Uniformly convex variational integrand ($F_{pp}(x,p)\ge \lambda I$)

A gradient integrand is uniformly convex in this elliptic variational sense when its Hessian matrix in the gradient variables has a common positive lower bound. Then $(F_p(x,p)-F_p(x,q))\cdot (p-q)\ge \lambda |p-q{|}^2$. Averaging $F_{pp}$ along a line segment in gradient space preserves the same lower bound.

For scalar minimizers of $\int F(Du)$ with $F\in C^2({\mathbb{R}}^n)$ and globally bounded uniformly positive Hessian matrix, no higher differentiability of $F$ is needed for interior gradient Hölder continuity. The weak Euler-Lagrange equation is $\mathrm{div}DF(Du)=0$. Difference quotients solve uniformly elliptic equations with coefficient matrices obtained by averaging $D^{2}F$ along gradient segments. The Caccioppoli inequality and difference quotient characterization give $u\in H_{\mathrm{loc}}^2$. The Sobolev chain rule then gives $\mathrm{div}(D^{2}F(Du)D(D_{k}u))=0$. The De Giorgi-Nash-Moser theorem makes each derivative Hölder continuous, with exponent depending only on dimension and the ellipticity ratio. This scalar conclusion should not be assumed for vector-valued elliptic systems.

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.