Sergiy A. Plaksa
Doctor of Sciences (Physics and Mathematics)
Institute of Mathematics of the National Academy of Sciences of Ukraine,
Tereshchenkivska Str. 3, 01601, Kiev, Ukraine
Phone (office): (38044) 234–51–50
Phone (home): (38044) 450–86–66
Fax: (38044) 235–20–10
Address: Tchornobylska Str., 4/56, apt. 70, Kiev 03179, Ukraine
Date and Place of Birth: 27 October 1962, Pershotravensk of the Zhitomir Region in Ukraine
Languages: Russian, Ukrainian, English, French
Zhitomir Pedagogical Institute, Faculty of Physics and Mathematics, 1979–1984, Master of science, diploma with honors;
Post-graduate studies at the Institute of Mathematics of the Ukrainian Academy of Sciences under the guidance of Professor P.M. Tamrazov, 1984–1989;
Ph.D. (in Physics and Mathematics, mathematical analysis), Institute of Mathematics of
the Ukrainian Academy of Sciences, 1989. Theses: Riemann boundary problem and singular integral equations;
Doctor of Sciences (in Physics and Mathematics, mathematical analysis), Institute of Mathematics of the Ukrainian Academy of Sciences, 2006. Dissertation Title: Monogenic functions in boundary problems for elliptic type equations with a degeneration on an axis
Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev,
Junior Research Fellow, 1989 – 1992
Research Fellow, 1992 – 1999
Senior Research Fellow, 1999 – 2006
Leading Research Fellow, 2006 – present
Honors and Grants:
1994 – 1995 ISF Grant UB 4000;
1995 – 1996 ISF Grant UB 4200;
1995 – 1998 INTAS Grant 94-1474;
1997 – 1998 Ukrainian – Polish Grant 2M/1401-97;
1999 ISAAC Award for outstanding research achievements in mathematics;
2000 – 2003 INTAS Grant 99-00089;
2008 Grant of London Mathematical Society
International Conferences and Congresses:
2nd European Congress of Mathematics (Budapest, Hungary, 1996);
Conference on Differential Equations and their Applications (Brno, Czech Republic, 1997);
XII International Conference on Analytic Functions (Lublin, Poland, 1998);
7-th International Colloquium on Finite or Infinite Dimensional Complex Analysis (Fukuoka, Japan, 1999);
Second International ISAAC Congress (Fukuoka, Japan, 1999);
International Conference dedicated to M.A. Lavrentyev on the occasion of his birthday centenary (Kiev, Ukraine, 2000), plenary lecture ”On the Lavrentyev’s problem: description of axial-symmetric potential fields by means of analytic functions”;
International Conference on Complex Analysis and Potential Theory (Kiev, Ukraine, 2001);
Ukrainian Congress of Mathematics (Kiev, Ukraine, 2001);
International Conference on Factorization, Singular Operators and Related Problems in Honour of Professor Georgii Litvinchuk (Funchal, Madeira, 2002);
International Workshop on Potential Flows and Complex Analysis (Kiev, Ukraine, 2002);
International Conference ”Complex Analysis and its Applications”, (Lviv, Ukraine, 2003);
International Workshop ”Potential Theory and Free Boundary Flows” (Kiev, Ukraine, 2003);
International Conference ”Analytic Methods of Analysis and Differential Equations” (Minsk, Belarus, 2003);
5-th International ISAAC Congress (Catania, Italy, 2005);
International Workshop ”Free boundary flows and related problem of analysis” (Kiev, Ukraine, 2005);
International Conference on Complex Analysis and Potential Theory (Gebze, Turkey, 2006);
6-th International ISAAC Congress (Ankara, Turkey, 2007);
International Conference ”Complex analysis and wave processes in mechanics” within the framework of Bogolyubov Readings (Zhitomir, Ukraine, 2007);
3-rd Schools on Abstract Differential Equations and Ordinary Differential Equations (Mostaganem, Algeria, 2008);
4-th International Conference on Complex Analysis and Dynamical Systems (Nahariya, Israel, 2009);
International Workshop ”Applied Nonlinear Analysis” (Holon, Israel, 2009);
International Conference ”Analytic methods of mechanics and complex analysis” dedicated to N.A. Kilchevskii and V.A. Zmorovich on the occasion of their birthday centenary (Kiev, Ukraine, 2009);
2nd Ukrainian Congress of Mathematics (Kiev, Ukraine, 2009);
3-rd International Workshop on Contemporary Problems of Mathematics and Mechanics (Minsk, Belarus, 2010);
6-th International Conference ”Finsler Extensions of Relativity Theory” (Moscow, Russia, 2010)
Complex and hypercomplex analysis;
Analytic function theory of complex variable;
Monogenic function theory in Banach algebras;
Boundary problems theory for monogenic functions;
Boundary problems of mathematical physics;
Singular integral equations and operators;
Perturbation theory of Noetherian and semi-Noetherian operators
Principal scientific results
1. Problem posed by M.A. Lavrentyev: to develop methods for investigation of spatial
potential solenoid fields analogous to analytic function methods in the complex plane for plane potential fields.
