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

P u = 0 \quad

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).

Contents

Second order operators

For expository purposes, we consider initially a second order linear partial differential operators of the form

P\phi = \sum_{k,j} a_{k j} D_k D_j \phi + \sum_\ell b_\ell D_{\ell}\phi +c \phi

where D_k = \frac{1}{i} \partial_{x_k} . Such an operator is called elliptic iff for every x the matrix of coefficients of the highest order terms

\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}

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

\vec{\xi} = (\xi_1, \xi_2, \ldots , \xi_n)

the following inequality holds:

\sum_{k,j} a_{k j}(x) \xi_k \xi_j > 0. \quad

Example. The negative of the Laplacian in Rn given by

- \Delta = \sum_{\ell=1}^n D_\ell^2

is an elliptic operator.

See also

External links

Bibliography

  • 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
  • A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9
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