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Interior differential operator

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with respect to a surface

A differential operator L(u) such that for any function for which it is defined its value at a point M\in\Sigma_m can be calculated from only the values of this function on the smooth surface \Sigma_m defined in the space E^n, m<n. An interior differential operator can be computed using derivatives in directions l which lie in the tangent space to \Sigma_m. If one introduces coordinates such that on \Sigma_m,

x_{m+1}=x_{m+1}^0,\dots,x_n=x_n^0,

then the operator L(u), provided it is interior with respect to \Sigma_m, will not contain, after suitable transformations, derivatives with respect to the variables x_{m+1},\dots,x_n (the so-called exterior or extrinsic derivatives). For instance, the operator

L(u)=\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z}

is an interior differential operator with respect to any smooth surface containing a straight line x-x_0=y-y_0=z-z_0, and with respect to any one of these lines. If the operator L(u) is an interior differential operator with respect to a surface \Sigma_{n-1}, then \Sigma_{n-1} is said to be a characteristic of the differential equation L(u)=0.

An operator is sometimes called interior with respect to a surface \Sigma_m if, at the points of this surface, the leading order of the extrinsic derivatives is lower than the order of the operator.

How to Cite This Entry:
Interior differential operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Interior_differential_operator&oldid=33241
This article was adapted from an original article by B.L. Rozhdestvenskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article