Semilinear differential operator

This article defines a property that can be evaluated for a differential operator on a differential manifold (viz a linear map from the space of differentiable functions to itself)

Definition

A semilinear differential operator on a differential manifold $M$ is a map $C^{\mathrm{\infty }}(M)\to C^{\mathrm{\infty }}(M)$ that can be expressed as:

$f\mapsto Lf+g\circ f$

where $L$ is a linear differential operator and $g$ is an element of $C^{\mathrm{\infty }}(M)$.

Relation with other properties

Stronger properties

• Linear differential operator

Weaker properties

• Quasilinear differential operator

Relation with differential equations

A differential equation of the form $Sf=0$ where $S$ is a semilinear differential operator, is termed a semilinear differential equation.

External links

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