Partial Regularity of the Gradient for Subsolutions

For any subharmonic function u , we prove that |∂ ⱼ u| | ∂ j u | ( j=1, … , n j = 1 , … , n ) is upper semi-continuous, provided that the super-level sets of u can be touched from the exterior by uniform C1,Dini C 1 , Dini domains at every point. This idea extends to a class of general operators, as well as to the boundary behaviour of the gradient of solutions of the Dirichlet problem in a domain whose boundary satisfies this geometric condition.