An Efficient Abstract Method For The Study of An Initial Boundary Value Problem On Singular Domain

TL;DR

This paper develops an efficient abstract method to analyze boundary value problems for second order linear differential equations on singular domains, establishing regularity results in Hölder spaces.

Contribution

It introduces a novel abstract approach for studying boundary value problems on singular domains, linking them to elliptic equations with variable coefficients.

Findings

Established regularity results in Hölder spaces for the boundary value problem.

Connected the singular domain problem to an elliptic abstract differential equation.

Provided a framework for analyzing similar problems on singular domains.

Abstract

The present work is devoted to the study of a boundary value problem for second order linear differential equation set on singular cylindrical domain. This problem can be regarded via a natural change of variables as an elliptic abstract differential equation with variable operators coefficients subject to some anti-periodic conditions. The complete study of this abstract version allows us to establish some interesting regularity results for our problem. The study is performed in the framework of H\"{o}lder spaces.