Partial differential equations in generalized Sobolev spaces

广义 Sobolev 空间中的偏微分方程

• 批准号: 405603184
• 负责人: Professor Dr. Robert Denk
• 金额: --
• 依托单位: The State Fund for Fundamental Researches
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 无数据
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/405603184?language=en
• 关键词: Partial; differential; equations; generalized; Sobolev

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.

该项目的目的是发展广义Sobolev空间中抛物和椭圆微分方程的现代理论。这些空间的正则性的阶由一个足够一般的权函数给出，与阶为数字的经典Sobolev空间相反。L. Hörmander（1963）在广义Sobolev空间中对偏微分方程的应用起着重要的作用。我们计划在Hörmander空间中获得关于一般抛物型初边值问题解的可解性和正则性的新结果。我们还计划证明关于负Sobolev和Hörmander空间中一般椭圆边值问题的Fredholm性质的新定理，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型微分算子特征值问题的研究。