Elliptic operator (nonfiction): Difference between revisions
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[[File:Laplace's equation on an annulus.jpg|thumb|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 equation (nonfiction)|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. | [[File:Laplace's equation on an annulus.jpg|thumb|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 equation (nonfiction)|partial differential equations]], '''elliptic operators''' are differential operators that generalize the [[Laplace operator (nonfiction)|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. | 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. | ||
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== Nonfiction cross-reference == | == Nonfiction cross-reference == | ||
* [[Laplace operator (nonfiction)]] | |||
* [[Mathematics (nonfiction)]] | * [[Mathematics (nonfiction)]] | ||
* [[Partial differential equation (nonfiction)]] | * [[Partial differential equation (nonfiction)]] | ||
Revision as of 16:25, 22 May 2019
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.
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External links:
- Elliptic operator @ Wikipedia