A Remark on the Kelvin Transform for a Quasilinear Equation

TL;DR

This paper investigates the limitations of the Kelvin transform's applicability to p-harmonic functions for various exponents p, highlighting its special case for p=2 and the absence of a general counterpart.

Contribution

The study demonstrates that the Kelvin transform cannot be generally extended to p-harmonic functions for arbitrary p, except in the linear case p=2.

Findings

Kelvin transform preserves p-harmonic functions only when p equals the space dimension.

No suitable generalization of the Kelvin transform exists for p ≠ 2.

The linear case p=2 uniquely allows the Kelvin transform to correct invariance issues.

Abstract

The p-harmonic functions are preserved under reflections in spheres only if the exponent p > 1 is equal to the dimension of the underlying Euclidean space. In the linear case p = 2 the Kelvin transform corrects this lack of invariance. We shall show that the Kelvin transform has no reasonable counterpart for general values of the exponent p.