Generalization of Rellich-Kondrachov theorem and trace compacteness in the framework of irregular and fractal boundaries

TL;DR

This paper surveys recent functional analysis results that extend classical PDE solution techniques to domains with irregular, fractal-like boundaries, broadening the scope of boundary value problem solvability.

Contribution

It generalizes the Rellich-Kondrachov theorem and trace compactness to domains with non-Ahlfors regular d-set boundaries, enabling PDE analysis in more complex geometries.

Findings

Extended Rellich-Kondrachov theorem for irregular boundaries

Proved trace operator compactness in fractal boundary domains

Established existence and uniqueness of weak solutions in generalized domains

Abstract

We present a survey of recent results of the functional analysis allowing to solve PDEs in a large class of domains with irregular boundaries. We extend the previously introduced concept of admissible domains with a d-set boundary on the domains with the boundaries on which the measure is not necessarily Ahlfors regular d-measure. This gives a generalization of Rellich-Kondrachov theorem and the compactness of the trace operator, allowing to obtain, as for a regular classical case the unicity/existence of weak solutions of Poisson boundary valued problem with the Robin boundary condition and to obtain the usual properties of the associated spectral problem.