Elliptic Operator - Definitions

Definitions

A linear differential operator L of order m on a domain in Rd given by

is called elliptic if for every x in and every non-zero in Rd,

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

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

A nonlinear operator

is elliptic if its first-order Taylor expansion with respect to u and its derivatives about any point is a linear elliptic operator.

Example 1

The negative of the Laplacian in Rd given by

is a uniformly elliptic operator. The Laplace operator occurs frequently in electrostatics. If ρ is the charge density within some region Ω, the potential Φ must satisfy the equation

Example 2

Given a matrix-valued function A(x) which is symmetric and positive definite for every x, having components aij, the operator

is elliptic. This is the most general form of a second-order divergence form linear elliptic differential operator. The Laplace operator is obtained by taking A = I. These operators also occur in electrostatics in polarized media.

Example 3

For p a non-negative number, the p-Laplacian is a nonlinear elliptic operator defined by

A similar nonlinear operator occurs in glacier mechanics. The stress tensor of ice, according to Glen's flow law, is given by

for some constant B. The velocity of an ice sheet in steady state will then solve the nonlinear elliptic system

where ρ is the ice density, g is the gravitational acceleration vector, p is the pressure and Q is a forcing term.