Partial differential equations Definition

Quasilinear PDE

Builds on Partial differential equation Order of a PDE Multi-index notation

Definition

Let $\Omega$ be an open set in $\R^n$ and let $m \ge 1$ . A partial differential equation of order $m$ in $\Omega$ is quasilinear if it has the following form. $\sum_{\abs{\alpha} = m} a_\alpha\left(D^{m-1} u, \dots, D u, u, x\right) \partial^\alpha u + a_0\left(D^{m-1} u, \dots, D u, u, x\right) = 0.$ Here the functions $a_\alpha$ and $a_0$ are given. Here $\alpha$ runs over multi-indices, and $D^j u$ denotes the collection of all partial derivatives of order $j$ . So the equation is affine in the derivatives of order $m$ , and its coefficients depend at most on the derivatives of order less than $m$ .

Remarks

Example: the minimal surface equation $\operatorname{div}\left(\nabla u / \sqrt{1 + \abs{\nabla u}^2}\right) = 0$ in two variables expands to $(1 + u_y^2) u_{xx} - 2 u_x u_y u_{xy} + (1 + u_x^2) u_{yy} = 0$ . The coefficients of $u_{xx}$ , $u_{xy}$ , $u_{yy}$ depend only on first derivatives, so it is quasilinear of order $2$ . Non-example: the Monge–Ampère equation $u_{xx} u_{yy} - u_{xy}^2 = f(x, y)$ is not quasilinear, because it contains products of second derivatives; it is fully nonlinear. Every semilinear PDE is quasilinear. For first-order equations, quasilinear means $\sum_{i} a_i(u, x) \partial_i u + a_0(u, x) = 0$ .

Tag	`DV92N2YWZJ`
URI	`m2://DV92N2YWZJ`
Kind	definition
Writing	W0 stub
MSC	35A99, 35J62, 35K59
Defines	Quasilinear PDE
Uses	Partial differential equation, Order of a PDE, Multi-index notation

Quasilinear PDE

Definition

Remarks

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