The Relation Between A-harmonic System And A-dirac System

A-harmonic equation express quite accurate in describing the electromagnetic field, the theory of relativity, the theory of elasticity and nonlinear potential theory. A-Dirac equation is an important extend of quasi-linear elliptic equations and Dirac Laplace equations. For the study of A-harmonic equation with A-Dirac equation can not only promote the development of the partial differential equations, harmonic analysis and related fields of nonlinear potential theory, but also provide a tools for studying in related fields of natural science and engineering systems related field, which make math and engineering linked.This paper studies under controllable growth conditions, the correlation between A-Dirac systems and A-harmonic systems. Combined the method of removes and some techniques used in proving partial regularity theory, we can get the following main results.Theorem3.1Let E be a relatively closed subset of Ω. Suppose that u∈Llocp(Ω) has distributional first derivatives in Ω, u is a solution to the scalar part of the A-Dirac systems-D A(x, Du)=f(x, Du) under controllable growth condition in Ω\E,and u is of p,k-oscillation in Ω\E. If for each compact Subset K of Ethen u extends to a solution of the A-Dirac systems in Ω.Theorem4.1Let E be a relatively closed subset of Ω. Suppose that u∈LLtocp(Ω) has distributional first derivatives in Ω, u is a solution to the scalar part of the A-Dirac equation-D A(x,u,Du)=f(x,u,Du) under controllable growth condition in Ω\E,and u is of p,k-oscillation in Ω\E. If for each compact Subset K of E then u extends to a solution of the A-Dirac systems in Ω.