Counterexample to C1 boundary regularity of infinity harmonic functions

In this short note, we construct a planar infinity harmonic function u in a C-1 domain Omega with smooth boundary data g; however Du is not continuous on the boundary. Changyou Wang and Yifeng Yu (2012) proved the C-1 boundary regularity of planar infinity harmonic functions provided that partial derivative Omega and g are C-2. They asked the question whether the result holds when partial derivative Omega and g are assumed to be C-1. 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. (C) 2014 Elsevier Ltd. All rights reserved.