A Liouville theorem for infinity harmonic functions

Abstract

We proved that a Lipschitz entire infinity harmonic function on $$\mathbb {R}^n$$ must asymptotically tend to a plane at infinity and it must be a plane provided the length of its gradient is continuous at infinity. We also prove that if the length of the gradient of an infinity harmonic function is continuous at some interior point then the gradient of the function is continuous at this point. All the proofs of the above conclusions are based on some modifications of Evans-Smart’s argument.