Interior differential operator

with respect to a surface $\Sigma_m$

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.