We constructed analytic functions of vector variable with values in an infinite-dimensional commutative Banach algebra and proved that components of these functions generate axial-symmetric potential functions and Stokes flow functions. In such a way new integral expressions for these functions were obtained and new method for investigation of spatial axial-symmetric potential solenoid fields was found. The suggested method is analogous to analytic function method in the complex plane and is the solution of the Lavrentyev problem for spatial axial-symmetric potential fields. Using new integral expressions for axial-symmetric potential functions and Stokes flow functions, we developed a functional-analytic method for effective solving boundary problems for axial-symmetric potential solenoid fields.
2. We developed an algebraic-analytic approach to equations of mathematical physics. An idea of such an approach means a finding of commutative Banach algebras such that monogenic functions defined on them form an algebra and have components satisfying to beforehand given equations with partial derivatives.
We obtained constructive descriptions of monogenic functions taking values in commutative algebras associated with the two-dimensional biharmonic equation and the three-dimensional Laplace equation by means of analytic functions of the complex variable. For the mentioned monogenic functions we established basic properties analogous to properties of analytic functions of the complex variable: the Cauchy integral theorem and integral formula, the Morera theorem, the uniqueness theorem, the Taylor and Laurent expansions.
3. The classical theorems on stability of the Noetherian properties of operators in complete spaces are well-known. However, attempts to eliminate the requirement of completeness of spaces from theorems on stability of the Noetherian property faced essential difficulties of topological nature.
To overcome these difficulties we have developed algebraic methods for proving theorems on stability of Noetherian and semi-Noetherian properties of operators in incomplete vector spaces. These algebraic methods are indifferent with respect to topological properties of the given spaces and operators.
Noether operators in incomplete spaces appear in the theory of singular integral equations on curves in the complex plane for extended classes of equation coefficients and curves. We studied the Noetherian property for singular integral Cauchy operators in incomplete spaces of quickly oscillating functions on closed Jordan regular rectifiable curves by the mentioned methods of algebraic nature.
4. Solvability of boundary problems for analytic functions theory in domains with boundary not being piece-wise smooth rectifiable depends on combined properties both of the given functions and the boundary. Difficulties increase if the index of a boundary problem is infinite. To overcome these difficulties we developed methods for constructing asymptotic expansions for the Cauchy-type integrals on non-regular rectifiable curves (in particular, on spiral-form curves). Using these asymptotic expansions, we have solved explicitly some boundary problems for analytic functions with infinite index in domains with both regular and non-regular rectifiable boundary. We have also solved explicitly some Riemann boundary problems with quickly oscillating coefficients on closed Jordan rectifiable curves.
(total number is more than 100):
1. Mel’nichenko I.P., Plaksa S.A. ”Commutative algebras and spatial potential fields”, Kiev, Institute of Mathematics of the National Academy of Sciences of Ukraine, 2008, 230 p. (in Russian).
2. Mel’nichenko I.P., Plaksa S.A. ”Potential fields with axial symmetry and algebras of monogenic functions of a vector variable”, Ukr. Math. J., Vol. 48, No. 11, 1717–1730 (1996); Vol. 48, No. 12, 1916–1926 (1996); Vol. 49, No. 2, 253–268 (1997).
3. Plaksa S. ”Algebras of hypercomplex monogenic functions and axial-symmetrical potential fields”, in: Proceedings of the Second ISAAC Congress, Fukuoka, August 16 – 21, 1999, Kluwer Academic Publishers, Vol. 1 (2000), 613–622.
4. Plaksa S. ”Boundary properties of axial-symmetrical potential and Stokes flow function”, in: Finite or Infinite Dimensional Complex Analysis. Lecture Notes in Pure and Applied Mathematics. Marcel Dekker Inc., 214 (2000), 443–455.
