In mathematics, an elliptic operator is one of the major types of differential operator. It can be defined on spaces of complex-valued functions, or some more general function-like objects. What is distinctive is that the coefficients of the highest-order derivatives satisfy a positivity condition. The usual partial differential equations involving time, such as the heat equation and the Schrodinger equation, also contain elliptic operators involving the spatial variables, as well as time derivatives. Elliptic operators are typical of potential theory. Their solutions (harmonic functions of a general kind) tend to be smooth functions (if the coefficients in the operator are continuous). More simply, steady-state solutions to hyperbolic and parabolic equations generally solve elliptic equations.

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