Researches On Weak Solutions To A-dirac Equations For Clifford-valued Functions

In recent decades, theoretical researches on A-harmonic equations have made con-siderable headway and have attracted the interests of many mathematicians. Meanwhile, as a generalization of Laplace equations and the second order linear partial differential equations, it draws extensive attentions of the researchers from the fields of physics and engineering technique. Clifford algebras are a mathematical tool which originates form physics, and have been widely applied on the partial differential equations, theory of func-tion space and differential geometry, etc. In2011, Nolder raised the A-Dirac equations DA(x, Du)=0which based on the Clifford-valued function space. A-Dirac equation-s are a generalization of A-harmonic equations in higher dimensions, which makes the A-harmonic equations have more widespread applications and can be applied to more complex practical problems in physics and engineering. Therefore, it has the theoretical and actual meaning for us to study A-Dirac equations, and it will greatly promote the progress of A-harmonic equations.In this dissertation, we mainly study the properties of solutions to a class of A-Dirac equations with Dirichlet boundary data. The specific content of this dissertation is as follows:Firstly, Poincare type inequalities are proved in Clifford-valued Sobolev space W01,(Ω, C(?)n). And by means of the relationship between differential forms and Clifford-valued functions, we get the Poincare-type integral inequalities which related to the mono-genic function for the uncoupled Clifford-valued function in the space of W1,P(Ω, C(?)n).Secondly, we study the properties of weak solutions to A-Dirac equations in Clifford-valued function space W1,P(Ω, C(?)n). Through the researches on the obstacle problem and the relationship between the solutions to the scalar part of the equation and solutions to the whole, we get the existence of solutions to the scalar part of A-Dirac equations. Consequently, the existence and uniqueness of the solutions to A-Dirac equations are proved. Furthermore, higher integrability and stability of solutions to obstacle problems are proved.On this basis, we studied the properties of very weak solutions to A-Dirac equations. Assuming that operator A satisfies some structure conditions, by means of the theorem of direct decomposition of the Lebesgue space for Clifford-valued functions, regularity of very weak solutions is proved. And when the boundary functions are a convergence Clifford-valued function sequence, the convergence of very weak solutions is proved.Finally, we discussed two classes of functions, the nonstandard growth class Young functions and the G(p, q, C)-class Young functions. We obtain the Poincare type inte-gral inequalities for the Clifford-valued functions which were acted the two classes of functions in Orlicz space respectively. Also, we get the integral inequalities of the maxi-mal operator, Teodorescu operator and the composite operator which were related to two classes of Young functions in Orlicz space.