Partial differential equations in generalized Sobolev spaces

该项目的目的是在广义Sobolev空间中发展抛物型和椭圆型微分方程的现代理论。这些空间的正则性是由一个足够一般的权函数给出的，这与经典的Sobolev空间的阶是数字的不同。L·Hörmander(1963)引入的内积空间在广义Sobolev空间中具有重要的应用价值。我们计划得到关于Hörmander空间中一般抛物型初边值问题解的可解性和正则性的新结果。我们还计划在负的Sobolev和Hörmander空间中证明关于一般椭圆边值问题的Fredholm性的新定理，并证明关于这些问题广义解的局部先验估计和局部正则性的定理。我们打算将这些结果应用于自伴椭圆微分算子的特征值问题的研究。

The purpose of the project is to develop the modern theory of parabolic and elliptic differential equations in generalized Sobolev spaces. The order of regularity for these spaces is given by a general enough weight function, in contrast to the classical Sobolev spaces whose orders are numbers. The inner product spaces introduced by L. Hörmander (1963) play an important role among generalized Sobolev spaces as to applications to partial differential equations. We plan to obtain new results about the character of solvability and regularity of solutions of general parabolic initial-boundary value problems for systems in Hörmander spaces. We also plan to prove new theorems on the Fredholm property of general elliptic boundary-value problems in negative Sobolev and Hörmander spaces and to prove theorems about local a priori estimates and local regularity of generalized solutions to these problems. We intend to apply these results to the investigation of eigenvalue problems for self-adjoint elliptic differential operators.