Quasilinear

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A quasilinear equation is an equation of the form

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

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

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

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

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

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.

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Quasilinear

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