Regularity for an anisotropic equation in the plane

TL;DR

This paper provides a straightforward proof of the $C^1$ regularity for $p$-anisotropic functions in the plane, establishing a logarithmic modulus of continuity for derivatives and utilizing their monotonicity.

Contribution

It introduces a simple proof technique for regularity of anisotropic functions, including the case with two exponents, expanding understanding of their smoothness properties.

Findings

Proves $C^1$ regularity for $p$-anisotropic functions in the plane.

Establishes a logarithmic modulus of continuity for derivatives.

Includes the case with two exponents.

Abstract

We present a simple proof of the $C^1$ regularity of $p$-anisotropic functions in the plane for $2\leq p<\infty$. We achieve a logarithmic modulus of continuity for the derivatives. The monotonicity (in the sense of Lebesgue) of the derivatives is used. The case with two exponents is also included.