Elliptic operator

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In mathematics, an elliptic operator is one of the major types of differential operator P. It will be defined on spaces of complex-valued functions, or some more general function-like objects. What is distinctive is that 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 Schrödinger equation also involve elliptic operators (on the LHS, say) as well as a time derivative (as RHS). Elliptic operators are typical of potential theory. Their solutions (harmonic functions of a general kind) tend to be smooth functions, and the boundary conditions of the Dirichlet problem can be fulfilled (as would be expected from the simple physical interpretation in electrostatics).

Contents 1 Second order operators 2 See also 3 References 4 External links

Second order operators

For expository purposes, we consider initially 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.

See also

• Hyperbolic partial differential equation

References

• L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998. ISBN 0-8218-0772-2

• D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York, 1983. ISBN 3-540-41160-7

External links

• Linear Elliptic Equations at EqWorld: The World of Mathematical Equations.

• Nonlinear Elliptic Equations at EqWorld: The World of Mathematical Equations.