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If you switch the second and third rows of your system, the differential operator is the same as the linearization of equation (4.3) in Existence of elastic deformations with prescribed principal stra …

COMMENT: The answer below is just the proof of the Frobenius theorem (https://en.wikipedia.org/wiki/Frobenius_theorem_(differential_topology) applied to this specific case. The arguments below are als …

ADDDED: Since equation (1) below is a first order linear ODE,it has an explicit solution (with an integral). It follows that the solution to the entire system of equations can be written explicitly as …

Moser iteration proceeds roughly like this: If the dimension $n$ is greater than $2$ and we assume homogeneous Dirichlet , we can proceed as follows: Using the Sobolev inequality on $\mathbb{R}^n$, $$ …

If you are interested in a real scalar linear PDE with constant coefficients, then it has all been worked out, primarily by Ehrenpreis, using the Fourier transform. Unfortunately, I don't know of a de …

Hormander showed that there is a generic set of scalar linear PDE's that can be studied using general techniques, known as microlocal analysis. This can be linked to algebraic geometry as follows: Any …

If you assume that the coefficients $a^{ij}$ are smooth functions and let $$b^{ij} = \frac{1}{2}(a^{ij} + a^{ji}),$$ then the PDE can be written as $$ b^{ij}\partial^2_{ij}u + \partial_ia^{ij}\partial …

Assuming that $P^{-1}$ is a right inverse and $\Omega$ an open subset of $\mathbb{R}^n$ or an open manifold, then you can proceed as follows: 1) An operator $Q: L^2(\Omega) \rightarrow L^2\Omega$ is …

It goes something like this (I can't promise that what I've written below is completely correct. It is only to help you read the rigorous details in more definitive reference): The initial observatio …