Uniqueness for uniformly convex gradient Dirichlet problems ($\mathrm{div}{F}_p(x,Du)=\mathrm{div}{F}_p(x,Dv)=0,u{|}_{\partial \Omega }=v{|}_{\partial \Omega }⟹u=v$)

On a bounded domain, classical solutions continuous on the closure have difference $w=u-v$ satisfying $D_i(A_{ij}{D}_{j}w)=0$, where $A={\int }_0^1{F}_{pp}(x,Dv+tDw)dt$. The coefficients are uniformly elliptic and locally smooth. Apply the weak maximum principle on inner domains whose closures lie in the original domain, then let their boundary distances tend to zero. The common continuous boundary values make their boundary differences tend uniformly to zero, proving uniqueness without assuming global bounds on derivatives of the linearized coefficients.