AnalysisAtInfinity · Analysis at Infinity: Integral Equations, Limit Operators and Beyond

Analysis at Infinity: Integral Equations, Limit Operators and Beyond

Differential equations, or more precisely boundary value problems, are used to model a large portion of physical phenomena in contemporary science. Solving these boundary value problems or a least proving the existence of a unique solution is therefore one of the most important tasks of contemporary mathematics. One of the most popular approaches is via potential theory where the (linear) differential equation is transformed to an integral equation. In case the integrand stays bounded, the resulting integral equation is well understood, but in many applications the integrand turns out to be unbounded. The purpose of this project was to investigate these so-called singular integral equations and the associated operators, and apply the results to concrete operators such as Toeplitz operators and double layer potentials (DLPs). Our novel approach was to transfer limit operator methods, which originate from the study of infinite matrices, to the theory of integral operators. The main idea of limit operator theory is that an infinite matrix can only contain finite information in finite space. To get the whole picture, one needs to look "at infinity". To access the information at infinity one has to shift the matrix along the integers and take the limit at infinity. This idea does not have a straightforward generalization to operators on continuous domains like surfaces. Just to name an obvious issue: it is not always clear what "infinity" exactly means, especially if we are dealing with operators on bounded domains. However, we managed to find a way to generalize these ideas to integral operators by reinterpreting some of the ingredients of limit operator theory. For example, to address the previously mentioned issue, we interpreted the boundary of a domain as "infinity" or conversely, infinity is interpreted as the boundary of the integers. The main principles and assumptions of limit operator theory were formalized in an algebraic language involving analytic and geometric terms so that it can be applied to a variety of contexts. In the second part of the project we obtained a variety of new results in the theory of Toeplitz, Hankel and Toeplitz+Hankel operators. Moreover, we are currently finalising a paper concerning DLP operators. Our original expectation was to disprove a long-standing conjecture regarding the spectral radius of the DLP operator by considering a domain with a peculiar type of singularity. However, our analysis showed that the conjecture still holds in this case and thus further evidence for the validity of the conjecture is obtained. Further investigations on this will be needed in the future.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз