The Relation Between A-dirac Systems And A-harmonic Equations Under Natural Growth Condition

A-Dirac systems are the important promotion of quasilinear elliptic equation-divA(x,▽u)=0and Dirac Laplace equation. They are applied widely in transport theory, partial differential equations and nonlinear analysis. A-harmonic systems are an important development of p-harmonic equation. It is a bridge connecting mathematics and natural sciences, engineering technology and the other branches.In this paper, we consider the relation between A-Dirac systems and A-harmonic systems. By the method of removable theorem and regularity theory, we establish that, under the natural growth condition, the relation between A-Dirac systems-DA(x,Du)=f(x,Du) and A-harmonic systems-divA(x,▽u)=f(x,▽u). Furthermore, we even extend this result to the case of the operator A also depending on weak solution u, and obtain that, under natural growth condition, the relation between A-Dirac systems-DA(x,u,Du)=f(x,u,Du) and A-harmonic systems-divA(x,u,▽u)=f(x,u,▽u). In other words, we get the following main results in this paper.Theorem3.1.1Let E be a relatively closed subset of Ω. Suppose that u∈Llocm(Ω)(?)L∞(Ω) has distributional first derivatives in Q, u is a solution to the scalar part of A-Dirac systems-DA(x, Du)=f(x, Du) under growth condition f(x,▽u)≤C1|ζ|m+C2in Ω\E,and u is of m,k-oscillation in Ω\E. If for each compact subset K of E,∫K(1)\Kd(x,K)m(k-1)-kdx<∞then u extends to a solution of the A-Dirac systems in Ω.Theorem4.1.1Let E be a relatively closed subset of Ω. Suppose that u∈Llocm(Ω)(?)L∞(Ω) has distributional first derivatives in Q, u is a solution to the scalar part of A-Dirac systems-DA(x,u,Du)=f(x,u,Du) under natural growth condition (H1)-(H4) in Ω\E, and u is of m, k-oscillation in Ω\E. If for each compact subset K of E, ∫K(1)\Kd(x,K)m(k-1)-kdx<∞then u extends to a solution of the A-Dirac systems in Ω.