Some Boundary Value Problems For High-order Complex Equation And Partial Differential Equations In Higher Dimensional Domains

The complex method is a powerful tool for studying partial differential equations. In this paper, we mainly study some boundary value problems for high-order complex equation and partial differential equations in higher dimensional domains, and generalize some results in recent literature.Firstly, k-regular function (the solution of the equation in the complex plane is studied. We obtain Cauchy theorem ,Morera theoremand extension theorem for this function . Applying these properties and Plemelj formula, we study its Riemann boundary value problem . Moreover, the mathematical formulation of a class of Riemann boundary value inverse problem for k-regular function is presented ,then we transfer this inverse problem into Riemann boundary value problem.Secondly, we define a class of generalized holomorphic function forseveral complex variables(the solution of the equationand study its some properties. Moreover, applying the method of singular integral equation and the Schauder fixed-point theorem, we prove the existence of the solution for a nonlinear boundary value problem and get the integral representation of its solution.Finally, we obtain some properties of 2-regular function in realClifford analysis(the solution of the equationd2f = 0, where— dd =^9, +e2d2+--- + endn,di =—,i=l,---n), such as its representation Nax,.Cauchy type integration Plemelj formulaN extension theorem. Then, westudy its certain Riemann boundary value problem. The solvability of this problem and the integral representation of its solution are obtained.