Regularity of the derivatives of $p$-orthotropic functions in the plane for $1<p<2$

TL;DR

This paper proves that in the plane, $p$-orthotropic functions for $1<p<2$ are continuously differentiable, providing explicit bounds on their derivatives' continuity based on monotonicity properties.

Contribution

It establishes $C^1$ regularity for $p$-orthotropic functions in the plane for the first time in this range, with explicit logarithmic modulus of continuity.

Findings

Proves $C^1$ regularity of $p$-orthotropic functions for $1<p<2$

Derives explicit logarithmic modulus of continuity

Uses monotonicity of derivatives as a key tool

Abstract

We present a proof of the $C^1$ regularity of $p$-orthotropic functions in the plane for $1<p<2$, based on the monotonicity of the derivatives. Moreover we achieve an explicit logarithmic modulus of continuity.