Transmission problems: regularity theory, interfaces and beyond

TL;DR

This paper reviews the development and current state of transmission problems in mathematical analysis, focusing on regularity issues, interface conditions, and recent advances in the field.

Contribution

It provides a comprehensive overview of transmission problems, highlighting recent progress in regularity theory and the treatment of free transmission problems.

Findings

Recent advances match minimal interface regularity requirements

Solutions achieve optimal regularity under certain conditions

The survey includes developments from the 1950s to present

Abstract

Modelling diffusion processes in heterogeneous media requires addressing inherent discontinuities across interfaces, where specific conditions are to be met. These challenges fall under the purview of Mathematical Analysis as \emph{transmission problems}. We present a pa\-no\-ra\-ma of the theory of transmission problems, encompassing the seminal contributions from the 1950s and subsequent developments. Then we delve into the discussion of regularity issues, including recent advances matching the minimal regularity requirements of interfaces and the optimal regularity of the solutions. A discussion on free transmission problems closes the survey.