Elliptic operator
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In mathematics, an Elliptic operator is a major type of differential operator P defined on spaces of complex-valued functions, or some more general function-like objects, such 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 Schrodinger equation also involve elliptic operators (on the LHS, say) as well as a time derivative (as RHS).
Second order operators
For expository purposes, we consider initially a 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.
External links
- Linear Elliptic Equations (http://eqworld.ipmnet.ru/en/solutions/lpde/lpdetoc3.pdf) at EqWorld: The World of Mathematical Equations.
- Nonlinear Elliptic Equations (http://eqworld.ipmnet.ru/en/solutions/npde/npde-toc3.pdf) at EqWorld: The World of Mathematical Equations.
Bibliography
- A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002.