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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 $n=2$.

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There is probably a typo, shouldn't it be $q/2$ on the left hand-side? –  Jung Wen Chen Jan 3 at 9:57
1  
Does the argument need the Sobolev-Poincare with the Sobolev exponent $\frac{2n}{n-2}$? Perhaps any one between $2$ and this is sufficient. –  Kelei Wang Jan 3 at 10:34
    
I don't know how Giaquinta and Giusti prove this, but I'm pretty sure it can be proved using Moser iteration (where you inductively prove $L^p$ bounds for an increasing sequence of $p$ (determined by the exponents involved in the Sobolev inequality) on a decreasing sequence of balls. The inductive step uses integration by parts, the Hölder inequality, and the Sobolev inequality. And, as Kelei says, I don't believe the sharp version of Sobolev is needed for for this. –  Deane Yang Jan 3 at 16:23
    
@ Athanagor...Thanks yes, there was a small typo. I've fixed it now. There is no q/2 on the LHS, however. –  Nirav Jan 4 at 1:07
    
@Kelei and @ Deane- I've just gone over my workings and you are right. Giaquinta's and Giusti's proof is valid for n=2. I'll look up Moser's iteration method anyhow...Thanks –  Nirav Jan 4 at 3:27

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