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We give examples of systems of Partial Differential Equations that admitnon-trivial, Lipschitz and one-homogeneous solutions in the form $u(R,\theta) =Rg(\theta)$, where $(R,\theta)$ are plane polar coordinates and $g:\mathbb{R}^{2} \to \mathbb{R}^{m}$, $m \geq 2$. The systems are singular in thesense 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. Suchsolutions cannot exist when $|x|D_{F}W(x,F) \to 0$ as $|x| \to 0$, so thecondition is optimal. The associated analysis exploits the well-knownFefferman-Stein duality. We also discuss conditions for the uniqueness of theseone-homogeneous solutions and demonstrate that they are minimizers of certainvariational functionals.

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