= Solution
\b[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 $0<\lambda\leq M<\infty$ with $\lambda I\leq D^2F(p)\leq MI$ for all $p$. The following argument gives <interior gradient regularity for uniformly convex autonomous energies>.
The <first variation> at the minimizer exists because $|DF(p)|\leq C(1+|p|)$, and its weak <Euler-Lagrange equation> is
$$
\int_\Omega DF(Du)\cdot D\varphi=0\qquad(\varphi\in C_c^\infty(\Omega)).
$$
For a coordinate difference quotient $\delta_hu$, subtract the translated equation and the original equation. The <averaged linearization of a nonlinear divergence-form equation> gives
$$
\operatorname{div}(A_hD\delta_hu)=0,\qquad A_h(x)=\int_0^1D^2F\bigl((1-t)Du(x)+tDu(x+he_k)\bigr)\,dt.
$$
These measurable symmetric matrices satisfy $\lambda I\leq A_h\leq MI$ independently of $h$. The <Caccioppoli inequality> and the standard bound $\|\delta_hu\|_2\leq\|D_ku\|_2$ bound $D\delta_hu$ on every smaller interior set. Quoting the <Sobolev characterization by bounded difference quotients> yields $u\in H^2_{\mathrm{loc}}$.
Now $DF$ is globally <Lipschitz continuous>, so the <Sobolev chain rule> and commutation of distributional derivatives allow differentiation of the weak equation. Each $v_k=D_ku\in H^1_{\mathrm{loc}}$ satisfies
$$
\operatorname{div}(A Dv_k)=0,\qquad A(x)=D^2F(Du(x)),\qquad\lambda I\leq A(x)\leq MI.
$$
The <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 $D_ku\in C^{0,\beta}_{\mathrm{loc}}$ 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>. Hence
$$
\boxed{u\in C^{1,\beta}_{\mathrm{loc}}(\Omega).}
$$
The higher-regularity conclusion comes from differentiating an autonomous scalar equation; it does not rely on boundary regularity or a higher derivative such as $D^3F$.
Back to article page
