| Quasiregular mappings are generalizations of complex functions (Analytic functions or regular functions). They have more broad applications than analytic functions in mathematics, physics and engineering ([Bek][Fa4] etc.). Where quasiregular mappings are generalized solutions of Beltrami system with single characteristic matrix G (x) or Beltrami system with double characteristic matrix G (x) and H (x) It is known that when n =2, system (I) and (II) are equivalent to Beltrami equation with characteristic μ(z) and Beltrami equation with double characteristic μ1 (z) and μ2 (z) w z = μ1 ( z)wz+μ2(z)wz, μ1 ( z )+ μ2(z)≤k2<1 (IV) respectively. Since equations (III) and (IV) are linear, quasiregular mappings in the plane and their generalizations —generalized analytic functions were developed greatly in the 1950's and 1960's, and have got great applications in differential geometry, partial differential equations and mechanics (i.e. thin crust). Quasiregular mappings in space have also got developed since 1960's and 1970's. But the development is slow because of their strong nonlinearity. Quasiregular mappings got breakthrough since the work of S.K.Donalson and D.P.Sullivan "Quasiconformal 4-manifolds"[DS ]. T.Iwaniec and G.Martin studied the properties of generalized solutions of system (I) with single characteristic matrix recurred to the idea of [DS], and obtained their sharp Liouville's Theorem, regularity and removability of the "linearized"Beltrami equation (I). T.Iwaniec [I 4] obtained also the p-harmonic equation and regularity of quasiregular mappings in all dimensions recurred to nonlinear partial differential equations and the modern theory of harmonic analysis. For system (II) in high dimension, T.Iwaniec studied the stability of solutions with the elliptic condition. Cheng Jinfa studied the regularity and removability of solutions in the condition that H (x) is diagonal. Zheng Shenzhou studied the regularity of the solutions in all dimensions with the condition that H (x) is Ho&& lder continuous. So there are a lot of problems to be studied in quasiregular mappings, especially Beltrami systems with double characteristic matrix. This paper studies many problems thereinto. In Chapter 1, we obtain the p-harmonic equation of Beltrami equation (II) with double characteristic matrix using outer differential andouter algebra of matrix, and get rid of the conditions of H (x) of Cheng Jinfa and Zheng Shenzhou. In the final, we obtain the weak monotonicity of component functions of weak solutions of Beltrami system in certain conditions. Weak monotonicity property is the generalization of monotonicity of functions, which have important applications in the theory of regularity of partial differential equations. In 1973, F.W.Gehring obtained the Reverse Ho&& lder inequality and Lp ( p> n) integrability of space quasiconformal mappings. The method and results have got broad applications in partial differential equations and harmonic analysis [Gia1 ]. It is natural to ask that if it has the same results for quasiregular mappings, which are the generalizations of quasiconformal mappings (that is, getting rid of the homeomorphisim)? In 1992, T.Iwaniec melted the Beltrami system into p-harmonic equation, and gave the Reverse Ho&& lder inequality of the equation. But the Reverse Ho&& lder inequality for quasiregular mappings has not been obtained. In Chapter 2 of this paper, we found a different method to obtain the Reverse H o&& lder inequality, regularity and H o&& lder continuity of quasiregular mappings with the help of some results obtained by B.Bojarski and T.Iwaniec when they studied space quasiconformal and quasiregular mappings. Next, we obtained also the weak monotonicity of component functions of quasiregular mappings when its integrability exponent is near to the space dimension.In 1977, L.Simon studied the ( K 1 ,K2) quasiconformal mappings between surfaces of two-dimension of R 3 and established their Ho&& lder continuity formulas. In [GT], D.Gilbarg and N.S.Trudinger obtained the 1,αC loc prior estimate of quasilinear elliptic equations of two variables and established the existence theorem of the solutions of Dirichlet boundary value problem using the H o&& lder continuity method established by studying the ( K 1 ,K2)quasiregular mappings in the plane. Because of the important significance of ( K 1 ,K2)quasiregular mappings in plane in the prior estimates of nonlinear PDEs, in 1998, Zheng Shenzhou and Fang Ainong [ZF 1] extended the concept of ( K 1 ,K2)quasiregular mappings to R n space ( f ∈Wlo1,c n(?)was needed) with the help of outer differential forms, obtained the Lp ( p> n) integrability, and get rid of the limitary assumption of unknown functions. It is natural to ask that if it is true for us if we weaken the integrability condition of f to f ∈Wlo1,c n? ε(?),0<ε<1 (in the later part we will call such f weak quasiregular)? This paper solved this problem partly. In Chapter 3, we obtained the regularity result of weak quaisregular mappings in the conditions that: if K 2 ≠0, then there exist some i ,1 ≤i≤n, so that df i≥ε0 >0; if K 2 =0, then J f ( x)>0. The method is also different from former researchers. Many properties of weak solutions of A-harmonic equation, which is closely related to quasiregualr mappings, have been obtained. A natural generalization of weak solution is very weak solution. The properties of...
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