拟齐性偏微分算子的普适理论

Some important problems and new problems in the general theory for.quasi-homogeneous linear partial differential operators (QHLPDO) are studied. The main results are as follows. The theorem for removable singularity and the general form of the fundamental solutions of the hypoelliptic QHLPDO are obtained. A new necessary and sufficient condition for global solvability in the polynomial space and the estimation of the.dimension of the kernel in the same space for QHLPDO are established. The.unsolvability in the Schwartz space for QHLPDO is proved. The existence or nonexistence results for the solutions of nonlinear subelliptic boundary values problem and eigenvalue problem for a class of QHLPDO are obtained. Some important inequalities and identities, such as the Hardy type.inequalities and Pohozaev type identity are set up on the homogeneous group, which are useful in the study of QHLPDE. Also, some new methods and tools are provided. Obtained results make the theory of the partial differential equations richer and.deeper, and so is the link between the partial differential equations and other subjectsof mathematics.

研究拟齐性偏微分算子的一般理论，包括亚椭圆性，基本解及拟基本解的性质，局部性质同整体性质间的关系，非线性边值问题等，本研究提供的观点，方法和结果，有助于深化和丰富偏微分方程的一般理论，有助于加强各数学分支间的联系。特别对于解决复合分析，复几何中某些困难问题会有所助益。