Elliptic operator (nonfiction)

A solution to Laplace's equation defined on an annulus. The Laplace operator is the most famous example of an elliptic operator.

In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is invertible, or equivalently that there are no real characteristic directions.

Elliptic operators are typical of potential theory, and they appear frequently in electrostatics and continuum mechanics. Elliptic regularity implies that their solutions tend to be smooth functions (if the coefficients in the operator are smooth). Steady-state solutions to hyperbolic and parabolic equations generally solve elliptic equations.

Fiction cross-reference

• Crimes against mathematical constants
• Gnomon algorithm
• Gnomon Chronicles
• Mathematician
• Mathematics

Nonfiction cross-reference

• Differential operator (nonfiction)
• Dirac operator (nonfiction)
• Elliptic complex (nonfiction)
• Hopf maximum principle (nonfiction)
• Hyperbolic partial differential equation (nonfiction)
• Laplace operator (nonfiction)
• Mathematics (nonfiction)
• Method of characteristics (nonfiction)
• Parabolic partial differential equation (nonfiction)
• Partial differential equation (nonfiction)
• Potential theory (nonfiction)
• Semi-elliptic operator (nonfiction)
• Smoothness (nonfiction)
• Symbol of a differential operator (nonfiction)
• Ultrahyperbolic wave equation (nonfiction)
• Weyl's lemma (nonfiction)

External links:

• Elliptic operator @ Wikipedia