Elliptic partial differential equation - Definition and Overview

In mathematics, an Elliptic operator is a major type of differential operator P defined on spaces of complex-valued functions, or some more general function-like objects, such that the coefficients of the highest-order derivatives satisfy a positivity condition. An important example of an elliptic operator is the Laplacian. Equations of the form

<math> P u = 0 \quad <math>

are called elliptic partial differential equations. Equations involving time, such as the heat equation or the Schrodinger equation also involve elliptic operators (on the LHS, say) as well as a time derivative (as RHS).

Second order operators

For expository purposes, we consider initially a second order linear partial differential operators of the form

<math> P\phi = \sum_{k,j} a_{k j} D_k D_j \phi + \sum_\ell b_\ell D_{\ell}\phi +c \phi <math>

where <math> D_k = \frac{1}{i} \partial_{x_k} <math>. Such an operator is called elliptic iff for every x the matrix of coefficients of the highest order terms

<math> \begin{bmatrix} a_{1 1}(x) & a_{1 2}(x) & \cdots & a_{1 n}(x) \\ a_{2 1}(x) & a_{2 2}(x) & \cdots & a_{2 n}(x) \\

\vdots & \vdots & \vdots & \vdots \\ a_{n 1}(x) & a_{n 2}(x) & \cdots & a_{n n}(x) \end{bmatrix}<math>

is a positive-definite real symmetric matrix. In particular, for every non-zero vector

<math> \vec{\xi} = (\xi_1, \xi_2, \ldots , \xi_n) <math>

the following inequality holds:

<math> \sum_{k,j} a_{k j}(x) \xi_k \xi_j > 0. \quad <math>

Example. The negative of the Laplacian in Rⁿ given by

<math> - \Delta = \sum_{\ell=1}^n D_\ell^2 <math>

is an elliptic operator.