Some Researches On The A-DIRAC Equation

This paper mainly concerns on the existence of the solution of the A-Dirac equation and some inequalities of the solution of the nonhomogeneous A-Dirac equation. A-Dirac is an important generalization of quasilinear elliptic equations -divA (x,(?)u)=0 and Dirac Laplace equation, and it has extensively applications in the field of potential theory, partial differential equation, nonlinear analysis, ect.At first, we introduced the source and present situation of the A-Dirac equation systematically, and we have got the relationship (Du)~#=du~#+d~*u~# between the exterior differential form which arises from the Grassmann algebra and the differential form which arises from the Clifford algebra. According to the relationship above, we proved the Poincaréinequality in the differential form which arises from the Clifford algebra, which generalized Poincaréinequality in the Sobolev space.Secondly, according to the definitions of the supersolution, subsolution and solution of quasilinear elliptic equations -divA (x,(?)u)=0, we defined those of the A-Dirac equation. Then we introduced the obstacle problem of the A-Dirac equation and have proved the existence of the solution. Consequently, we got the existence of the solution of the A-Dirac equation.Finally, we give the two inequalities, Caccioppoli inequality and weakly reverse H(?)lder inequality of the solution of the nonhomogeneous A -Dirac equation, and the corresponding inequalities of the solution in the weighted form, which expanded the scope of application of the two inequalities, these conclusions will be a nice foundation for further study of the A-Dirac equation.