Some Problems On The Solutions Of The Beltrami Equation

The Beltrami equation is applied widely on Hydrodynamics, Elastic mechanics and Modern control theory with the development of the Cauchy-Riemann equation. The Beltrami equation is divided two types from the expression.The firstThe secondμ(z) and a(z) are called the complex dilatations。For the long time, many mathematicians have been interested in the problem of the existence and uniqueness of the homeomorphisms solutions of the Beltrami equation which is one of the hot points. In Section 2 and Section 3, we discuss this question in detail.Sufficient conditions for the existence of a homeomorphic solution to the Beltrami equation are given in terms of the complex dilatations coefficient. The L₂ - homeomorphism solution of the first Beltrami equation is a quasiconformal mapping when ||μ||_∞ < 1. For the converse process, a quasiconformal mapping must be a L₂- homeomorphism solution of some aBeltrami equation . There is a harmonic function f which can map one given domain onto the other given domain when ||a|| < 1, that is to say , f does not collapse. In case of ||μ||_∞ = 1 or ||a||_∞ =1, the solutions of the Beltrami equation are complicated. For ||μ||_∞ =1, we add some new conditions to the function μ(z), so we obtain the sufficient and necessary conditions for the existence and uniqueness of the μ(z)-homeomorphic solutions to the Beltrami equation. For ||a||_∞ =1 , we discuss mainly the existent conditions and properties ofthe solutions when a(z) is a Blaschke product.In Section 4, we discuss the estimate of modulus on univalent harmonic function. We research it by some new knowledge combining the quasiconformal theory with the harmonic theory. A new estimate of modulus is given which is relation to the angular dilatation.