Let $1<p<\infty$, let $U\subset\mathbb R^n$ be open, and let $F:\mathbb R^{m\times n}\to\mathbb R$ be $C^2$. Assume that there exist constants $0<\lambda\le \Lambda<\infty$ such that

\begin{align*} \lambda |A|^p-\Lambda \le F(A)\le \Lambda(1+|A|^p) \end{align*}

for all $A\in\mathbb R^{m\times n}$, and

\begin{align*} |DF(A)|\le \Lambda(1+|A|^{p-1}),\qquad |D^2F(A)|\le \Lambda(1+|A|^2)^{(p-2)/2} \end{align*}

for all $A\in\mathbb R^{m\times n}$. Assume that $F$ is quasiconvex and that it is uniformly Legendre-Hadamard elliptic in the following quantified sense: for every $A\in\mathbb R^{m\times n}$, every $a\in\mathbb R^m$, and every $\xi\in\mathbb R^n$,

\begin{align*} D^2F(A)[a\otimes \xi,a\otimes \xi] \ge \lambda(1+|A|^2)^{(p-2)/2}|a|^2|\xi|^2. \end{align*}

For every open $V\subset\subset U$, define $I_V:W^{1,p}(V;\mathbb R^m)\to\mathbb R$ by

\begin{align*} I_V[v]=\int_V F(\nabla v)\,d\mathcal L^n. \end{align*}

Let $u\in W^{1,p}_{\mathrm{loc}}(U;\mathbb R^m)$ be a local minimizer in the sense that

\begin{align*} I_V[u]\le I_V[u+\varphi] \end{align*}

for every open $V\subset\subset U$ and every $\varphi\in W^{1,p}_0(V;\mathbb R^m)$. Then there is an exponent $\alpha\in(0,1)$, depending only on $n,m,p,\lambda,\Lambda$ and the modulus of continuity of $D^2F$ on bounded subsets of $\mathbb R^{m\times n}$, and an [open set](/page/Open%20Set) $U_0\subset U$ such that $u\in C^{1,\alpha}_{\mathrm{loc}}(U_0;\mathbb R^m)$. The singular set $U\setminus U_0$ has [Lebesgue measure](/page/Lebesgue%20Measure) zero.