Partial differential equations Definition

Fully nonlinear PDE

Builds on Partial differential equation Quasilinear PDE Order of a PDE

Definition

A partial differential equation $F\left(D^m u, \dots, D u, u, x\right) = 0$ of order $m$ is fully nonlinear if it is not quasilinear. That is, $F$ cannot be written as an affine function of the derivatives $D^m u$ of order $m$ whose coefficients depend only on $D^{m-1} u, \dots, D u, u, x$ .

Remarks

Example: the Monge–Ampère equation $\det D^2 u = f(x)$ in $\R^n$ with $n \ge 2$ is fully nonlinear of order $2$ . For $n = 2$ it reads $u_{xx} u_{yy} - u_{xy}^2 = f$ , which contains the product $u_{xx} u_{yy}$ of two second derivatives. Non-example: the minimal surface equation is not fully nonlinear, because it is affine in the second derivatives with coefficients depending on $\nabla u$ only. For first-order equations, the eikonal equation $\abs{\nabla u}^2 = 1$ is fully nonlinear.

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Tag	`MTQ3SGX56W`
URI	`m2://MTQ3SGX56W`
Kind	definition
Writing	W0 stub
MSC	35A99, 35J60
Defines	Fully nonlinear PDE
Uses	Partial differential equation, Quasilinear PDE, Order of a PDE

Fully nonlinear PDE

Definition

Remarks

Reading path

Further reading

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