Explicit examples of Lipschitz, one-homogeneous solutions of loglog-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)u(R,θ)=Rg(θ), where (R,θ)(R,θ) are plane polar coordinates and gg takes values in RmRm, m≥2m≥2. The systems are singular in the sense that they arise as the Euler–Lagrange equations of the functionals I(u)=∫BW(x,∇u(x))dx, where DFW(x,F)DFW(x,F) behaves like 1|x| as |x|→0|x|→0 and WW satisfies an ellipticity condition. Such solutions cannot exist when |x|DFW(x,F)→0|x|DFW(x,F)→0 as |x|→0|x|→0, so the condition is optimal. The associated analysis exploits the well-known Fefferman–Stein duality (Fefferman and Stein, 1972). We also discuss conditions for the uniqueness of these one-homogeneous solutions and demonstrate that they are minimizers of certain variational functionals.