S. Carrillo — The problem of multisummability in higher dimensions 1 The problem of multisummability in higher dimensions 18th June 2018 Sergio A. Carrillo. sergio.carrillo@univie.ac.at Universit¨ at Wien, Vienna, Austria. a a Supported by the Austrian Science Fund (FWF), project P 26735-N25: Differential Analysis: Perturbation and Quasianaliticity.
S. Carrillo — The problem of multisummability in higher dimensions Asymptotic Analysis and Borel summability in one variable 2 We have at our disposal a powerful summability theory useful in the study of formal solutions of analytic problems, e.g. ODEs at irregular singular points, families of PDEs, difference equations, conjugacy of diffeomorphisms of ( C , 0) , normal forms for vector fields, singular perturbation problems, normal forms of real-analytic hypersurfaces... ◮ Asymptotic expansions, Gevrey asymptotic expansions, k − summability. ◮ Borel and Laplace transformations. Tauberian theorems. ◮ Ecalle’s accelerator operators, Multisummability.
S. Carrillo — The problem of multisummability in higher dimensions Asymptotics in several variables 3 For several variables there are different approaches. In this framework we can mention: ◮ Strong Asymptotic Expansions, (Majima, 1984). ◮ Composite Asymptotic Expansions (Fruchard-Sch¨ afke, 2013). ◮ Asymptotic Expansions in a monomial or in an analytic function (Mozo-Sch¨ afke, 2007, 2017). We will focus in the item and pose the problem of multisummability for those methods.
S. Carrillo — The problem of multisummability in higher dimensions The scope of applications 4 ◮ (1990 Ramis, Sibuya, Braaskma) Multisummability of non-linear equations x p +1 d y dx = F ( x, y ) . When ∂ F ∂ y (0 , 0 ) is invertible the unique formal power series solution is p − summable. ◮ (2003 Luo, Chen, Zhang) Summability in the variable x of solutions of PDEs of the form t∂ t u = F ( t, x, u, ∂ x u ) , u (0 , x ) = 0 , under certain conditions on F . ◮ (2007 Costin, Tanveer) Existence, uniqueness and asymptotic in several variables of solutions of PDEs of the form u t + P ( ∂ j x ) u + g ( x , t, { ∂ j x u } ) = 0 , u ( x , 0) = u I ( x ) , where the principal part of the constant coefficient n − th order differential operator P is subject to a cone condition.
S. Carrillo — The problem of multisummability in higher dimensions 5 ◮ (2007 Canalis-Duran, Mozo, Sch¨ afke) 1 − x p ε q − summability of the unique formal power series solution of the doubly singular equation ε q x p +1 ∂ y ∂x = F ( x, ε, y ) , when ∂ F ∂ y (0 , 0 , 0 ) is invertible. ◮ (2018 -) 1 − x α ε α ′ − summability of the unique formal power series solution of the singularly perturbed PDE x α ε α ′ � � λ 1 x 1 ∂ y ∂x 1 + · · · + λ n x n ∂ y = F ( x , ε , y ) , ∂x n where x ∈ C n , ε ∈ C m , α ∈ ( N + ) n , α ′ ∈ ( N + ) m , λ = ( λ 1 , . . . , λ n ) ∈ ( R + ) n and F analytic at the origin and ∂ F ∂y (0 , 0 , 0 ) is invertible.
S. Carrillo — The problem of multisummability in higher dimensions 6 The theory in one variable
S. Carrillo — The problem of multisummability in higher dimensions Example: Euler’s equation 7 Consider Euler’s equation: x 2 y ′ + y = x. We can solve it for x > 0 to get � + ∞ e − ξ/x y ( x ) = ce 1 /x + 1 + ξ dξ. 0 But it also has the formal power series solution ∞ � ( − 1) n n ! x n +1 . y ( x ) = ˆ n =0
S. Carrillo — The problem of multisummability in higher dimensions The notion of asymptotic expansion 8 Let us fix a complex Banach space ( E, � · � ) . We work in sectors at the origin S = S ( a, b ; r ) = { x ∈ C | 0 < | x | < r, a < arg ( x ) < b } . Definition f = � ∞ n =0 a n x n ∈ E [[ x ]] as asymptotic expansion on We say f ∈ O ( S, E ) has ˆ f on S .) if for every subsector S ′ ⊂ S and N ∈ N we can find S ( f ∼ ˆ C N ( S ′ ) > 0 such that � � � � N − 1 � � � a n x n � ≤ C N ( S ′ ) | x | N , x ∈ S ′ . � � f ( x ) − � n =0
S. Carrillo — The problem of multisummability in higher dimensions Basic properties 9 f = � ∞ n =0 a n x n and g ∼ ˆ Assume f ∼ ˆ g on S . The following properties hold: f ( n ) ( x ) for any subsector S ′ . 1. a n = lim x → 0 n ! x ∈ S ′ dx ∼ d ˆ 2. f + g ∼ ˆ g , fg ∼ ˆ d f f f + ˆ f ˆ g , dx on S . 3. (Borel-Ritt) Given any ˆ f ∈ E [[ t ]] and S there is f ∈ O ( S, E ) such that f ∼ ˆ f on S .
