Clifford分析中的算子有界性研究及其在高阶边值问题中的应用

Clifford analysis essentially is a higher dimensional function theory offering both a generalization of the theory of holomorphic functions in the complex plane and a refinement of classical harmonic analysis. Clifford analysis technique is also the most powerful tool which generalize boundary value problems in complex plane to higher dimensional space. Using the methods of Clifford algebra , we will construct higher order kernel functions. Firstly, based on higher order kernel functions, we will present quasi-higher order Borel-Pompeiu formula in Clifford analysis. Secondly, we will establish generalized Teodorescu transforms, singular integral operators with higher order kernel functions. We will investigate the boundedness of generalized Teodorescu transforms, singular integral operators on some function space for instance Hölder space, Lp space, Sobolev space and generalized Morrey space. Members of this project will establish some boundary value problems and nonlinear boundary value problems for poly-Helmholtz equations in higher dimensional space. Finally, The explicit of representation of solutions for Riemann, Hilbert, Dirichlet, mixed boundary value problem for poly-Helmholtz equations will be presented. The existence, uniqueness, norm estimates for solutions of nonlinear boundary value problems for poly-Helmholtz equations in Clifford analysis will be studied.

Clifford 分析是单复变函数论的高维推广，提炼了经典的调和分析. Clifford分析技术也是复边值问题推广到高维空间的最有力的工具. 本课题组拟利用Clifford代数方法构造高阶核函数，利用复分析思想，建立Clifford分析框架下的拟高阶Borel-Pompeiu公式，进而建立高阶广义Teodorescu变换，带高阶核的奇异积分算子，研究这些积分算子在函数空间的有界性，如在Clifford分析体系下，Hölder空间，Lp空间，Sobolev空间以Morrey空间算子的有界性. 在Clifford体系下，拟研究多Helmholtz方程几类边值问题，非线性边值问题，给出Riemann边值问题，Hilbert边值问题，Dirichlet边值问题，混合边值问题解的表达式，非线性边值问题解的存在性和唯一性条件及解的范数估计.

Clifford分析是单复变函数论的高维推广，提炼了经典调和分析. Clifford分析技术也是复边值问题推广到高维空间的最有力工具..本课题组利用泛Clifford代数方法构造了二阶、四阶核函数，利用复分析思想，建立了Clifford分析框架下的二阶、四阶Borel-Pompeiu公式，.进而建立了二阶广义Teodorescu变换，带四阶核函数的奇异积分算子，研究了这些算子在Clifford值Holder空间和Clifford值Lp空间上的算子有界性.及其边界行为. 基于上述理论研究，在Clifford分析体系下，解决了修正Helmholtz方程的Riemann-Hilbert边值问题，证明了广义Riemann-Hilbert边值问题的可解性.并且研究了解的Holder范数估计. 利用类似的思想，我们证明一类四阶椭圆方程的Liouville定理，解决了四阶椭圆Riemann型边值问题并且得到解的表达式.