2.3: More than 2D

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In more than two dimensions we use a similar definition, based on the fact that all eigenvalues of the coefficient matrix have the same sign (for an elliptic equation), have different signs (hyperbolic) or one of them is zero (parabolic). This has to do with the behavior along the characteristics, as discussed below.

Let me give a slightly more complex example

\[x^2\frac{\partial^2 u}{\partial x^2} + y^2\frac{\partial^2 u}{\partial y^2} + z^2\frac{\partial^2 u}{\partial z^2}+2 xy\frac{\partial^2 u}{\partial x \partial y}+2 xz\frac{\partial^2 u}{\partial x \partial z}+2 yz\frac{\partial^2 u}{\partial y \partial z}=0. \nonumber \]

The matrix associated with this equation is \[\left(\begin{array}{lll} x^2 & xy & xz \\ xy & y^2 & yz \\ xz & yz & z^2 \end{array}\right) \nonumber \]

If we evaluate its characteristic polynomial we find that it is \[\lambda^2 (x^2-y^2+z^2-\lambda)=0. \nonumber \] Since this has always (for all \(x,y,z\)) two zero eigenvalues this is a parabolic differential equation.

Characteristics and Classification

A key point for classifying equations this way is not that we like the conic sections so much, but that the equations behave in very different ways if we look at the three different cases. Pick the simplest representative case for each class, and look at the lines of propagation.