Classification of Second-Order Linear PDEs

1Introduction

This lecture explains how to classify a second-order linear PDE in two variables pointwise from its highest-order derivatives and how to determine its characteristic directions and canonical form. The terms elliptic, parabolic, and hyperbolic suggest typical equilibrium, diffusion, and propagation behavior, but the type alone does not guarantee existence, uniqueness, or stability.

The discriminant classification presented here applies to two independent variables. In higher dimensions, it is generalized by counting the signs of the eigenvalues of the principal coefficient matrix.

2General form and principal part

On a domain $$\mathrm{\Omega }\subset R^2$$\Omega\subset\mathbb R^2, consider an equation with real-valued continuous coefficients,

Here $$A,\dots ,F,G$$A,\ldots,F,G are prescribed functions of $$(x,y)$$(x,y) and do not depend on $$u$$u or its derivatives; this is the meaning of linearity in $$u$$u. The {principal part} used for classification is

with principal coefficient matrix

The terms $$D,E,F,G$$D,E,F,G affect the particular form and well-posedness of solutions, but the type is determined by $$M$$M alone.

3Discriminant and the three types

At a point $$(x,y)$$(x,y), set

At points where $$M\ne 0$$M\ne0, equivalently where $$A,B,C$$A,B,C are not all zero, the classification is as follows.

If $$A=B=C=0$$A=B=C=0, then the second-order principal part is absent, so the equation is not called parabolic even though $$\delta =0$$\delta=0. When the coefficients vary, the classification is pointwise rather than automatically valid throughout the domain.

4Computations for representative equations

For Laplace's equation

one has $$A=C=1$$A=C=1 and $$B=0$$B=0, so $$\delta =-1<0$$\delta=-1<0 and the equation is elliptic.

Write the heat equation in the variable order $$(x,t)$$(x,t) as

Then $$A=-\kappa$$A=-\kappa and $$B=C=0$$B=C=0, so $$\delta =0$$\delta=0 and $$M$$M has rank 1; the equation is parabolic. The term $$u_t$$u_t is first order and therefore does not enter the classification, although it is essential in determining the direction of time evolution.

For the wave equation

one has $$A=-c^2$$A=-c^2, $$B=0$$B=0, and $$C=1$$C=1, so $$\delta =c^2>0$$\delta=c^2>0 and the equation is hyperbolic.

5Why the type is preserved under coordinate changes

Let the new coordinate functions $$(\xi ,\eta )=\left(\xi \right(x,y),\eta (x,y\left)\right)$$(\xi,\eta)=(\xi(x,y),\eta(x,y)) be of class $$C^2$$C^2, and let $$J$$J be their Jacobian matrix. At a point where $$\det J\ne 0$$\det J\ne0, the inverse function theorem makes them local coordinates, and the principal coefficient matrix in the new coordinates is transformed by congruence:

Consequently,

so the sign of $$\delta =-\det M$$\delta=-\det M is unchanged. Transforming a variable-coefficient equation also produces lower-order terms from second derivatives of the coordinate functions, but the type of the principal part is preserved by a nonsingular coordinate change. This conclusion is unavailable for a singular transformation.

6Relation to characteristic curves

The normal direction to a curve $$\varphi (x,y)=\text{constant}$$\phi(x,y)=\text{constant} is the gradient $$({\varphi }_x,{\varphi }_y)$$(\phi_x,\phi_y). The curve is {characteristic} for the principal part when

Writing the tangent direction as $$(dx,dy)$$(dx,dy) gives the equivalent homogeneous condition

This formulation also includes vertical curves. If the curve is written as $$y=y\left(x\right)$$y=y(x) and $$m=dy/dx$$m=dy/dx, its normal is proportional to $$(-m,1)$$(-m,1), and hence

When $$A\ne 0$$A\ne0, this is a quadratic equation for the finite slope $$m$$m, with discriminant $$4(B^2-AC)=4\delta$$4(B^2-AC)=4\delta. If $$A=0$$A=0, the quadratic term disappears; the equation is linear when $$B\ne 0$$B\ne0 and constant when $$B=0$$B=0. A vertical direction $$dx=0$$dx=0 may then be invisible in the slope coordinate $$m$$m. One must return to the homogeneous equation or use a representation $$x=x\left(y\right)$$x=x(y). Counting all projective directions yields the numbers of real characteristic directions in the classification table.

For the heat equation, $$m=dt/dx=0$$m=dt/dx=0 is a repeated root, so $$t=\text{constant}$$t=\text{constant} is characteristic for the second-order principal part. Nevertheless, one initial value can be prescribed at $$t=0$$t=0 for forward evolution because the first-order term $$u_t$$u_t remains and creates the parabolic evolution structure. The non-characteristic condition for a first-order quasilinear PDE in the preceding lecture must not be identified with this characteristic condition for a second-order principal part.

7Entry point to canonical forms

At a single point, a nonsingular linear coordinate transformation, together with multiplication of the entire equation by $$-1$$-1 when necessary, reduces the principal part to one of the following forms.

Reducing a variable-coefficient equation to canonical form throughout a neighborhood additionally requires sufficient coefficient regularity, constancy of type, and the existence of suitable local coordinates. A global reduction does not follow automatically.

8What the type does and does not predict

The type is an entry point for selecting a problem formulation and a method of solution. The following correspondences hold only under representative additional assumptions.

{Uniform ellipticity} on a domain means that there are a fixed $$\sigma \in \{1,-1\}$$\sigma\in\{1,-1\} and constants $$0<{\lambda }_0\text{[PARSE ERROR: Undefined("Command("le")")]}{\mathrm{\Lambda }}_0<\mathrm{\infty }$$0<\lambda_0\le\Lambda_0<\infty such that, at every point and for every $$\zeta \in R^2$$\zeta\in\mathbb R^2,

This condition makes the eigenvalue signs consistent throughout the domain and keeps their magnitudes uniformly separated from both zero and infinity. Such uniformity is important when estimates associated with a type are applied over an entire domain.

For example, the forward and backward heat equations have the same parabolic principal-part classification, but their stability properties differ. Likewise, ellipticity alone does not imply that arbitrary boundary conditions determine a unique solution. Initial conditions, boundary conditions, and well-posedness must be assessed for the complete problem separately from its type.

9Counterexample and limitation

For an equation whose coefficients vary with position, the type need not be constant throughout the domain. For example,

has $$A=y$$A=y, $$B=0$$B=0, and $$C=1$$C=1, so $$\delta =-y$$\delta=-y. It is elliptic for $$y>0$$y>0, hyperbolic for $$y<0$$y<0, and parabolic of rank 1 for $$y=0$$y=0. The line $$y=0$$y=0 is a type-change line where uniformity is lost; estimates and methods valid on one side cannot automatically be applied across it.

10Common errors

• Substituting the wrong coefficient into $$B^2-AC$$B^2-AC without checking the convention $$2B{u}_{xy}$$2B u_{xy}. Under the convention $$B{u}_{xy}$$B u_{xy}, the off-diagonal entries of the principal coefficient matrix are $$B/2$$B/2.
• Calling a degenerate equation with $$A=B=C=0$$A=B=C=0 parabolic merely because $$\delta =0$$\delta=0.
• Concluding well-posedness from the type without examining boundary conditions, time direction, or coefficient regularity.
• Extending a pointwise classification to an entire domain while ignoring type changes or loss of uniformity.

11Related links