In Giaquinta-Giusti's (1978) paper "Nonlinear Elliptic Systems with Elliptic Growth" (thm 1.1) they consider the following system:

\begin{equation} \sum_{i, j=1}^{n}\sum_{\alpha, \beta=1}^N\frac{\partial}{\partial x_i}\left(A^{ij}_{\alpha\beta}(x)\frac{\partial u^{\beta}}{\partial x_j}\right)=f_{\alpha}(x, u, Du),\quad\alpha=1,\dots, N, \end{equation} where the coefficients are bounded, measurable and satisfy the strong ellipticity condition:

\begin{equation} \sum_{i, j=1}^{n}\sum_{\alpha, \beta=1}^N A^{ij}_{\alpha\beta}(x)\xi_{i}^{\alpha}\xi_{j}^{\beta}\geq\lambda|\xi|^2, \quad\xi\in \mathbb{R}^{nN} \end{equation}with $\lambda>0$ and \begin{equation} |A^{ij}_{\alpha\beta}(x)|\leq L. \end{equation}

Lastly, $f$ satisfies: \begin{equation} |f(x, u, p)|\leq a|p|^2+b \end{equation}whenever $|u|\leq M$.

They show that if $u\in H^1(\Omega)$ is a weak solution to the above system in $B_2$, $|u|\leq M$ and $2Ma<\lambda$ then there exists a $q>2$ such that $|Du|\in L^q(B_1)$ and \begin{equation} \int_{B_1}(b+|Du|^2)^q\ \mathrm{d}x \leq K\left\{\int_{B_2}(b+|Du|^2)\ \mathrm{d}x\right\}^{\frac{q}{2}}. \end{equation}

In their proof they use the Sobolev-Poincare inequality at a point (thus their result is true for $2<n$). I was wondering if it is possible to get a similar result for $n=2$.

In Giaquinta-Giusti's (1978) paper "Nonlinear Elliptic Systems with Elliptic Growth" (thm 1.1) they consider the following system:

\begin{equation} \sum_{i, j=1}^{n}\sum_{\alpha, \beta=1}^N\frac{\partial}{\partial x_i}\left(A^{ij}_{\alpha\beta}(x)\frac{\partial u^{\beta}}{\partial x_j}\right)=f_{\alpha}(x, u, Du),\quad\alpha=1,\dots, N, \end{equation} where the coefficients are bounded, measurable and satisfy the strong ellipticity condition:

\begin{equation} \sum_{i, j=1}^{n}\sum_{\alpha, \beta=1}^N A^{ij}_{\alpha\beta}(x)\xi_{i}^{\alpha}\xi_{j}^{\beta}\geq\lambda|\xi|^2, \quad\xi\in \mathbb{R}^{nN} \end{equation}with $\lambda>0$ and \begin{equation} |A^{ij}_{\alpha\beta}(x)|\leq L. \end{equation}

Lastly, $f$ satisfies: \begin{equation} |f(x, u, p)|\leq a|p|^2+b \end{equation}whenever $|u|\leq M$.

They show that if $u\in H^1(\Omega)$ is a weak solution to the above system in $B_2$, $|u|\leq M$ and $2Ma<\lambda$ then there exists a $q>2$ such that $|Du|\in L^q(B_1)$ and \begin{equation} \int_{B_1}(b+|Du|)^q\ \mathrm{d}x \leq K\left\{\int_{B_2}(b+|Du|)^2\ \mathrm{d}x\right\}^{\frac{q}{2}}. \end{equation}

In their proof they use the Sobolev-Poincare inequality at a point (thus their result is true for $2<n$). I was wondering if it is possible to get a similar result for $n=2$.