Classification of Second-Order Linear PDEs

The classification is the organizing spine of every introductory and advanced PDE course, but its reach extends far beyond pure analysis. In numerical analysis, the type of a PDE dictates which discretization schemes are stable: explicit time-stepping for parabolic and hyperbolic equations must satisfy CFL-type stability conditions tied directly to characteristic speeds, while elliptic equations are solved by relaxation or spectral methods requiring boundary data on the entire domain boundary. In fluid dynamics, the Euler equations for compressible flow change type across the sound barrier — subsonic flow is governed by elliptic-type behavior and supersonic flow by hyperbolic behavior, with the transonic regime described by mixed-type equations like the Tricomi equation from Example 1. In general relativity, the Einstein field equations' hyperbolic character (in suitable gauges) underlies the finite propagation speed of gravitational waves, detected a century after Einstein's prediction. In mathematical finance, the Black–Scholes equation is a parabolic PDE (after a change of variables it becomes the heat equation exactly), and its parabolicity is what guarantees the smoothing and uniqueness properties options pricers rely on. In image processing, elliptic and parabolic PDEs (such as anisotropic diffusion) are used for denoising and edge-preserving smoothing, exploiting the same discriminant structure to control how information spreads across an image. In control theory and inverse problems, distinguishing type determines whether a problem is well-posed forward-in-time (parabolic, hyperbolic) or requires boundary-value formulations (elliptic), directly shaping which numerical solvers and regularization techniques apply.

Ask the tutor: Can you walk through classifying a PDE whose coefficients depend on both x and y, and finding where its type changes?

The coefficients $A$, $B$, $C$ of the second-derivative terms define a symmetric bilinear (quadratic) form, exactly the way the coefficients of $Ax^2 + Bxy + Cy^2$ define a quadratic form on $\mathbb{R}^2$. Formally, associate to the PDE the symmetric matrix

whose determinant is $AC - B^2/4 = -\Delta/4$. The eigenvalues of $M$ determine the signature of this quadratic form exactly as they do for a conic section. If $\det M > 0$ (i.e. $\Delta < 0$), the eigenvalues have the same sign and the form is definite — this is the elliptic case, and the level curves $Ax^2+Bxy+Cy^2 = \text{const}$ are ellipses. If $\det M < 0$ ($\Delta>0$), the eigenvalues have opposite signs — an indefinite form, giving hyperbolas. If $\det M = 0$ ($\Delta=0$), one eigenvalue vanishes — a degenerate form, giving parabolas. This is precisely why the names "elliptic," "parabolic," "hyperbolic" were borrowed from conic sections: the discriminant test is the same test, applied to the leading coefficients of the PDE instead of to the coefficients of a plane curve. The deeper reason this algebraic fact controls the PDE's qualitative behavior is the theory of characteristics: curves along which the PDE fails to determine the second derivatives of $u$ uniquely from Cauchy data, and along which singularities or discontinuities in the data can propagate. Requiring a curve $\phi(x,y) = \text{const}$ to be characteristic leads to an ODE for its slope, $A(\phi_y)^2 - B\phi_x\phi_y + C(\phi_x)^2=0$, whose solvability in real numbers is governed by exactly the same discriminant $B^2-4AC$. Two real, one real (double), or zero real solution families of this ODE correspond precisely to the hyperbolic, parabolic, and elliptic cases.

Example 1 — A variable-coefficient equation of mixed type (Tricomi-type). Consider the Euler–Poisson–Darboux-flavored equation

on $\mathbb{R}^2$. Here $A=y$, $B=0$, $C=1$, so

This is a genuinely mixed-type equation:

• For $y>0$: $\Delta<0$, elliptic.
• For $y=0$: $\Delta=0$, parabolic (a curve, not just a point, of degeneracy — the $x$-axis).
• For $y<0$: $\Delta>0$, hyperbolic.

Let's find the characteristics explicitly in the hyperbolic region $y<0$. The characteristic ODE is

Separating variables ($y<0$, so $\sqrt{-y}$ is real): $\sqrt{-y}\,dy = \mp dx$. Integrating,

So the two characteristic families are

This example — the Tricomi equation $yu_{xx}+u_{yy}=0$ (in the closely related sign convention $u_{yy}=yu_{xx}$) — is the classical model for transonic flow: $y<0$ (subsonic-analogue) is elliptic, $y>0$ is hyperbolic in that convention, illustrating that a single physically meaningful equation can genuinely change type across a region, which is why the classification in the Theorem is stated locally at a point, not globally for an equation.

Example 2 — Reducing a non-diagonal constant-coefficient hyperbolic equation to canonical form. Consider

Here $A=3,\ B=7,\ C=2$, so

confirming this is hyperbolic everywhere (constant coefficients, so the type never changes). The characteristic ODE is

By the quadratic formula,

giving two roots $dy/dx = 2$ and $dy/dx = 1/3$. Integrating each:

So take

Check the Jacobian: $\xi_x=-2,\xi_y=1,\eta_x=-1,\eta_y=3$, so $J=\xi_x\eta_y-\xi_y\eta_x = (-2)(3)-(1)(-1) = -6+1=-5\neq0$, confirming invertibility. Now apply the transformation formulas from Step 1 of the proof to double-check the reduction:

Indeed $A^*=C^*=0$ and $B^*=-25\ne0$ as predicted, so the equation becomes, after dividing by $-25$,

This can now be solved directly: integrating $u_{\xi\eta}=0$ in $\eta$ gives $u_\xi = f(\xi)$ for arbitrary $f$, and integrating in $\xi$ gives $u(\xi,\eta) = F(\xi)+G(\eta)$ for arbitrary twice-differentiable $F,G$. Translating back,

which one can verify directly solves $3u_{xx}+7u_{xy}+2u_{yy}=0$ for any choice of $C^2$ functions $F,G$ — exhibiting the general solution as a superposition of waves propagating along the two characteristic directions, the hallmark of hyperbolic behavior.