Research On The DIRAC-SOBOLEV Function Space

A-harmonic analysis theory is to study of A-harmonic equations in classical Sobolev spaces. The solution of such equations is defined by distribution functions In recent years, the classical Sobolev space show its limitation in many nonlinear problems. Expecially, when Professor C.Nolder generalized the famous A-harmonic equation to A-Dirac equation of multidimensional form, since its solution is usually a Clifford-valued function in Rn. Therefore, the study of Clifford-valued functions spaces have its meaning and value. The main object of this thesis is Dirac-Sobolev space.In this paper, we mainly discuss the theory of spaces of Clifford-valued functions, and the Dirac-Sobolev spaces. The main contents are as follows:(i) We establish the theory spaces of Clifford-valued functions. We introduce Lebesgue spaces of Clifford-valued functions Lp(Ω,C(?)n) and two kinds of Dirac-Sobolev spaces W D,P(Ω,C(?)n)、W1,p(Ω,C(?)n) and we define their own norms. We discuss the properties of Lp(Ω,C(?)n), for example, reflexivity、completeness、 dense, etc. Further, we prove that both W D,p(Ω,C(?)n) and W1,p(Ω,C(?)n) are reflective Banach and C∞(Ω,C(?)n)∩W1,p(Ω,C(?)n) is dense in W1,p(Ω,C(?)n).(ii)Basis on (i) as an application of Dirac-Sobolev spaces, this paper studied the higher-order integrability and stability of a double obstacle problem in the Spaces of Clifford-valued functions.