Partial differential equations in generalized Sobolev spaces

点击翻译按钮获取中文摘要

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.