Regularity For Weak Solutions Of A-Harmonic Equations

A-harmonic equation is a highly crucial type of partial differential equation that has been widely applied in various physical scenarios.In recent years,research on weak solutions to A-harmonic equation mostly focus on the local regularity,while those focusing on the global regularity are very scarce.The article to study the global regularity of weak solutions to A-harmonic equation,and the details are as follows.Chapter 1 began with an introduction to the problem's background and significance,proceeded to describe the research status of A-harmonic equation and its weak solutions(very weak solutions),before finally introducing the overall structure and main work.Chapter 2 introduced the related research on the regularity of weak solutions to Aharmonic equation.At the same time,the research progress on the regularity of weak solutions(very weak solutions)to A-harmonic equation,such as integrability,continuity,and removable singularity,etc.,were also described.In Chapter 3,the div A(x,?u)(28)0 boundary value problem of homogeneous Aharmonic equations was studied.The global regularity of very weak solutions is proven by means of Hodge decomposition theorem and Sobolev space analysis under controlled growth conditions.In Chapter 4,the div A(x,?u)(28)f(x)boundary value problem of A-harmonic equations was studied.This chapter also shows how,under controlled growth conditions,the global regularity of very weak solution is obtained using the Sobolev space analysis method,Hodge decomposition theorem and Gehring lemma.In Chapter 5,the div A(x,?u)(28)B(x,?u)boundary value problem of nonhomogeneous A-harmonic equation was discussed,and showed that the nonhomogeneous term on the right satisfied the controlled growth condition.In this chapter,a new test function is also constructed based on Hodge decomposition theorem,and the global regularity of weak solutions to non-homogeneous A-harmonic equation is proven by combining H?lder inequality,Young inequality,and other methods.Figure 0;Table 0;Reference 56...