Function spaces of generalized smoothness and elliptic boundary problems

• Applicant: Dr. Anna Anop
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 530831274

The objective of this project is to further the development of the theory of function spaces of generalized smoothness and to apply it to elliptic boundary value problems. A specific focus will be on the interpolation in scales of distribution spaces. It is planned to introduce and investigate wide classes of Sobolev, Besov and Triebel--Lizorkin normed and quasi-normed spaces of generalized smoothness on manifolds and to show that they possess interpolation properties with respect to their classical counterparts. We will discuss the solvability of general elliptic differential and pseudodifferential boundary-value problems in these function spaces. It is planned to establish elliptic regularity results and to derive a priori estimates for generalized solutions to the problems under consideration. Special attention will be paid to elliptic problems with low regularity (rough) data, e.g., Dirac distributions or white noise in right-hand sides. These particular applications arise in the physical and engineering sciences. To this end, it is planned to introduce and study various variants of spaces of negative generalized smoothness suitable for the investigation of such problems.

• DFG Programme: WBP Position