Abstract

The content of the report is related to the project “Perturbation problems and asymptotics for elliptic differential equations: variational and potential theoretic methods” developed at the Department of Mathematics, Padova university.

The report is addressed to the recent results in the factorization of matrix-function as well as in the development of the Wiener-Hopf technique. Partly it is related to the results presented during a one-month research programme “Bringing pure and applied analysis together via the Wiener–Hopf technique, its generalisations and applications” organized by the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK.

Among the theoretical results we mention perturbation analysis applied to the study of stability of partial indices and further development of the factorization of the triangular matrix-functions. A few applications are outlined in the report. First, we discuss the usage of the factorization technique to solve the R-linear conjugation problem, which is important for the study of different problems of elasticity and composite materials. Second, application to electromagnetic problems is briefly outlined. At last, we present some procedures which show how the Wiener-Hopf technique is working in analysis of static and dynamic problems of fracture mechanics.

The talk is within the frame of the Project funded by the EuropeanUnion – Next Generation EU under the National Recovery and Resilience Plan (NRRP), Mission 4 Component 2 Investment 1.1 – Call PRIN 2022 No. 104 of February 2, 2022 of Italian Ministry of University and Research; Project 2022SENJZ3 (subject area: PE – Physical Sciences and Engineering) “Perturbation problems and asymptotics for elliptic differential equations: variational and potential theoretic methods”.