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2d

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System of linear first order PDE with constant coefficients
Finally, if the system is degenerate, then you should post it here, and someone can help you figure out what, if any, compatibility conditions are needed. 
2d

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System of linear first order PDE with constant coefficients
You should check the references in the MO answer I linked to. They might handle the linear first order system case explicitly. Or you can compose your operator on the left by the cofactor matrix of your operator. This results in a higher order diagonal operator, which can then be analyzed using the results on higher order scalar operators. 
2d

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System of linear first order PDE with constant coefficients
A compatibility condition can arise only if the system is degenerate. The system is non degenerate if there exists a nonzero $(t_1, \dots, t_n)$ such that the matrix $t_1M_1 + \cdots + t_nM_n$ is invertible. If this holds, there are no compatibility conditions, and the system has a solution for suitable $b$. 
2d

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System of linear first order PDE with constant coefficients
You might try the answers to mathoverflow.net/questions/186779/… 
2d

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System of linear first order PDE with constant coefficients
Could you say on what domain you want a solution? A bounded domain, all of $\mathbb{R}^n$, in a neighborhood of a point, or something else? 
2d

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Moser's iteration for non homogeneous quasilinear elliptic PDE
And if you've spent many, many hours (or days) trying without success, you should post a more specific question explaining what you've been able to do so far and pointing out where you got stuck. 
2d

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Moser's iteration for non homogeneous quasilinear elliptic PDE
Usually, any technique for analyzing PDE's has to be adapted to a specific situation. You should learn how Moser iteration works in principle and see if you can use interpolation estimates (such as the GagliardoNirenberg inequalities) or whatever else you can find to make it work in this situation. 
Dec 14 
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The definition of $W_0^{1,\infty}$
And Lipschitz functions are continuous. 
Dec 14 
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The definition of $W_0^{1,\infty}$
For one thing $W^{1,\infty}_0 \subset W^{1,p}_0$ 
Dec 14 
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The definition of $W_0^{1,\infty}$
The answer is yes. And you still have trace equal to zero. 
Dec 14 
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The definition of $W_0^{1,\infty}$
You get Lipschitz functions. 
Dec 14 
reviewed  Close Interpolating (tangent)vectors on a sphere 
Dec 14 
reviewed  Close Time estimate to determine if a number is prime 
Dec 14 
reviewed  Close The singular value of $F(\theta)=\sin\theta\int_{a}^{a}e^{ikz\cos\theta}f(z)dz.$ 
Dec 14 
reviewed  Close On a paper by Yoneda 
Dec 13 
reviewed  Close Continuously dependent on parameters 
Dec 13 
reviewed  Close second fundamental form of boundary of convex subset nonnegative? 
Dec 12 
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$\lVert u\rVert_{W^{2,p}}$ is bounded above by $\lVert \Delta_p u\rVert_{L^2}$ for $u \in W^{1,p}_0 \cap W^{2,p}$?
The question assumes the function is compactly supported, so boundary regularity is not an issue. 
Dec 11 
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Uhlenbeck's theorem novelty
I'm sure there are other references by now, but one is Instantons and FourManifolds. by Freed and Uhlenbeck. 
Dec 11 
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A sumofdeterminants identity
It's disappointing that you didn't get a quick answer on math.stackexchange.com. 