---
tag: J53A2A04P7
kind: definition
name: Classification of second-order PDEs
msc: [35A99, 35J15, 35L10]
defines: 5FC6BMM1NA
uses: [6DZETZ5T9G, F31T650DKY, 0VR671PHE5, 0BQAA2BK3Q]
sources:
  - url: https://en.wikipedia.org/wiki/Partial_differential_equation
---

## Definition

Let $\Omega$ be an open set in $\R^n$ and consider a [[linear PDE|6DZETZ5T9G]] of order $2$ in $\Omega$.
$$\sum_{i,j=1}^{n} a_{ij}(x) \, \partial_i \partial_j u + \sum_{i=1}^{n} b_i(x) \, \partial_i u + c(x) \, u = f(x).$$
Its coefficients are real and $a_{ij} = a_{ji}$, so that $A(x) = (a_{ij}(x))$ is a real [[symmetric matrix|0VR671PHE5]].
At a point $x \in \Omega$ the equation is called:

1. *elliptic* if all [[eigenvalues|0BQAA2BK3Q]] of $A(x)$ are nonzero and have the same sign;
2. *hyperbolic* if all eigenvalues of $A(x)$ are nonzero, and exactly one of them has a sign opposite to the others;
3. *ultrahyperbolic* if all eigenvalues are nonzero, and at least two are positive and at least two are negative;
4. *parabolic* if exactly one eigenvalue of $A(x)$ is zero and the others are nonzero and have the same sign.

The equation has one of these types in $\Omega$ if it has that type at every point of $\Omega$.

## Remarks

The quadratic form $\xi \mapsto \sum a_{ij}(x) \xi_i \xi_j$ is the [[principal symbol|F31T650DKY]] at $x$, so the type depends only on the second-order part.
In two variables, write the second-order part as $A u_{xx} + 2B u_{xy} + C u_{yy}$.
The equation is elliptic where $AC - B^2 > 0$, hyperbolic where $AC - B^2 < 0$, and parabolic where $AC - B^2 = 0$ and $(A, B, C) \ne (0, 0, 0)$.
Example: the [[Tricomi equation|21DD2334NR]] $u_{yy} + y \, u_{xx} = 0$ has $AC - B^2 = y$, so it is elliptic for $y > 0$, hyperbolic for $y < 0$ and parabolic on the line $y = 0$.
Non-example: for $n = 3$, a matrix $A(x)$ with eigenvalues $1, 0, 0$ fits none of the four clauses; such points are degenerate.
The type is invariant under a smooth change of variables, by [[Sylvester's law of inertia|0WFED3K3DQ]] applied to the transformed coefficient matrix.

## Explanations

### L1 human

The type of a second-order equation is read off from the signs of the eigenvalues of its matrix of second-order coefficients.