Game-theoretic p-Laplace operator involving the gradient

The game-theoretic p-Laplacian operator is a version of classical variational p-Laplacian which is in connection with stochastic games called Tug-of-War with noise. The existence of positive singular and Holder continuous solutions of the game-theoretic p-Laplace operator involving the gradient in a small C2 perturbation of the unit ball in Rn are proved. Finally, a more case problem is introduced.