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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.

No additional hypothesis on $F$ is needed here for scalar interior $C^{1,\beta }$ regularity. In the elliptic sense of uniform convexity, the assumptions already give constants $0<\lambda \le M<\infty$ with $\lambda I\le D^{2}F(p)\le MI$ for all $p$. The following argument gives interior gradient regularity for uniformly convex autonomous energies.

The first variation at the minimizer exists because $|DF(p)|\le C(1+|p|)$, and its weak Euler-Lagrange equation isFor a coordinate difference quotient ${\delta }_{h}u$, subtract the translated equation and the original equation. The averaged linearization of a nonlinear divergence-form equation givesThese measurable symmetric matrices satisfy $\lambda I\le A_h\le MI$ independently of $h$. The Caccioppoli inequality and the standard bound $\Vert {\delta }_{h}u{\Vert }_2\le \Vert D_{k}u{\Vert }_2$ bound $D{\delta }_{h}u$ on every smaller interior set. Quoting the Sobolev characterization by bounded difference quotients yields $u\in H_{\mathrm{loc}}^2$.

Now $DF$ is globally Lipschitz continuous, so the Sobolev chain rule and commutation of distributional derivatives allow differentiation of the weak equation. Each $v_k=D_{k}u\in H_{\mathrm{loc}}^1$ satisfiesThe De Giorgi-Nash-Moser theorem states that weak solutions of this scalar divergence-form equation with bounded measurable uniformly elliptic coefficients are locally Hölder continuous. It gives a common $\beta \in (0,1)$ depending only on $n$ and the ellipticity ratio $M/\lambda$, with $D_{k}u\in C_{\mathrm{loc}}^{0,\beta }$ for every $k$. A Sobolev function whose weak gradient is continuous has a $C^1$ representative, as follows by local mollification and integration of its gradient. HenceThe higher-regularity conclusion comes from differentiating an autonomous scalar equation; it does not rely on boundary regularity or a higher derivative such as $D^{3}F$.

Fix nested relatively compact subsets of $\Omega$ and let $M$ bound the gradient on the larger one. The fundamental theorem of calculus along a line segment gives $|w_h|\le M$ on the smaller one. Part (i) gives coefficient bounds independent of small $h$, so the De Giorgi-Nash-Moser theorem gives uniform interior Hölder continuity estimates for these weak solutions. Since $Du$ is continuous, the difference quotients $w_h$ converge locally uniformly to $D_{\ell }u$. Passing to the limit in the Hölder seminorm estimate, and repeating for every coordinate, yieldsThe first derivatives are locally Hölder continuous. Here the initially obtained exponent depends on $n$ and the local gradient bound. This is the interior interpretation of the printed regularity conclusion; a uniform global estimate does not follow from $u\in C^1(\Omega )$ alone.

Assume $u\in W^{1,2}(B_2(0))$ is nonnegative almost everywhere, and that the real coefficient matrix $A=(a^{ij})$ is measurable and satisfies, for almost every $x$ and every $\xi \in {\mathbb{R}}^n$,The operator norm bound controls all coefficients, including any antisymmetric part. For a symmetric matrix, these assumptions are exactly the usual lower and upper quadratic-form bounds. The weak solution condition isThen the Harnack inequality for uniformly elliptic divergence-form equations isA nonnegative weak solution has its supremum controlled by its infimum on the smaller ball. The De Giorgi-Nash-Moser theorem supplies a Hölder continuous function representative, for which ordinary supremum and infimum can replace the essential supremum and essential infimum. Nonnegativity and the larger ball are essential hypotheses; there is no differentiability requirement on the coefficients.

Work first with essential bounds. On a Euclidean ball $B_R(x_0)⋐B_1$, put $M={\mathrm{esssup}}_{B_R}u$, $m={\mathrm{essinf}}_{B_R}u$ and $\omega =M-m$. The functions $M-u$ and $u-m$ are nonnegative weak solutions, hence weak supersolutions. In $B_{R/2}$, at least one of the sets $\{u\le (M+m)/2\}$ and $\{u\ge (M+m)/2\}$ has at least half the measure.