5. Plaksa S. ”Singular and Fredholm integral equations for Dirichlet boundary problems for axial-symmetric potential fields”, in: Factotization, Singular Operators and Related Problems: Proc. of Conference in Honour of Prof. Georgui Litvinchuk, Funchal, January 28 – February 1, 2002, Kluwer Academic Publishers (2003), 219–235.
6. Plaksa S. ”On integral representations of an axisymmetric potential and the Stokes flow function in domains of the meridian plane”, Ukr. Math. J., Vol. 53, No. 5, 726–743 (2001); Vol. 53, No. 6, 938–950 (2001).
7. Plaksa S. ”Dirichlet problem for an axisymmetric potential in a simply connected domain of the meridian plane”, Ukr. Math. J., Vol. 53, No. 12, 1976–1997 (2001).
8. Plaksa S. ”On an outer Dirichlet problem solving for the axial-symmetric potential”, Ukr. Math. J., Vol. 54, No. 12, 1634–1641 (2002).
9. Plaksa S. ”Dirichlet problem for the Stokes flow function in a simply connected domain of the meridian plane”, Ukr. Math. J., Vol. 55, No. 2, 241–281 (2003).
10. Mel’nichenko I.P., Plaksa S.A. ”Outer boundary problems for the Stokes flow function and steady streamline along axial-symmetric bodies”, in: Complex Analysis and Potential Theory, Kiev, Institute of Mathematics of the National Academy of Sciences of Ukraine (2003), 82–91.
11. Mel’nichenko I.P., Plaksa S.A. ”Commutative algebra of hypercomplex analytic functions and solutions of elliptic equations degenerating on an axis”, in: Transactions of the Institute of Mathematics of the National Academy of Sciences of Ukraine, Vol. 1 (2004), No. 3, 144–150.
12. Plaksa S. ”Commutative algebras of hypercomplex monogenic functions and solutions of elliptic type equations degenerating on an axis”, in: More progress in analysis: Proc. of 5th International ISAAC Congress, Catania, July 25 – 30, 2005, World Scientific (2009), 977–986.
13. Grishchuk S.V., Plaksa S.A. ”On construction of generalized axial-symmetric potentials by means components of hypercomplex analytic functions”, in: Transactions of the Institute of Mathematics of the National Academy of Sciences of Ukraine, Vol. 2 (2005), No. 3, 67–83.
14. Grishchuk S.V., Plaksa S.A. ”Integral representations of generalized axially symmetric potentials in a simply connected domain”, Ukr. Math. J., Vol. 61, No. 2, 195–213 (2009).
15. Plaksa S. ”Harmonic commutative Banach algebras and spatial potential fields”, in: Complex Analysis and Potential Theory: Proc. of Conference Satellite to ICM-2006, Gebze Institute of Technology, Turkey, September 8 – 14, 2006, World Scientific (2007), 166–173.
16. Plaksa S. ”An infinite-dimensional commutative Banach algebra and spatial potential fields”, in: Further progress in analysis: Proc. of 6th International ISAAC Congress, Ankara, August 13 – 18, 2007, World Scientific (2009), 268–277.
17. Grishchuk S.V., Plaksa S.A. ”Monogenic functions in a biharmonic algebra”, Ukr. Math. J., Vol. 61, No. 12, 1865–1876 (2009).
18. Plaksa S.A., Shpakivskyi V.S. ”Constructive description of monogenic functions in a harmonic algebra of the third rank”, Ukr. Math. J., Vol. 62, No. 8, 1078–1091 (2010).
19. Shpakivskyi V.S., Plaksa S.A., ”Integral theorems in a commutative three-dimensional harmonic algebra”, in: Progress in analysis and its applications: Proc. of 7th International ISAAC Congress, London, July 13 – 18, 2009, World Scientific (2010), 977–986.
20. Plaksa S.A., ”Singular integral operators in spaces of oscillating functions on a rectifiable curve”, Ukr. Math. J., Vol. 55, No. 9, 1457–1471 (2003).
21. Plaksa S. ”Differentiation of singular integrals with piecewise continuous density”, in: Analytic Methods of Analysis and Differential Equations. Cottenham: Cambridge Scientific Publishers (2006), 199–208.
22. Vasil’eva Yu.V., Plaksa S.A. ”Piecewise-continuous Riemann boundary value problem on a rectifiable curve”, Ukr. Math. J., Vol. 58, No. 5, 616–628 (2006).
23. Plaksa S.A., Kudjavina Yu.V., ”Riemann boundary value problem on an open Jordan rectifiable curve”, Ukr. Math. J., Vol. 62, No. 11, 1511–1522 (2010); Vol. 62, No. 12, 1659–1671 (2010).