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2 Hormander->Hörmander, principle->principal

I am unsure of the etiquette surrounding multi-part questions. Here are answers to two sub-parts. Since your Q5. invites opinion, I've addressed that in a comment instead.

Q1: yes, the definition of ellipticity via the non-vanishing of the principle symbol is a useful characterization for elliptic PDE, see Hormander's Hörmander's book. All manners of existence and regularity properties can be examined from here.

Q2: This is murkier. If the principle principal symbol of a linear PDE, order q, with smooth coefficients is a hyperbolic polynomial, then the PDE is hyperbolic. This doesn't generalize easily to nonlinear cases, and is not an easy condition to check. See an extensive discussion here: http://math.stackexchange.com/questions/21525/mathematical-precise-definition-of-a-pde-being-elliptic-parabolic-or-hyperbolic

L.C. Evans, in his preface to his AMS text on PDE, mentions that he finds it unsatisfactory to classify PDE, since it creates the false impression that a general classification is available. Several equations change type (eg. Tricomi's equation) and many PDE of interest are highly nonlinear.

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I am unsure of the etiquette surrounding multi-part questions. Here are answers to two sub-parts. Since your Q5. invites opinion, I've addressed that in a comment instead.

Q1: yes, the definition of ellipticity via the non-vanishing of the principle symbol is a useful characterization for elliptic PDE, see Hormander's book. All manners of existence and regularity properties can be examined from here.

Q2: This is murkier. If the principle symbol of a linear PDE, order q, with smooth coefficients is a hyperbolic polynomial, then the PDE is hyperbolic. This doesn't generalize easily to nonlinear cases, and is not an easy condition to check. See an extensive discussion here: http://math.stackexchange.com/questions/21525/mathematical-precise-definition-of-a-pde-being-elliptic-parabolic-or-hyperbolic

L.C. Evans, in his preface to his AMS text on PDE, mentions that he finds it unsatisfactory to classify PDE, since it creates the false impression that a general classification is available. Several equations change type (eg. Tricomi's equation) and many PDE of interest are highly nonlinear.