Partial Schauder estimates for second-order elliptic and parabolic equations

TL;DR

This paper develops Schauder estimates for second-order elliptic and parabolic equations, focusing on H"older semi-norms with respect to only some variables, advancing the understanding of partial regularity in these equations.

Contribution

It introduces partial Schauder estimates for elliptic and parabolic equations, considering semi-norms with respect to a subset of variables, a novel approach in regularity theory.

Findings

Established Schauder estimates with respect to some variables

Extended regularity results to divergence and non-divergence form equations

Provided new tools for partial regularity analysis in PDEs

Abstract

We establish Schauder estimates for both divergence and non-divergence form second-order elliptic and parabolic equations involving H\"older semi-norms not with respect to all, but only with respect to some of the independent variables.