It is known that in the case of more than two independent variables, it is usually not possible (especially in the case of PDE with the variable coefficients) to reduce a linear partial differential equation of the second order to a canonical form throughout a domain. I read the above result, for example, in Renuka Ravindran's book, Partial Differential Equations, Page 58~59. Since I have not found any detailed explanation of this result, I want to construct a counterexample of PDE in at least three independent variables, like this $$\sum_{i,j=1}^n a_{ij}(x)u_{x_ix_j}+\sum_{i=1}^n b_{i}(x)u_{x_i}+c(x)u(x)=0, x\in\Omega\subset\mathbb{R}^n, $$ where $n\geq3,$ such that which can not be reduced to canonical form globally, that is the form of the following $$\sum_{i,j=1}^n \alpha_{ij} u_{\xi_i\xi_j}+\text{lower oder terms}=0, ~ \xi\in\Omega'\subset\mathbb{R}^n,$$ where $$\alpha_{i,j}=0,\pm 1.$$ But I do not how. Can anyone help me, or tell me where I can find these counterexamples?
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$\begingroup$ What is your notion of canonical form? $\endgroup$– Igor KhavkineCommented Mar 27, 2014 at 5:29
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2$\begingroup$ The point is that, when $n\ge3$, the symmetric quadratic $(2,0)$-form $\alpha = \alpha^{ij}\partial_i\partial_j$ (which is the principal symbol of the equation) will typically have conformal invariants when it is nondegenerate, i.e., there will usually not be any change of coordinates that reduces it to having constant coefficients. $\endgroup$– Robert BryantCommented Mar 27, 2014 at 8:52
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