S. Carrillo — The problem of multisummability in higher dimensions Gevrey type asymptotic expansions 10 If f ∼ ˆ f on S and we can choose C N ( S ′ ) = CA N N ! 1 /k , then we say that the asymptotic expansion is of type 1 /k − Gevrey ( f ∼ 1 /k ˆ f on S ). Then ˆ � a n � ≤ CA n n ! 1 /k , f ∈ E [[ x ]] 1 /k , i.e. the space of 1 /k − Gevrey series in x . ◮ f ∼ 1 /k 0 on S if and only if for every S ′ ⊂ S , we can find K, M > 0 � f ( x ) � ≤ K exp( − M/ | x | k ) . ◮ (Borel-Ritt-Gevrey) If b − a < π/k given any ˆ f ∈ E [[ x ]] 1 /k and S ( a, b, r ) there is f ∈ O ( S, E ) such that f ∼ 1 /k ˆ f on S . ◮ (Watson’s Lemma) If b − a > π/k and f ∼ 1 /k 0 on S ( a, b, r ) then f ≡ 0 .
S. Carrillo — The problem of multisummability in higher dimensions k − Sumability 11 Definition Let ˆ f ∈ E [[ x ]] 1 /k and θ ∈ R a direction. ◮ ˆ f is k − summable in a direction θ if we can find f ∈ O ( S, E ) , π 2 k + ε, r ) such that f ∼ 1 /k ˆ π S = S ( θ − 2 k − ε, θ + f . ◮ ˆ f is k − summable if it is k − summable in all directions, up to a finite number of them, mod. 2 π . We will use the notation E { x } 1 /k,θ and E { x } 1 /k for the corresponding sets.
S. Carrillo — The problem of multisummability in higher dimensions Borel-Laplace method 12 f = � ∞ n =0 a n x n ∈ E [[ x ]] 1 /k is called k − Borel-summable in The series ˆ direction θ if � � a n B k ( ˆ � Γ( n/k ) ξ n − k , f − a n x n ) := n ≤ k n>k can be analytically continued, say as ϕ , and � ϕ ( ξ ) � ≤ C exp( M | ξ | k ) , for some C, M > 0 . Its Borel sum is defined by � a n x n + L k ( ϕ )( x ) f ( x ) = n ≤ k � e iθ ∞ � a n x n + ϕ ( ξ ) e − ( ξ/x ) k d ( ξ k ) . = 0 n ≤ k
S. Carrillo — The problem of multisummability in higher dimensions 13 The Borel-Laplace analysis exploits the isomorphism between the following structures � � � � E [[ x ]] 1 /k , + , × , x k +1 d � B k ξ − k E { ξ } , + , ∗ k , kξ k ( · ) − − → , dx where × denotes the usual product and ∗ k stand for the convolution product � 1 ( f ∗ k g )( ξ ) = ξ k f ( ξτ 1 /k ) g ( ξ (1 − τ ) 1 /k ) dτ. 0
S. Carrillo — The problem of multisummability in higher dimensions Euler’s equation II 14 Applying the 1 − Borel transformation to Euler’s example: ∞ ∞ � � 1 � ( − 1) n ξ n = B 1 ( − 1) n n ! x n +1 y ( x ) = ˆ − − → Y ( ξ ) = 1 + ξ , n =0 n =0 x 2 y ′ + y = x � B 1 − − → ξY + Y = 1 . Using the Laplace transform we get the solution � + ∞ e − ξ/x y ( x ) = 1 + ξ dξ, Re ( x ) > 0 . 0
S. Carrillo — The problem of multisummability in higher dimensions Example of non-linear ODEs 15 Consider the differential equation � x p +1 d y A I ( x ) y I , dx = F ( x, y ) = b ( x ) + A ( x ) y + | I |≥ 2 where p ∈ N + , y ∈ C N , F is analytic in a neighborhood of (0 , 0 ) ∈ C × C N and F (0 , 0 ) = 0 . Using ˆ B = ˆ B p , ∗ = ∗ p we obtain the convolution equation � ( pξ p I N − A 0 ) Y = B ( b ) + B ( A − A 0 ) ∗ Y + B ( A I − A I (0)) ∗ Y ∗ I | I |≥ 2 � A I (0) Y ∗ I . + | I |≥ 2 We ask for pξ p I N − A 0 to be invertible, therefore we work on domains inside Ω := { ξ ∈ C | pξ p � = λ j for all j = 1 , . . . , N } , where λ j are the eigenvalues of A 0 .
S. Carrillo — The problem of multisummability in higher dimensions 16 Theorem If A 0 = ∂ F ∂ y (0 , 0 ) is invertible then the previous ODE has a unique formal y ∈ C [[ x ]] N . Furthermore ˆ power series solution ˆ y is p − summable. For µ > 0 consider A N µ ( S ) := { f ∈ O ( S, C N ) | f (0) = 0 , � f � N,µ := max 1 ≤ j ≤ N � f j � µ < + ∞} , | f ( ξ ) | (1 + | ξ | 2 p ) e − µ | ξ | p , � f � µ := M 0 sup f ∈ O ( S ) . ξ ∈ S S := S R = S ( θ, 2 ǫ ) ∪ D R ⊂ Ω , � 1 dτ s> 0 s (1 + s 2 ) I ( s ) ≈ 3 . 76 , M 0 = sup I ( s ) := (1 + s 2 τ 2 )(1 + s 2 (1 − τ ) 2 ) . 0
S. Carrillo — The problem of multisummability in higher dimensions 17 For general linear systems it is enough to consider equations of type xd y dx = b ( x ) + A ( x ) y , � � r r � � x − k h A h + x 1 − k h I h A ( x ) = A + ( x ) , h =0 h =0 where 0 = k 0 < k 1 < · · · < k r , k h ∈ N + , N = n 0 + · · · + n r , A h is a n h × n h invertible matrix , I h is the identity matrix of size n h , A + and g analytic at the origin. Theorem (Braaksma-Balser-Ramis-Sibuya) If the system posses a formal solution ˆ y then it is k − multisummable, where k = ( k 1 , . . . , k n ) .
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