Properties Of Quasiregular Mappings And Very Weak Solutions To A Harmonic Equations

Quasiregular mappings are generalizations of complex functions(analytic functions or regular functions). They have more broad applications than analytic functions in mathe-matics, physics and engineering, where quasiregular mappings are gerneralized solutions of Beltrami system with one (or two) characteristic matrices.In Chapter 2, we obtain the Caccioppoli type estimate of degenerate weakly quasireg-ular mappings by using its definition. As we know, one can easily obtain the weakly reverse Holder inequality from the Caccioppoli inequality, and then we can obtain the regularity result of degenerate weakly quasiregular mappings. We use the point-wise inequality of Sobolev functions, Mcshane extension theorem and some closely related results to com-plete our proof. The method used in this paper is different from the classical one of Hodge decomposition, which makes the computations and the estimates of exponents in the proof more easilier.Many properties of weak solutions of A-harmonic equation div A(x,▽u(x))=0 which is closely related to quasiregular mappings, have been obtained. A natural generaliza-tion of weak solution is very weak solution. The properties of very weak solution, especialy regularity, existence and uniqueness, have been paid attention to and studied widely.In Chapter 3, we obtain the Comparison Priciple of very weak solutions of A-harmonic equation. If two very weak solutions u1,u2 satisfiing u1≥u2 on boundary ofΩ, then u1≥u2 a.e. inΩwith certain conditions. If the integrability exponent of very weak solutions is equal to the natural one, then this result is nothing but the classical one. Meanwhile, a direct result of Comparison Principle is Maximal Principle.In Chapter 4, we investigate the property of very weak solutions to single obstacle prob-lem associated with A-harmonic equations. By using the point-wise inequality of Sobolev functions, we constrcut a gloable continuous function to act as test fucntion in the definition of very weak solutions. Then we obtain the quasiminimizer property and higher integrabil-ity result of very weak solutions by Mcshane extension theorem and some closely related results. The results are coincide with the classical ones. Meanwhile, we also obtain the similar results of very weak solutions of single obstacle problem associate with nonhomo- geneous A-harmonic equation. div A(x,▽u(x))=divF(x)In Chapter 5, we study the properties of very weak solutions to double obstacle prob-lems associate with A-harmonic equations. By using the point-wise inequality of Sobolev functions, Mcshane extension theorem and some closely related results, we also obtain the quasiminimizer property and regularity result of very weak solution. At same time, we ob-tain related results of nonhomogeneous double obstacle problem, such as quasiminimizer property and regularity result. In the end, we discuss the relation between single obstacle problems and double obstacle problems, and obtain a convergence property of very weak solution to double obstacle problems.