Lecture Notes of TICMI, Vol.13, 2012 Giorgi Akhalaia, Grigory Giorgadze, Valerian Jikia, Neron Kaldani, Giorgi Makatsaria, Nino Manjavidze ELLIPTIC SYSTEMS ON RIEMANN SURFACES © Tbilisi University Press Tbilisi Abstract. Systematic investigation of Elliptic systems on Riemann surface and new results from the theory of generalized analytic functions and vectors are pre- sented. Complete analysis of the boundary value problem of linear conjugation and Riemann-Hilbert monodromy problem for the Carleman-Bers-Vekua irregular systems are given. 2000 Mathematics Subject Classi¯cation: 57R45 Key words and phrases: Generalized analytic function, Generalized analytic vector, irregular Carleman-Bers-Vekua equation, linear conjugation problem, holo- morphic structure, Beltrami equation Giorgi Akhalaia, Grigory Giorgadze*, Valerian Jikia, Neron Kaldani I.Vekua Institute of Applied Mathematics of I. Javakhishvili Tbilisi State University 2 University st., Tbilisi 0186, Georgia Giorgi Makatsaria St. Andrews Georgian University 53a Chavchavadze Ave., Tbilisi 0162, Georgia Nino Manjavidze Georgian Technical University 77 M. Kostava st., Tbilisi 0175, Georgia * Corresponding Author. email: [email protected] Contents Preface 5 1 Introduction 7 2 Functional spaces induced from regular Carleman-Bers-Vekua equations 15 2.1 Some functional spaces . 15 2.2 The Vekua-Pompej type integral operators . 16 2.3 The Carleman-Bers-Vekua equation . 18 2.4 The generalized polynomials of the class Ap;2(A; B; C), p > 2 . 20 2.5 Some properties of the generalized power functions . 22 2.6 The problem of linear conjugation for generalized analytic functions . 26 3 Beltrami equation 28 4 The pseudoanalytic functions 36 4.1 Relation between Beltrami and holomorphic disc equations . 38 4.2 The periodicity of the space of generalized analytic functions . 40 4.3 Almost complex structure . 44 4.4 The holomorphic discs equation . 46 5 Irregular Carleman-Bers-Vekua equation with weak singularity at infinity 48 5.1 The functional spaces induced from irregular Carleman-Bers-Vekua equations . 56 5.2 The Liouville type theorems for Carleman-Bers-Vekua irregular equa- tions . 67 5.3 The generating triple . 75 5.4 The problem of linear conjugation for some classes of Carleman-Bers- Vekua equation . 76 6 Singular points of solutions of some elliptic systems on the plane 80 6.1 Degenerate elliptic systems . 80 6.2 Quasilegular Carleman-Bers-Vekua equations . 82 6.3 Correct boundary value problems for some classes of singular elliptic differential equations on a plane . 86 6.4 One class of two-dimensional third kind Fredholm integral equation . 104 7 Some problems of 2n-elliptic systems on the plane 107 7.1 Maximum modulus theorem . 107 7.2 Generalized Beltrami equation . 109 7.3 The solvability of the problem (V ) . 110 3 7.4 Boundary value problems for elliptic system on the plane with angular points . 115 8 Monodromy of generalized analytic functions 122 8.1 System of elliptic equations on the Riemann surfaces . 122 8.2 Associated bundles with regular connections . 126 9 Carleman-Bers-Vekua equation on Riemann surfaces 131 9.1 Cauchy-type integral on Riemann surfaces . 131 9.2 Hypercomplex function . 134 10 Appendix 137 10.1 On the analytic solutions of the system degenerated in the point . 137 10.1.1 Some auxiliary statements . 137 10.1.2 The existence of the analytic solutions . 139 10.2 Darboux transformation and Carleman-Bers-Vekua system . 143 10.3 Topological properties of generalized analytic functions . 146 References 148 4 Preface The idea of using the methods of complex analysis for a wider class of functions then the space of analytic functions takes its origin from the end of 19th century and actually is connected with the period of conception of complex analysis as the independent branch of science. First attempts for the extension of the complex methods were connected with the generalization of Cauchy-Riemann system and the investigation of the properties of the space of solutions such a system (Picard, Hilbert, Carleman, Teodorescou). Later on, since the 1950-s, after the appearance of the monographs of I.Vekua and L.Bers, generalized analytic functions due to the terminology of Vekua [124] and pseudo-analytic functions due to Bers [11] these problems are the subject of investigation by many scientists. In the frame of Vekua-Bers theory the geometric (topological) properties of the solutions of elliptic systems on the plane became clear: in particular, the natural relation between the pseudoanalytic functions and the quasiconformal mappings was established. Here we mean that Carleman - Vekua and Beltrami equations always exist in parallels. Concepts of Bers and Vekua are similar in the cases of Bers normalized pair and Carleman-Bers-Vekua