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

We give examples of systems of Partial Differential Equations that admit non-trivial, Lipschitz and one-homogeneous solutions in the form u(R, θ) = Rg(θ), where (R, θ) are plane polar coordinates and g takes values in R, m ≥ 2. The systems are singular in the sense that they arise as the EulerLagrange equations of the functionals I(u) = ∫ B W (x,∇u(x)) dx, where DFW (x, F ) behaves like 1 |x| as |x| → 0 and W satisfies an ellipticity condition. Such solutions cannot exist when |x|DFW (x, F ) → 0 as |x| → 0, so the condition is optimal. The associated analysis exploits the well-known Fefferman-Stein duality [5]. We also discuss conditions for the uniqueness of these one-homogeneous solutions and demonstrate that they are minimizers of certain variational functionals.