A Study On Quasi-linear Elliptic Equation

In the theoretical research of partial differential equations, the second-order quasi-linear elliptic partial differential equations are very important. It is closely relative with the industrial, economic, medical, and it has been widely applied in information science, biology, engineering, and aerodynamics, meteorology, fluid mechanics in physics. Such as: many of the problems in physics can be attributed to second-order quasi-linear elliptic partial differential equations and systems, meanwhile the A-harmonic type equation has a wide range of applications in quasi-regular mappings, elasticity and physics.Well-poseness of solutions of the quasi-linear elliptic partial differential equation of second-order, includes existence of solutions, uniqueness and stability and so on. From the study of classical solutions, one start to study weak solutions and very weak solutions. The so-called weak solutions of second-order quasi-linear elliptic equationThe definition of very weak solutions of elliptic equations was first given in 1992 by T. Iwaniec [1], and the very weak solutions of the second-order quasi-linear elliptic equations means: is called very weak solutions of the equation. At present, under the different conditions, well-poseness of weak solutions and very weak solutions of the quasi-linear partial differential equations of second-order has been extensively researched in different space. For example, there are many conclusions of existence and uniqueness of weak solutions and very weak solutions. It will be related in detail in chapter one. However we find seldom results of stability with respect to weak solutions and very weak solutions to above mentioned equations. The stability of solutions is the continuous dependence of the solution, and it may be depends on the index of operators, a boundary condition, or other parameters. In this paper, based on the known results, for a class of second-order quasi-linear elliptic equation , we give a uniform estimate of the gradient of weak solutions and the stability of solutions with respect to domain. And we study the weak solutions of the obstacle problem of a class of second-order quasi-linear elliptic equation divA(x,u,â–³u) = 0. With the conditions of some restrictions on the lower-order items, we obtain a uniform estimate by applying the capacity theory in nonlinear potential under the conditions of uniform p-thick regions, and by the virtue of Sobolev inequality and some capacity inequalities. Whereas the uniform estimate of solution of equation is the base of the stability of solution in certain conditions. With regard to the stability of weak solutions of equations, when the outside of regions satisfy uniform p-thick conditions, we show that the weak solutions of equations are stable by using regional variation and Hardy inequality. For the obstacle problem of second-order quasi-linear elliptic equation, we obtain a property of weak solutions of one-side obstacle problem mainly using Hodge decomposition, Sobolev space theory and some inequalities. In our studies, the main tools are real analysis, differential geometry and analysis methods of Sobolev spaces and nonlinear potential theory, harmonic analysis as well as others. In the paper, H?lder inequality, Young's inequality and the Sobolev embedding theorem are extensively applied in the proofs. Under some weaker assumptions about A(x,u,Du) than those in [9, 7, 17], we obtain some prescribed results. Indeed, in order to assure uniqueness and stability of weak solutions and very weak solutions, we need to restrict the integrable index of u, such that Sobolev embedding theorem can be applied.This paper consists of four parts. The first part is the introduction, the second part is a uniform estimates of gradient of weak solutions of a class of elliptic equations, the third part is the stability of weak solutions with the domain to a class of quasi-linear elliptic equations, at last, a property of very weak solutions of a obstacle problem for a class of second-order quasi-linear elliptic equations.