Let $n,m\in\mathbb N$, let $U\subset\mathbb R^n$ be open, let $x_0\in U$, and let $R>0$ be such that $B(x_0,R)\subset U$. Let $1<p<\infty$ and let $\lambda,\Lambda>0$. Equip $\mathbb R^{m\times n}$ with the Frobenius norm $|A|:=\left(\sum_{i=1}^m\sum_{j=1}^n A_{ij}^2\right)^{1/2}$. Suppose that $F:\mathbb R^{m\times n}\to\mathbb R$ is convex and satisfies

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

for every $A\in\mathbb R^{m\times n}$.

Let $u\in W^{1,p}(B(x_0,R);\mathbb R^m)$. Let $\operatorname{Tr}:W^{1,p}(B(x_0,R);\mathbb R^m)\to L^p(\partial B(x_0,R),\mathcal H^{n-1};\mathbb R^m)$ denote the Sobolev trace operator, and define the affine Dirichlet class

\begin{align*} \mathcal A_u:=\{v\in W^{1,p}(B(x_0,R);\mathbb R^m):\operatorname{Tr}v=\operatorname{Tr}u\text{ }\mathcal H^{n-1}\text{-a.e. on }\partial B(x_0,R)\}. \end{align*}

Define the functional $I:\mathcal A_u\to\mathbb R$ by

\begin{align*} I[v]=\int_{B(x_0,R)}F(\nabla v)\,d\mathcal L^n. \end{align*}

If $u$ minimizes $I$ over $\mathcal A_u$, then there exists a constant $C>0$, depending only on $n,m,p,\lambda,\Lambda$, such that for every $0<r<R$ and every $c\in\mathbb R^m$,

\begin{align*} \int_{B(x_0,r)}|\nabla u|^p\,d\mathcal L^n \le \frac{C}{(R-r)^p}\int_{B(x_0,R)}|u-c|^p\,d\mathcal L^n + C R^n. \end{align*}