Explicit examples of Lipschitz, one-homogeneous solutions of log-singular planar elliptic systems. (September 2015) - Bevan, Jonathan J.
Abstract: We give examples of systems of Partial Differential Equations that admit non-trivial, Lipschitz and one-homogeneous solutions in the form u ( R, θ ) = R g ( θ ), where ( R, θ ) are plane polar coordinates and g takes values in R m, m ≥ 2 . The systems are singular in the sense that they arise as the Euler–Lagrange equations of the functionals I ( u ) = ∫ B W ( x, ∇ u ( x ) ) d x, where D F W ( x, F ) behaves like 1 | x | as | x | → 0 and W satisfies an ellipticity condition. Such solutions cannot exist when | x | D F W ( x, F ) → 0 as | 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.