INVESTIGATIONS IN ANALYSIS, DIFFERENTIAL EQUATIONS AND APPLICATIONS

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1 International Conference Advances and Perspectives of Basic Sciences in Caucasus and Central Asian Region Tbilisi, November 1-3, 2011 MATHEMATICS IN GEORGIA, PART 2: INVESTIGATIONS IN ANALYSIS, DIFFERENTIAL EQUATIONS AND APPLICATIONS ROLAND DUDUCHAVA I. Javakhishvili State University, A. Razmadze Mathematical Institute, Georgian Mathematical Union

2 Main topics Let me survey main achievements of the Mathematicians in Georgia in recent years in the following topics: Analysis; Ordinary differential equations (ODEs); Partial differential equations (PDEs); Applications to: 2D and 3D Elasticity theory; scattering of EM-waves; shell theory. These investigations are traditional in Georgia and are motivated strongly by the applications.

3 Research groups Besides A. Razmadze Mathematical Institute, Tbilisi State University (TSU) strong research groups are present at: TSU, Department of Mathematics - Differential Equations, Analysis TSU, I.Vekua Institute of Applied Mathematics - Differential and Integral Equations, Elasticity, Shell Theory Georgian Technical University (GTU) - Differential and Integral Equations, Elasticity theory, Analysis GTU, N.Muskhelishvili Institute of computational Mathematics - Integral Equations A.Tsereteli State University, Kutaisi Fourier a nalysis

4 Analysis I Analysis delivers essential tools for investigations in Mathematics and applications (Physics, Elasticity ). In Georgia is performed, traditionally, an Intensive research in Analysis. A very short list of Main achievements: Theory of integral operators in nonstandard Banach function spaces: variable exponent weighted spaces, including grand Lebesgue spaces GNSF, INTAS & NATO; Solution of trace problems for Riesz potentials, fractional maximal functions and potentials with product kernel in Grand Lebesgue Spaces GNSF & INTAS;

5 Analysis II Solution of two-weighted problems for singular integral operators on traditional spaces and for potential type operators and integral transforms defined on: nilpotent Lie groups, fractals quasisim- metric spaces SOROS, GNSF & EPRSC, UK; New effective matrix spectral factorization method with connection to wavelet theory JSPS & MIF Japan; Solution of the Dirichlet, Riemann-Hilbert-Poincaré etc. problems in domains with non-smooth boundaries in the class Hardy and Smirnov classes with variable exponents GNSF & INTAS.

6 Ordinary Differential Equations I Differential equations encounter almost in all mathematical models in Mechanics, Physics, Biology etc. It is divided in two: ordinary (ODE) and partial (PDE). A short list of investigations in ordinary differential equations: Construction of the theory of boundary value problems for linear and nonlinear ordinary differential equations and systems with nonintegrable singularities - GNSF;

7 Ordinary Differential Equations II Elaboration of new methods for investigation of non-classical initial-boundary value problems for multi-dimensional nonlinear partial differential equations of hyperbolic type - GNSF; Criteria for the existence of so-called proper and blow-up solutions to higher order essentially nonlinear non- autonomous ordinary differential equations. Detailed Asymptotics of such solutions - GNSF.

8 Partial Differential Equations I A big majority of mathematical models in Elsticity, Physics, Biology etc. lead to Boundary Value Prblems (BVPs) for Partial Differential Equations (PDEs). A big challenge at the turn of 20-th and 21-st century was the development of the theory of BVPs for PDEs in domains slit by open surfaces (crack and screen -type problems). The results apply to the investigation of cracks problems in 3D elastic bodies, scattering (diffraction) of electromagnetic waves by open surfaces, antennas and surfaces of military and civil objects etc.

9 Partial Differential Equations II The challenge was to establish not only the solvability properties, but develop tools for analysis of singularities and efficient numerical methods. A short list of contribution of the Georgian mathematicians in the theory of PDEs: Full Hp-theory of solvability of crack-type BVPs for Partial differentil equations - Soros; Full asymptotics of solutions to BVPs for cracktype problems near edges (with extension to pseudodifferential equations) - DFG;

10 Partial Differential Equations III Theory of general PDEs: Green formulae, Layer potentials, Calderon projections, Reduction to boundary pseudodifferential equations - DFG; Calculus of tangential PDEs on surfaces (Gunter s & Stoke s derivatives), enabling essential simplification of BVPs on surfaces, including equations for shells Milnor & DFG. New method of numerical solutions of BVPs for PDEs with variable coefficients based on the localized parametrix method - EPRSC, UK; Application of the calculus of Gunter s & Stocke s derivatives to BVPs on surfaces: Simplified form of Laplace- Beltrami Navier-Stocke s etc. equations DFG & GNSF.

11 Applications: 2D Elasticity theory A contact problems of deformable bodies, of infinite compound and infinite orthotropic plates with elastic inclusions of variable rigidity - GNSF. asymptotic analysis of unknown contact stresses at the boundary of elastic inclusions - GNSF. Investigation of related third kind singular integral equations, Riemann and Riemann-Hilbert- Carleman BVPs using the methods of complex analysis - GNSF.

12 Applications: 3D Elasticity theory I Solvability and uniqueness of solutions to the 3D interface crack problems (ICPs) for metallic piezoelastic bodies with thermal effects DFG & GNSF; ICPs with non-classical boundary conditions (mixed type problems, Dirichlet-Neumann etc.) EPRSC; Full asymptotics of corresponding thermo-mechanical and electric fields in metallic piezo-elastic bodies with thermal effects near the crack edges and near the curves where different boundary conditions collide DFG & GNSF;

13 Applications: 3D Elasticity theory II Efficient algorithms for calculation of stress singularity exponents are written. Is shown a clear dependence of the stress singularity exponents on the material parameters - EPRSC & GNSF; Consideration of basic boundary and transmission problems of elasticity for hemitropic Cosserat type solids - GNSF. The contact problems with and without friction conditions are studied by using a variational inequality technique GNSF.

14 Applications: scattering of EM-waves Solvability, uniqueness of solutions and numerical methods for scattering of electro-magnetic (EM) waves by anisotropic multi-layered media- Dornje (Germany) & GNSF; Uniqueness of solutions to Maxwell s equation in anisotropic media (Sommerfeld type conditions) - GNSF; Full asymptotic of solutions to BVPs for Maxwell s equation, describing scattering and diffraction of EM waves by open surfaces in anisotropic media - GNSF.

15 Applications: shell theory Application of the calculus of Gunter s and Stocke s derivatives to shell theory essentially simplified equations of shells, similar to Lame equations, which allow effective investigation of solvability and numerical treatment - GNSF. I. Vekua s hierarchical model is extended and worked out for cusped beams and prismatic shells - GNSF.

16 International cooperation Kings College & Bruenel University, London, University of Delaware & University of Missuri USA, University Rennes 1, France, University of Rome, Italy, Stuttgart & Saarbruecken University, Germany, Technical University of Lisbon, Portugal, University of Faro & University of Aveiro, Portugal, University of Athens, Greece, University of Mexico, Mexico, Lulea University of Technology, Sweden, Universities in Japan, Chech Rep. Spain, Hungary,

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