aa r X i v : . [ m a t h . C V ] M a r On a Carleman formula for lunes
D. Fedchenko Abstract
In this paper we consider a simple formula for analytic continuationin a domain D ⊂ C of special form. Introduction
Integral representations of holomorphic functions solve the classical problem onrestoring a holomorphic function in a domain D by its values on ∂D . Connectedwith this problem, there is another one: restore holomorphic function in D byits values on a set Γ ⊂ ∂D .The first result of such type was obtained by Carleman [C] for a plain domain D of special form. His idea of using extinguishing functions was developed inthe article of Goluzin and Krylov [GK] and by Fok and Kuni [FK] for simplyconnected plain domains of special form. This method provides an extinguishing function for any subset Γ of ∂D of positive measure. Another method wasproposed by Lavrent’ev in 1956 [LRS].All these formulas and some their applications can be found in the book ofAizenberg [A].In this notice we show a simple trick to construct Carleman formula forlunes. Let C be the complex plane of z = x + ıy , where x, y ∈ R . For an open set D , denote by O ( D ) the space of functions holomorphic in D . Write ∂ for theCauchy-Riemann operator ∂ = 12 (cid:18) ∂∂x + ı ∂∂y (cid:19) in C .Let Γ be a subset of ∂D . The problem of analytic continuation from Γinto D consists in the following. Given a function u ∈ C (Γ), find a function u ∈ O ( D ) ∩ C ( D ∪ Γ) such that Institute of Mathematics, Siberian Federal University, Svobodny Prospect 79, Krasno-yarsk, 660041, Russia. E-mail: [email protected] research of the author was done in the framework of the Mikhail Lomonosov Fellowshipwhich is supported by the Russian Ministry of Education and the Deutsche Forschungsge-meinschaft.
Key words : Analytic continuation, Carleman formulas.2000
Mathematics Subject Classification : 35Cxx. ∂u = 0 in D,u = u on Γ . One easily specifies this problem within Cauchy problems for solutions of ellipticequations.
Let Γ be a smooth curve in the unit disk B (0 ,
1) dividing B (0 ,
1) into twodomains. Denote by D those of these domains which does not contain theorigin. Such domains D are referred to as lunes. And let γ ( t ) = ( x ( t ) , y ( t )) , t ≥ ,γ (0) = 0be a smooth curve with end point the origin, which lies in B (0 , \ D (seeFig. 1). Let Γ and γ intersect transversally at the origin. Fix now any curve γ with these properties. D Γ0 γ ( t )Fig. 1. Example of D Denote by B ( γ ( t )) the circle in C with center at the point γ ( t ) and withradius dist( γ ( t ) , ∂B (0 , Lemma 2.1.
For any z ∈ D , there exists point γ ( t ) such that z ∈ B ( γ ( t )) .Proof. Proof by contradiction: suppose there is a point z ∈ D such that z / ∈ B ( γ ( t )) for all t ≥
0, then (cid:26) | z − γ ( t ) | ≥ dist( γ ( t ) , ∂B (0 , , for all t ≥ | z | < . Let t →
0. Then we have (cid:26) | z | ≥ , | z | < . Obtained contradiction proofs this lemma. (cid:3)
Take any numerical sequence γ ( t N ) = ( x ( t N ) , y ( t N )), N ∈ N , t N ≥ N →∞ γ ( t N ) = 0. 2 heorem 2.2. If u ∈ O ( D ) ∩ C ( D ) , then the formula u ( z ) = 12 πı lim N →∞ Z Γ u ( ζ ) ζ − z (cid:18) z − γ ( t N ) ζ − γ ( t N ) (cid:19) N +1 dζ, holds for any point z ∈ D , where the convergence is uniform in z on compactsubsets of D .Proof. Fix z ∈ D . Choose N large enough, so that z ∈ B ( γ ( t N )). Expandthe Cauchy kernel as Laurent series in the variable ζ in the complement of thedisk B ( γ ( t N )) by 1 ζ − z = ∞ X k =0 ( z − γ ( t N )) k ( ζ − γ ( t N )) k +1 . Consider the sequence of kernels C N ( ζ, z ) = 12 πı ζ − z − N X k =0 ( z − γ ( t N )) k ( ζ − γ ( t N )) k +1 ! (2.1)which we call Carleman kernel.Using the geometric sum formula, we make the following transformation ofCarleman kernels (2.1) C N ( ζ, z ) = 12 πı ζ − z (cid:18) z − γ ( t N ) ζ − γ ( t N ) (cid:19) N +1 . Let N → ∞ . Write12 πı lim N →∞ "Z ∂D u ( ζ ) ζ − z (cid:18) z − γ ( t N ) ζ − γ ( t N ) (cid:19) N +1 dζ − Z ∂D \ Γ u ( ζ ) ζ − z (cid:18) z − γ ( t N ) ζ − γ ( t N ) (cid:19) N +1 dζ =12 πı lim N →∞ Z Γ u ( ζ ) ζ − z (cid:18) z − γ ( t N ) ζ − γ ( t N ) (cid:19) N +1 dζ. (2.2)Fix any compact K ⊂ D . We see that the second integral from the left-handside in (2.2) tends to zero uniformly on K , because q K = max z ∈ K,ζ ∈ ∂D \ Γ | z || ζ | < u ( ζ ) (cid:16) z − γ ( t N ) ζ − γ ( t N ) (cid:17) N +1 , for all N ∈ N ,and form the Cauchy formula we conclude that the integral over ∂D equals u ( z )in the domain D .We thus arrive at the desired formula u ( z ) = 12 πı lim N →∞ Z Γ u ( ζ ) ζ − z (cid:18) z − γ ( t N ) ζ − γ ( t N ) (cid:19) N +1 dζ. (cid:3) xample 2.3. Let D = {| z | < , ℜ z > } and Γ = {| z | < , ℜ z = 0 } . As γ ( t N ) we take the sequence ( − /N, , N ∈ N . If u ∈ O ( D ) ∩ C ( D ) , then for anypoint z ∈ D ∪ Γ the formula u ( z ) = 12 πı lim N →∞ Z Γ u ( ζ ) ζ − z (cid:18) N z + 1
N ζ + 1 (cid:19) N +1 dζ holds, where the integral converges uniformly on compact subsets of D . Remark 2.4. If z = 0 does not belong to D then our Carleman formula recoversone of the formulas in [A]. References [C] T. Carleman,
Les fonctions quasianalytiques , Paris: Gauthier-Villars.(1926).[GK] G. Goluzin and V. Krylov,
Generalized Carleman formula and its ap-plication to analytic continuation of functions
Mat. Sb., , 144 - 149(1933).[FK] V. Fok and F. Kuni, On the cutting function in dispersion relations ,Dokl. Akad. Nauk SSSR (1959), 1195-1198 (Russian)[LRS] M. M. Lavrent’ev, V. G. Romanov and S. P. Shishatskii.
Ill-PosedProblems of Mathematical Physics and Analysis [in Russian], Nauka,Moscow (1980); English transl.: Amer. Mathem. Soc., Providence Vol.64 (1986).[A] L. Aizenberg,