Conservation Laws for p-Harmonic Systems with Antisymmetric Potentials and Applications

Abstract

We prove that p-harmonic systems with antisymmetric potentials of the form

$$\begin{aligned} -\,\text{ div }\left( (1+|\nabla u|^2)^{\frac{p}{2}-1}\,\nabla u\right) =(1+|\nabla u|^2)^{\frac{p}{2}-1}\,\Omega \cdot \nabla u, \end{aligned}$$

(\(\Omega \) is antisymmetric) can be written in divergence form as a conservation law

$$\begin{aligned} -\text{ div }\left( (1+|\nabla u|^2)^{\frac{p}{2}-1}\,A\,\nabla u\right) =\nabla ^\perp B\cdot \nabla u. \end{aligned}$$

This extends to the p-harmonic framework the original work of the second author for \(p=2\) (see Rivière in Invent Math 168(1):1–22, 2007). We give applications of the existence of this divergence structure in the analysis \(p\rightarrow 2\).