The Properties And Boundary Value Problem Of Isotonic Functions In Complex Clifford Analysis

The Isotonic functions studied in this paper are the functions defined in the even dimen-sional Euclidean space R2m with values in the complex Clifford algebra C0,m and satisfying theIsotonic system. Isotonic Clifford analysis is a new and active field in Clifford analysis. It is thesubject which focuses on the integral representation of the Isotonic function and its applicationsto different fields. This subject may be regarded as an elegant generalization to the complexClifford algebra of holomorphic functions of several complex variables, and now it is closelyrelated to a new branch of Clifford analysis, which focuses on Hermitian monogenic functions.So studying the Isotonic function can generalize the further applications of Clifford analysisto the fields in mathematics and other different subjects. Thus, it is significant in theory andvaluable in application.In chapter 1, the basic structure of the complex Clifford algebra and its important opera-tions are introduced. And the definition of the Isotonic function is given. Furthermore, the firstchapter also gives some important lemmas and proves two estimate formulas about module,which play a key role in the following integral estimate value.In chapter 2, the integral representation of functional solution satisfying the Isotonic sys-tem is given. And inspired by some properties of regular functions in complex analysis andClifford analysis, we prove some basic properties of the sequence of the Isotonic function, suchas completeness, equicontinuity, sequential compactness and convergence. And these propertieslay the foundations of helping us understand the Isotonic functions more clearly.In chapter 3, several essential theorems of the Isotonic functions defined in the boundeddomain are mainly studied. On the basis of the integral representation of the Isotonic functionsin chapter 2, we define Cauchy type integral of the Isotonic functions in this chapter. Apply-ing the method of local generalized sphere coordinates transformation, we also prove that theCauchy type integral has the meaning of Cauchy principal value over the boundary. To studythe boundary value's characteristics of the Cauchy type integral further, we give Plemelj for-mula, which is also a basic tool to study boundary value problems of the Isotonic functions. Inthe process of proving the Plemelj formula, we first divide the Cauchy type integral into twoparts, and thus we just have to prove the continuity of one part when the point in the domainapproaches the boundary. Then we divide the ways of the point in the domainΩ(?) R^2m ap-proaching the point on the boundary ?Ωinto two cases and discuss them respectively. The firstcase is that the approaching way of the point in the domain is not along the direction of the tangent plane of the point on the boundary. The solution is to divide the integral surface intotwo parts. One has singular point and the other has no. The second is that the approachingway of the point in the domain is along the direction of the tangent plane of the point on theboundary. On the basis of the former case, the proof of the second one is very clear. Applyingthe Plemelj formula, we can obtain the Ho¨lder continuity of the boundary value of the Cauchytype integral and the Privalov theorem. And the proof of the Ho¨lder continuity of the boundaryvalue is based on the respective discussion of the integral singularity. At last, we divide theproof of the Privalov theorem of the Cauchy type integral into three cases. The first is that thetwo arbitrary points are both on the boundary, and this case can be transformed into the discus-sion of the Ho¨lder continuity of the boundary value of the Cauchy type integral. The secondis that between the two arbitrary points, one point is on the boundary and the other is in thedomain. And this case can be proved by combining the properties of the bounded domain andthe integral representation of the boundary value as well as the Cauchy type integral. The thirdis that the two arbitrary points are both in the domain. Similar to the proof of the second case,this case can be proved according to the properties of the bounded domain. Then, taking theabove discussion into consideration, we can prove the Privalov theorem.In chapter 4, we first give a kind of nonlinear boundary value problem of the Isotonic func-tions namely Problem IN and its boundary conditions. Then we discuss and give its solvableconditions, under which, the paper mainly uses the classical methods of dealing with boundaryvalue problems to prove the existence of the solution for this kind of boundary value problem.We first transform this kind of boundary value problem into an integral equation problem byapplying the Plemelj formula of the Cauchy type integral of the Isotonic function in Chapter3. Then we prove the solvability of the integral equation by the Schauder fixed point theorem.However, the paper estimates theβmodule which is newly defined by using some inequalitiesbefore the proof. And the result lays the foundations of proving the operator in the integralequation to satisfy the conditions of the Schauder fixed point theorem. Thus, under the solv-able conditions, the existence of the solution for this integral equation is proved and then thesolvability of the boundary value problem IN is also obtained. Lastly, the conditions for theuniqueness of the solution are given and the uniqueness is also proved in the paper.