Theory And Applications Of One-Sided Coupled Operator Matrices

TL;DR

This paper surveys the theory of one-sided coupled operator matrices, emphasizing their applications to initial-boundary value problems with unbounded boundary feedbacks, and discusses well-posedness and stability of related systems.

Contribution

It provides a comprehensive overview of the recent theory and demonstrates its application to complex boundary value problems with unbounded feedbacks.

Findings

Well-posedness of a diffusion-transport system with dynamical boundary conditions.

Analysis of stability and solution properties for systems with unbounded boundary feedbacks.

Establishment of well-posedness for a wave equation with dynamical boundary conditions.

Abstract

The theory of one-sided coupled operator matrices, recently introduced by K.-J. Engel, is an abstract framework for concrete initial value problems and allows complete information on well-posedness, and stability of solutions. These notes are meant as a survey on this rich theory, with a particular stress on applications to initial-boundary value problems with unbounded boundary feedbacks. A diffusion-transport system with dynamical boundary conditions is discussed, and its well-posedness and various other properties are investigated. As a by-product, the well-posedness of a wave equation with dynamical boundary condition is also obtained.