In the first case, the Weak Harnack inequality applied to $M-u$ givesIn the second case, applying it to $u-m$ gives the same lower bound for the improvement of the minimum. Thus the oscillation decay estimate isThis measure argument works even when $q<1$, avoiding a false use of the triangle inequality in that range.

Iteration yields ${\mathrm{osc}}_{B_r(x_0)}u\le C(r/R{)}^{\mu }{\mathrm{osc}}_{B_R(x_0)}u$, with, for example, $\mu =\min \{1/2,-\log \tau /\log 2\}\in (0,1)$. At Lebesgue points the shrinking oscillations define a unique continuous representative, and the estimate extends to every point by continuity. For $x,y\in B_{1/4}$ at distance less than $1/4$, apply the estimate to a Euclidean ball centred at $x$ with initial radius $1/2$ and final radius comparable to $|x-y|$. This Euclidean ball stays in $B_1$. Larger distances use the trivial bound $2\Vert u{\Vert }_{\infty }$. We obtainThe constants depend only on $n,\lambda ,\Lambda$. This derives the relevant conclusion of the De Giorgi-Nash-Moser theorem directly from part (a).

Write $u=\phi +w$ with $w\in W_0^{1,2}(B)$. On this space defineBoundedness of the coefficients makes $\mathcal{B}$ bounded, and ellipticity giveswhich is coercive by the Poincaré inequality. The functional $z\mapsto -\mathcal{B}(\phi ,z)$ is bounded, so the Lax-Milgram theorem gives a unique $w$ and hence a unique weak solution $u$.

The supplied global De Giorgi-Nash-Moser theorem and the weak maximum principle give $u\in C^{0,\beta }(\overline{B})$ for some $\beta =\beta (n,\gamma ,\Gamma )$. Finally, test the weak equation with $u{\psi }^2$. Ellipticity, the coefficient bound, and Young inequality yieldRearranging gives

For $v\in C^0(\overline{B})$, let $T(v)$ be the solution from part (i). The weak maximum principle givesOn this fixed bounded range of values, $\lambda$ and $\Lambda$ have common positive lower and finite upper bounds. The global De Giorgi-Nash-Moser theorem therefore bounds $T(v)$ uniformly in one Hölder space. The compact embedding $C^{0,\mu }(\overline{B})\hookrightarrow C^0(\overline{B})$ makes $T$ compact. Uniform convergence $v_k\to v$, the continuity of $a_{ij}$ on the relevant compact set, energy bounds, and uniqueness of the limiting linear problem show that $T(v_k)\to T(v)$ uniformly, so $T$ is continuous.

The same maximum-principle bound controls every solution of the Leray–Schauder homotopy after using boundary data $t\phi$. Thus the homotopy set is bounded, and the Leray-Schauder fixed point theorem supplies $u=T(u)$. By definition,and $u\in W^{1,2}(B)\cap C^0(\overline{B})$.

The De Giorgi-Nash-Moser theorem gives numbers $\alpha =\alpha (n,\lambda ,\Lambda )\in (0,1)$ and $C=C(n,\lambda ,\Lambda ,\theta )$ such that every weak solution satisfiesfor each fixed $0<\theta <1$. In particular, every such weak solution has a locally Hölder-continuous representative despite the coefficients being only bounded and measurable.

If $|Du|\le L$, the eigenvalues of $a^{ij}(Du)$ lie between $(1+L^2{)}^{-3/2}$ and $1$, so the differentiated equations are uniformly elliptic with bounded measurable coefficients depending only on $L$. Apply the De Giorgi-Nash-Moser theorem to each $D_{k}u$. Since $|D_{k}u|\le L$, for every $\theta <1$,for some $\alpha =\alpha (n,L)\in (0,1)$. The mapis smooth with bounded derivative on $|p|\le L$. Composition therefore gives