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

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

where . Such an operator is called elliptic iff for every x the matrix of coefficients of the highest order terms

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

the following inequality holds:

Example. The negative of the Laplacian in Rⁿ given by

is an elliptic operator.