International Conference on Algebras and Lattices Prague, Czech Republic June 21 -25, 2010
Algebra of the solutions to the Beltrami equation
Algebra of the solutions to the Beltrami equation Eduard Yakubov H. I. T. - Holon Institute of Technology Israel Jointly with U. Srebro (Technion) and D. Goldstein (H. I. T. ) Prague-2010
The authors would like to thank the organizers for invitation and hospitality!
Notation:
Complex derivatives:
Complex dilatation:
Jacobian of
Sense-preserving mappings Sense-reversing mappings
The Beltrami Equation
Historical Remarks Gauss (1880) Isothermal Coordinates - Real Analytic case Korn (1914)&Lichtenstein (1916) - Holder Cont’ case Lavrent’ev (1935) - Continuous case Morrey (1938) - Measurable case Ahlfors, Bojarski, Vekua (mid 50’s) - Measurable case Singular Integral Methods
Among contributors to (B) theory Ahlfors, Andrean-Cazacu, Astala, Belinski, Bers, Bojarski, Brakalova, Danilyuk, David, Dzuraev, Earle, Gutlyanskii, Heinonen, Iwaniec, Jenkins, Krushkal, Kuhnau, Lavrent’ev, Lehto, Martin, Martio, Migliaccio, Miklyukov, Moscariello, Morrey, Mueller, Painvarinta, Pesin, Pfluger, Reich, Ricciardi, Ryazanov, Salimov, Sastry, Shabat, Sheretov, Srebro, Strebel, Sugava, Sullivan, Suvorov, Sverak, Teichmuller, Tukia, Vekua, Virtanen, Volkoviski, Vuorinnen, Walczak, Zhong Li, Yakubov, Zahirov, Zorich
Applications Uniformization Robotics Elasticity Control Theory String Theory Hydro - dynamics Kleinian Groups Riemann Surfaces Teichmuller Spaces Holomorphic Motions Complex Dynamics Differential Geometry Conformal Field Theory Quasi-conformal mappings Low Dimensional Topology
Classification of (B) The Classical Case: The Relaxed Classical Case: The Alternating Case:
Solutions in the measurable case A continuous mapping a solution to (B) if satisfy (B) a. e. is called and ACL mean absolutely continuous on a. e. horizontal and vertical lines
Open and discrete solutions of (B) A solution whenever is called open if is open. A solution is called discrete if discrete set for every. is open is a
Main Problems Existence Uniqueness Regularity Representation of the solution set Removability of isolated singularities Boundary behavior Mapping properties
Existence and uniqueness of solutions THEOREM Let be a measurable function with Then: 1. (B) has a homeomorphic solution. 2. is unique up to a post-composition by a conformal map 3. generates the set of all open, discrete and s. p. solutions, i. e. for every open, discrete and s. p. solution there is an analytic function such that
Algebra of solutions THEOREM 1 For every measurable function the set of the solutions of equation (B) is an algebra (with respect to point-wise addition & multiplication) Proof: Straightforward calculation
Algebra of solutions THEOREM 2 Let be a measurable function with. The set of open, discrete and s. p. solutions of equation (B) is a sub-algebra of. Proof: Based on Stoilow’s decomposition theorem
Algebra of solutions THEOREM 3 Let be measurable functions. If a. e. then.
Algebra of solutions THEOREM 4 Let be a measurable function with. Then the algebra of open, discrete and s. p. solutions of (B) is isomorphic to the algebra of analytic functions in. Proof: Based on Stoilow’s decomposition theorem
Algebra of solutions Remark: Similar results are also true in the alternating case for (B) which has : 1. Proper folding solutions 2. Proper (p, q)- cusp solutions
Algebra of solutions 3. Proper straight umbrella solutions
Uniqueness and Representation Properties in the alternating case 1. 2. 3. 4. 5. The following solutions are unique and generates the solution set: Proper folding solutions Proper (p, q)-cusp solution with |p-q|=2 Proper umbrella solution of degree 1 or -1 6. Corollary: (B) cannot have both folding solutions 7. and (3, 1)-cusp solutions
Algebra of solutions Open problems: 1. Characterize algebras of solutions in the relaxed classical case: 2. 3. 4. 5. What are connections between topological properties of solutions and corresponding algebras? Describe ideals of algebras of solutions according to Describe lattices of the solutions according to. .
Reference: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. U. Srebro and E. Yakubov, Beltrami Equations, Handbook in Complex Analysis, 2005, Elsevier 2. T. Iwaniec and G. Martin, The Beltrami Equation, 2008, Memories of AMS 3. O. Martio, V. Ryazanov, U. Srebro and E. Yakubov, Moduli in Modern Mapping Theory, 2009, Springer Monographs in Mathematics 4. V. Gutlyanskii, V. Ryazanov, U. Srebro and E. Yakubov, The Beltrami Equation: A Geometric Approach, Springer Monographs in Mathematics (to appear)
Thank you for your attention !
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