Counterexample to C1C1 boundary regularity of infinity harmonic functions

In this short note, we construct a planar infinity harmonic function uu in a C1C1 domain ΩΩ with smooth boundary data gg; however DuDu is not continuous on the boundary. Changyou Wang and Yifeng Yu (2012) proved the C1C1 boundary regularity of planar infinity harmonic functions provided that ∂Ω∂Ω and gg are C2C2. They asked the question whether the result holds when ∂Ω∂Ω and gg are assumed to be C1C1. Our counterexample answered this question negatively. Our construction of the counterexample is strongly inspired by the counterexample of Dongsheng Li and Lihe Wang (2006) for uniformly elliptic equations.