Explicit examples of Lipschitz, one-homogeneous solutions of log-singular planar elliptic systems

Abstract

We give examples of systems of Partial Differential Equations that admit non-trivial, Lipschitz and one-homogeneous solutions in the form $u(R,\theta )=Rg(\theta )$, where $(R,\theta )$ are plane polar coordinates and $g:{\mathbb{R}}^2\to {\mathbb{R}}^m$, $m\ge 2$. The systems are singular in the sense that they arise as the Euler-Lagrange equations of the functionals $I(u)={\int }_{B}W(x,\nabla u(x))dx$, where $D_{F}W(x,F)$ behaves like $\frac{1}{|x|}$ as $|x|\to 0$ and $W$ satisfies an ellipticity condition. Such solutions cannot exist when $|x|D_{F}W(x,F)\to 0$ as $|x|\to 0$, so the condition is optimal. The associated analysis exploits the well-known Fefferman-Stein duality. We also discuss conditions for the uniqueness of these one-homogeneous solutions and demonstrate that they are minimizers of certain variational functionals.

This paper was submitted to the Journal of the London Mathematical Society on 8 November 2012