## Bivariate Second-order PDE

Second-order PDE with two variables can be written as $A u_{xx} + 2B u_{xy} + C u_{yy} + D u_x + E u_y + F =0$. The coefficient matrix of the second order terms is $T = \begin{pmatrix} A & B \\ B & C \end{pmatrix}$, with eigenvalues $\lambda_{1,2} = ((A+C)\pm \sqrt{(A-C)^2+4B^2})/2$. Denote discriminant $\Delta = B^2 - AC$, then $\Delta = -\lambda_1 \lambda_2$.

Table: Classification of Bivariate Second-order PDEs

Type Criteria Alt Criteria Standard Form Alt Standard Form
Hyperbolic $\Delta > 0$ $(α,β,γ) = (1,1,0)$ $u_{xy} + \dots =0$ $u_{xx} - u_{yy} + \dots =0$
Elliptic $\Delta < 0$ $(α,β,γ) = (2,0,0)$ $u_{xx} + u_{yy} + \dots =0$
Parabolic $\Delta = 0$ $(α,β,γ) = (1,0,1)$ $u_{xx} + \dots =0$ $u_{yy} + \dots =0$

## Second-order PDE

Second-order PDE with n independent variables can be written as: (all coefficients are real) $$\sum_{i,j=1}^{n} a_{ij} \frac{\partial^2 u}{\partial x_i \partial x_j} + \sum_{k=1}^{n} b_k \frac{\partial u}{\partial x_k} + cu = f$$ The major part of the PDE is: $$\sum_{i,j=1}^{n} a_{ij} \frac{\partial^2 u}{\partial x_i \partial x_j} = \left( \frac{\partial}{\partial x_1}, \cdots , \frac{\partial}{\partial x_n} \right) \begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \cdots & a_{nn} \end{pmatrix} \begin{pmatrix} \frac{\partial}{\partial x_1} \\ \vdots \\ \frac{\partial}{\partial x_n} \end{pmatrix}$$

The coefficient matrix $T$ is a real symmetric matrix, and can be standardized into a diagonal matrix with entries -1, 0, or 1. Denote the multiplicity (here the geometric multiplicity equals the algebraic ones) of positive, negative and zero (real) eigenvalues as $\alpha, \beta, \gamma$ respectively, then $\alpha + \beta + \gamma = n$. Noticing that cases $(\alpha, \beta, \gamma)$ and $(\beta, \alpha, \gamma)$ are essentially the same, we ignore situations when $\alpha$ and $\beta$ switches.

Table: Classification of Second-order PDEs

Type Criteria Example
elliptic $(\alpha, \beta, \gamma) = (n,0,0)$ Laplace equation $u_{xx} + u_{yy} + u_{zz} =0$
parabolic $(\alpha, \beta, \gamma) = (n-1,0,1)$ heat equation $u_{t} = a^2 (u_{xx} + u_{yy} + u_{zz})$
hyperbolic $(\alpha, \beta, \gamma) = (n-1,1,0)$ wave equation $u_{tt} = a^2 (u_{xx} + u_{yy} + u_{zz})$
ultrahyperbolic $\alpha, \beta \geq 2, \gamma = 0$
elliptic-parabolic $\beta=0, \gamma \geq 2$
hyperbolic-parabolic $\alpha \beta \gamma \neq 0$