 1 − 1 / p , p ( I )
Dirichlet theorem R ( T ), ∃ unique f ∈ H p , 0 Thm For all ϕ ∈ L p such that a.e. on T : ν Re (tr f ) = ϕ moreover � f � H p ν ≤ c p ,ν � ϕ � L p ( T ) hence, Hilbert transform (conjugation op.) continuous L p ( T ): Re (tr f ) = u | T = ϕ H ν �→ Im (tr f ) = v | T = H ν ( ϕ ) + higher regularity results, H ν ctn on W 1 − 1 / p , p ( T ) then W 1 , p ( T ) Dirichlet-Neumann map Λ σ = ∂ θ H ν [AP], Calder´ on
Tools and ideas of the proof... Thm [BN] Let α ∈ L ∞ ( D ) ¯ ¯ ∂ν ∂σ 2 σ = ¯ ∂ log σ 1 / 2 α = − 1 − ν 2 = ⇒ w = f − ν f 1 − ν 2 = σ 1 / 2 u + i σ − 1 / 2 v ∈ G p f = u + i v ∈ H p ν ⇐ √ α Hardy spaces of solutions to ∂ w = α w (4) f ∈ W 1 , p ( D ) solves (2) ⇔ w ∈ W 1 , p ( D ) solves (4) � σ 1 / 2 Im tr w d θ = 0 G p , 0 : with normaliz. cond. α T w solves Schr¨ odinger equ.
... Tools and ideas of the proof... Thm [BN] Every w ∈ G p α admits in D a representation w = e s F for s ∈ W 1 , q ( D ) , ∀ q ∈ (1 , + ∞ ) and F ∈ H p further � s � L ∞ ( D ) ≤ c � α � L ∞ ( D ) s can be chosen such that Re s = 0 on T (or Im s = 0) (hence s ∈ C 0 ,γ ( D ) , ∀ γ ∈ (0 , 1); also w ∈ W 1 , q loc ( D ) , ∀ q ∈ (1 , + ∞ )) Proof: take r = α w / w if w � = 0 ( r = 0 if w = 0) and ¯ ∂ s = r in D
... Tools and ideas of the proof... Properties of G p α : similar to H p ν non tangential limit, Fatou, uniqueness Representation: for w ∈ G p α ( D ) and a.e. z ∈ D Cauchy-Green formula 1 � tr w ( ξ ) 1 �� α w ( ξ ) w ( z ) = ξ − z d ξ + ξ − z d ξ ∧ d ξ 2 π i 2 π i T D whence w = C (tr w ) + T α w boundedness properties of Cauchy operators C , T α w = ( I − T α ) − 1 C (tr w ) Re (tr w ) �→ tr w continuous on G p , 0 α
... Tools and ideas of the proof Dirichlet: for all ϕ ∈ L p R ( T ), ∃ unique w ∈ G p , 0 such that α Re (tr w ) = ϕ a.e. on T moreover � w � G p α ≤ c p ,ν � ϕ � L p ( T ) For H p fos: for all ϕ ∈ H p , ∃ unique w ∈ G p , 0 such that α P + (tr w ) = tr ϕ a.e. on T a.e. in D , w = ϕ + T α ( w ); moreover � w � G p α ≤ C p � ϕ � H p
Back to conductivity equation • solve Dirichlet problem with given data φ = u | T in L p ( T ) • Cauchy type issues? on I , from (noisy) data, get � ( σ∂ n ) u | I in L p ( I ) but f �∈ (tr H p f = u | I u + i ν ) | I in view of density: inf � f − h � L p ( I ) = 0 h ∈ tr H p ν while for such a sequence h n , � h n � L p ( T \ I ) ր ∞
Bounded extremal problems However, with norm constraint M > 0 the BEP: min � f − h � L p ( I ) h ∈ tr Hp ν � h � Lp ( T \ I ) ≤ M achieved by a unique h 0 ∈ tr H p ν such that � h 0 � L p ( T \ I ) ≤ M ∈ (tr H p further, if f / ν ) | I � h 0 � L p ( T \ I ) = M � solution to Cauchy problem
Further results in H p ν • orthogonal space and duality • higher regularity results
Conclusion... • simply connected smooth Ω: conformal mapping ψ : D → Ω ¯ ∂ ( f ◦ ψ ) = ( ν ◦ ψ ) ∂ ( f ◦ ψ ) • annulus A = D \ ̺ D : annular domains H p ν ( ̺ D ): Hardy space of solutions to (2) in C \ ̺ ¯ D ν i ( D ) ⊕ H p H p ν ( A ) = H p ν e ( ̺ D ) ν i ∈ W 1 , ∞ ( D ), ν e ∈ W 1 , ∞ ( C \ ̺ D ) such that ν i | A = ν e | A = ν as for classical Hardy spaces, with ν, ν i , ν e = 0 • related PDEs Schr¨ w in C ), Laplace (∆ U = 0 in R 3 ) odinger (∆ w = aw + b ¯
... Conclusion... • computation of solutions to (BEP) for p = 2: H ν ? � ⊥ projection L p ( T ) → tr H p ν ? in H 2 • application to plasma/tokamaks: σ ( x , y ) = 1 / ( x + x 0 ) • bases of H 2 ν ? families of Bessel functions? z + ¯ z + 2 x 0 − 2 ν ( z , ¯ z ) = z + ¯ z + 2 x 0 + 2 • geometrical issues: free boundary Bernoulli pb? • ITER
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