Partial Schauder estimates for second-order elliptic and parabolic equations: a revisit

TL;DR

This paper revisits Schauder estimates for second-order elliptic and parabolic equations, establishing results where coefficients are only measurable in some variables and semi-norms are with respect to a subset of variables.

Contribution

It introduces Schauder estimates allowing coefficients to be merely measurable in certain variables, broadening applicability in elliptic and parabolic equations.

Findings

Established Schauder estimates with partial semi-norms

Coefficients can be merely measurable in some variables

Results apply to divergence and non-divergence form equations

Abstract

Under various conditions, 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. A novelty of our results is that the coefficients are allowed to be merely measurable with respect to the other independent variables.