Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 12 / 1 / iv

Let $u,v$ be two critical points with the same classical boundary values, and put $w=u-v$. Since $F$ is independent of $z$, the Euler-Lagrange equation for each is $D_i{F}_{p_i}(x,Du)=0$. By the fundamental theorem of calculus along the segment between the gradients,Subtracting the equations gives $D_i(A_{ij}{D}_{j}w)=0$. The uniformly convex variational integrand makes $A$ symmetric and uniformly elliptic with the same positive lower bound. Since $F$ is smooth and $u,v\in C^2(\overline{\Omega })$, $A$ is continuously differentiable locally in $\Omega$, sois a classical nondivergence-form elliptic operator with no zeroth-order term.

To avoid assuming that derivatives of $F$ stay bounded all the way to the boundary, apply the proved weak maximum principle on inner open sets ${\Omega }_{\delta }=\{x\in \Omega :\mathrm{dist}(x,\partial \Omega )>\delta \}$. Their closures lie compactly in $\Omega$, so all linearized coefficients there are bounded. The maximum principle applies on each component and yieldsEvery point of the inner boundary is at distance $\delta$ from $\partial \Omega$. Since $w$ is continuous on the compact closure and equals zero on that boundary, the right-hand side tends uniformly to zero as $\delta \downarrow 0$. Every fixed interior point eventually belongs to ${\Omega }_{\delta }$, provingThis establishes uniqueness for uniformly convex gradient Dirichlet problems using precisely the boundary continuity and ellipticity needed for the comparison.