Global Hِlder continuity of weak solutions to quasilinear divergence form elliptic equations

We derive global Hölder regularity for the W 0 1 , 2 ( Ω ) -weak solutions to the quasilinear, uniformly elliptic equation div ( a i j ( x , u ) D j u + a i ( x , u ) ) + a ( x , u , D u ) = 0 over a C 1 -smooth domain Ω ⊂ R n , n ⩾ 2 . The nonlinear terms are all of Carathéodory type with coefficients a i j ( x , u ) belonging to the class VMO of functions with vanishing mean oscillation with respect to x, while a i ( x , u ) and a ( x , u , D u ) exhibit controlled growths with respect to u and the gradient Du.