Gradient Estimates For Very Weak Solutions Of The A-harmonic Equation And Its Obstacle Problems

Partial differential equation is an important part of modern mathematics.It is mainly used in modern physics,biology,fluid mechanics and other disciplines.Among them,Aharmonic equation,as a generalization of many classical harmonic equations,have been deeply studied,and many important results have been obtained.These research results have important theoretical and application value for natural science and engineering technology.The properties of very weak solutions for the A-harmonic equation are studied,mainly focuses on gradient estimates for very weak solutions of the A-harmonic equation and its obstacle problem.The paper is divided into four chapters,and the content of each chapter is organized as follows:In Chapter 1,the background and significance,the research development of the Aharmonic equation and the research scheme are introduced briefly.In Chapter 2,the symbols and relevant preliminary knowledges are introduced.In Chapter 3,the gradient estimates for very weak solutions of the A-harmonic equation are discussed.The classical result(gradient estimates for weak solutions of the Aharmonic equation in Orlicz space)is extended to very weak solution.The regularity estimates of very weak solutions in Sobolev space are obtained by using Hodge decomposition theorem to construct the test function and with some inequalities tools.Then the estimates are extended to Orlicz space by Jensen’s inequality and Orlicz space lemma,combining the new normalization method and the iterative covering method.In Chapter 4,the gradient estimates for very weak solutions of the obstacle problem are discussed.The results of the gradient estimates in Chapter 3 are extended to obstacle problems.The regularity estimates for very weak solutions of the obstacle problem in Orlicz space are obtained by constructing a permissible function,using Hodge decomposition theorem,Young inequality,H(?)lder inequality,Sobolev-Poincaré inequality,and combining with comparative estimates,new normalization method,and iterative covering lemma.Figure 0;Table 0;Reference 54...