Elliptic operator

In the theory of partial differential equations, elliptic operators are differential operators 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 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 functions (if the coefficients in the operator are smooth). Steady-state solutions to hyperbolic and parabolic equations generally solve elliptic equations.

Definitions

Let

L

{\displaystyle L}

be a linear differential operator of order m on a domain

Ω

{\displaystyle \Omega }

in Rn given by

L

u

=

∑

|

α

|

≤

m

a

α

(

x

)

∂

α

u

{\displaystyle Lu=\sum _{|\alpha |\leq m}a_{\alpha }(x)\partial ^{\alpha }u}

where

α

=

(

α

1

,

…

,

α

n

)

{\displaystyle \alpha =(\alpha _{1},\dots ,\alpha _{n})}

denotes a multi-index, and

∂

α

u

=

∂

1

α

1

⋯

∂

n

α

n

u

{\displaystyle \partial ^{\alpha }u=\partial _{1}^{\alpha _{1}}\cdots \partial _{n}^{\alpha _{n}}u}

denotes the partial derivative of order

α

i

{\displaystyle \alpha _{i}}

in

x

i

{\displaystyle x_{i}}

.

Then

L

{\displaystyle L}

is called elliptic if for every x in

Ω

{\displaystyle \Omega }

and every non-zero

ξ

{\displaystyle \xi }

in Rn,

∑

|

α

|

=

m

a

α

(

x

)

ξ

α

≠

0

,

{\displaystyle \sum _{|\alpha |=m}a_{\alpha }(x)\xi ^{\alpha }\neq 0,}

where

ξ

α

=

ξ

1

α

1

⋯

ξ

n

α

n

{\displaystyle \xi ^{\alpha }=\xi _{1}^{\alpha _{1}}\cdots \xi _{n}^{\alpha _{n}}}

.

In many applications, this condition is not strong enough, and instead a uniform ellipticity condition may be imposed for operators of order m = 2k:

(

−

1

)

k

∑

|

α

|

=

2

k

a

α

(

x

)

ξ

α

>

C

|

ξ

|

2

k

,

{\displaystyle (-1)^{k}\sum _{|\alpha |=2k}a_{\alpha }(x)\xi ^{\alpha }>C|\xi |^{2k},}

where C is a positive constant. Note that ellipticity only depends on the highest-order terms.

A nonlinear operator

L

(

u

)

=

F

(

x

,

u

,

(

∂

α

u

)

|

α

|

≤

m

)

{\displaystyle L(u)=F\left(x,u,\left(\partial ^{\alpha }u\right)_{|\alpha |\leq m}\right)}

is elliptic if its linearization is; i.e.