# Quasilinear

A quasilinear equation is an equation of the form

${\displaystyle F(u,Du,\ldots ,D^{k}u)=0}$

which is linear (and nontrivial) in the top order terms ${\displaystyle D^{k}u}$. Thus a quasilinear equation takes the schematic form

${\displaystyle F(u,Du,\ldots ,D^{k-1}u)D^{k}u=G(u,Du,\ldots ,D^{k-1}u).}$

By differentiating this equation up to ${\displaystyle k-1}$ times and working with the system of fields ${\displaystyle v:=(u,Du,\ldots ,D^{k}u)}$, one can place such equations in the slightly simpler form

${\displaystyle F(v)D^{k}v=G(v,Dv,\ldots ,D^{k-1}v)}$

though this trick comes at the cost of lowering the regularity of the fields.

Quasilinear equations are more nonlinear than semilinear ones, but less nonlinear than fully nonlinear equations.