Schauder theory in variable H\"older spaces

TL;DR

This paper extends Schauder theory to elliptic equations in variable Hölder spaces, providing interior and global a priori estimates, and establishing existence and uniqueness of solutions for Dirichlet problems.

Contribution

It introduces Schauder estimates and solvability results for elliptic equations in variable Hölder spaces, a novel generalization of classical theory.

Findings

Interior Schauder estimates established

Global Schauder estimates established

Existence and uniqueness of solutions proved

Abstract

We study elliptic equations on bounded domain of Euclidean spaces in the variable H\"{o}lder spaces. Interior a priori Schauder estimates are given as well as global ones. Moreover, the existence and the uniqueness of solutions to the Dirichlet boundary value problem is proved.