regular equations. In particular, the normalized generating pair induces the regular equation and vice versa, the generating pair, corresponding to the regular equation is normalized. In such conditions these approaches are com- pleting each other and we get the functional spaces with richer properties. In other cases when the equation is non-regular or Bers generating pair isn't normalized the corresponding space of the generalized analytic functions is studied less intensively and the theory itself is not as consistent as the Bers-Vekua theory. Besides, other interesting generalized elliptic equations of Cauchy-Riemann equations are known, for example, the equations which give the spaces of poly-analytic or p-analytic func- tions. The present work is an attempt to fill up this gap at least partially. Its main part is devoted to the study of the solutions of irregular elliptic systems, specifically to the development of the fundamental theorems of the theory of analytic functions and the corresponding boundary value problems in such spaces as well. The re- maining part deals with the extension of Bers-Vekua theory on 2n-elliptic systems, which has been soon developed independently by B.Bojarski [31], [26] and A.Douglis [45] and it is also the subject of the authors' interest. In particular, the geometric interpretation of the matrix Beltrami equation, the problem of the generating pairs, the Riemann-Hilbert monodromy problem etc. The authors think that the solution of the listed problems is very important for the study of differential - geometric connections of vector bundles and the complex structure of the moduli space, which in their turn are connected with the fundamental problems of the contemporary mathematical physics: topological and conformal field theories, the string and super string theories, supersymmetry, etc. Further development of the theories of B.Bojarski and A. Douglis is given in the 5 monographs of G. Manjavidze [90] and R. Gilbert and J.Buchanan [49]. Investigation of such non-regular systems is an open problem yet. This work paves the way for the study of these systems. Acknowledgements. The Authors would like to express their deepest gratitude to Prof. B. Bojarski for his permanent interest and many-sided support in above mentioned researches. The authors are also grateful to the participants of the seminar "Elliptic systems on Riemann surfaces" at the I.Javakhishvili Tbilisi State University for many discussions. This paper was partially supported by the Rustaveli National Science Founda- tion, Grant No 1-3/85 (2009-2012). 6 1 Introduction Let α, β, γ, δ be measurable functions of two variables (x, y), and a; b; c; d; e; f 2 Lp(U), p > 2, U ⊂ C. Consider the system of linear partial differential equations with respect to functions (u(x; y), v(x; y))of two variables: 8 > @v @u @u <> = α + β + au + bv + e @y @x @y (1.1) > @v @u @u : − = γ + δ + cu + dv + f @x @x @y The system (1.1) is elliptic if the inequalities α(x; y) > 0; 4α(x; y)δ(x; y) − (β(x; y) + γ(x; y))2 > 0 hold. Along with ellipticity, let us require uniform ellipticity of the system (1.1), which means that for a fixed constant k0 the functions α, β, γ, δ satisfy the inequality 2 4αδ − (β + γ) ≥ k0 > 0: Consider the functions occurring in the system (1.1) as functions of complex variable z = x + iy: Suppose w(z) = u(x; y) + iv(x; y); using appropriate simplifications and notations we get the equation @w(z) @w(z) @w − µ (z) − µ (z) = A(z)w(z) + B(z)w(z) + C(z); (1.2) @z¯ 1 @z 2 @z where 2q jqj2 + (1 + p)(1 − p¯) µ (z) = ; µ (z) = − ; 1 jqj2 − j1 + pj2 2 jqj2 − j1 + pj2 the functions q and p are expressed by the functions α, β, γ, δ as follows: α + δ + i(γ − β) α − δ + i(γ + β) q = ; p = ; 2 2 whereas the functions A, B, C are obtained from the functions a, b, c, d, e, f by 1 adding and multiplying by constants and by the function jqj2−|1+pj2 . Since the system is uniformly elliptic, the inequality jµ1j + jµ2j ≤ µ0 holds for any z 2 U. Moreover, the functions A, B, C are either bounded on U or belong to Lp(U). The equation (1.2) on the complex plane with natural restrictions on the func- tions µ1; µ2; A; B, contains many well-known equations: Cauchy-Riemann, Beltrami, Carleman-Bers-Vekua, holomorphic disc and other equations, which are obtained from (1.1) or (1.2) by an appropriate choice of the coefficients. All these equations are "deformations" of the Cauchy-Riemann equation and the properties of the space of solution of the corresponding equations are close to the properties of the spaces of analytic functions. 7 Lecture Notes of TICMI, vol. 13, 2012 It is well-known, that the system (1.1) is equivalent to the system of equations ux − Vy + au + bV = 0; uy + Vx + cu + dV = 0; under sufficiently general conditions.