Elliptic Partial Differential Equation

An elliptic partial differential equation is a partial differential equation whose principal symbol is nonzero at every nonzero real frequency in the linear case. For a nonlinear equation, this condition is imposed on its linearization at a solution. For a second-order equation in two variables of the form

(1)

ellipticity means that the matrix

(2)

is positive definite or negative definite, equivalently . Elliptic partial differential equations have applications in almost all areas of mathematics, from harmonic analysis to geometry to Lie theory, as well as numerous applications in physics. As with a general PDE, elliptic PDE may have non-constant coefficients and be non-linear. Despite this variety, the elliptic equations have a well-developed theory.

The basic example of an elliptic partial differential equation is Laplace's equation

(3)

in -dimensional Euclidean space, where the Laplacian is defined by

(4)

Other examples of elliptic equations include the nonhomogeneous Poisson's equation

(5)

and the non-linear minimal surface equation.

For example, boundary conditions are used to give the constraint on , where

(6)

holds in .

One property of constant coefficient elliptic equations is that their solutions can be studied using the Fourier transform. Consider Poisson's equation with periodic . The Fourier series expansion is then given by

(7)

where is the principal symbol up to sign. For , the multiplier is nonzero and determines . A periodic solution requires the compatibility condition , meaning that has zero mean. When this condition holds, the solution is determined up to an additive constant.

In general, a PDE may have non-constant coefficients or even be non-linear. A linear PDE is elliptic if its principal symbol, as in the theory of pseudodifferential operators, is nonzero away from the origin. For instance, (3) has as its principal symbol , which is nonzero for , and is an elliptic PDE.

A nonlinear PDE is elliptic at a solution if its linearization is elliptic at . One simply calls a non-linear equation elliptic if it is elliptic at any solution, such as in the case of harmonic maps between Riemannian manifolds.