Editing Elliptic operator (section)

{{Short description|Type of differential operator}}
{{More footnotes needed|date=September 2024}}
[[File:Laplace's equation on an annulus.svg|right|thumb|300px|A solution to [[Laplace's equation]] defined on an [[Annulus (mathematics)|annulus]]. The [[Laplace operator]] is the most famous example of an elliptic operator.]]

In the theory of [[partial differential equations]],  '''elliptic operators''' are  [[differential operator]]s that generalize the [[Laplace operator]]. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the [[principal symbol]] is invertible, or equivalently that there are no real [[Method of characteristics|characteristic]] directions.

Elliptic operators are typical of [[potential theory]], and they appear frequently in [[electrostatics]] and [[continuum mechanics]]. [[Elliptic regularity]] implies that their solutions tend to be [[smooth function]]s (if the coefficients in the operator are smooth).  Steady-state solutions to [[Hyperbolic partial differential equation|hyperbolic]] and [[Parabolic partial differential equation|parabolic]] equations generally solve elliptic equations.