Transmission, condition of

transmission condition

A condition on a pseudo-differential operator on a smooth manifold with boundary that guarantees that functions that after extension by zero remain smooth up to the boundary are taken by these operators to functions that are again smooth up to the boundary. Here the extension by zero is carried out in some neighbourhood of the original manifold, which is regarded as imbedded in a larger manifold without boundary, so that points of the boundary become interior points.

If the symbol of the given pseudo-differential operator has an asymptotic expansion in positive homogeneous functions $ a _ \alpha ( x, \xi ) $ (where $ \alpha $ is the order of homogeneity) in local coordinates in a neighbourhood of the boundary, then the transmission condition can be written in the form of the following condition on the function $ a _ \alpha $:

$$ \partial _ {x} ^ \gamma \ \partial _ {\xi ^ \prime } ^ \beta [ a _ \alpha ( x, \xi ^ \prime , \xi _ {n} ) - e ^ {- i \pi \alpha } a _ \alpha ( x, - \xi ^ \prime , - \xi _ {n} )] \mid _ {x _ {n} = 0 , \xi ^ \prime = 0 } = 0, $$

where $ \gamma $, $ \beta $ are any multi-indices; the coordinates of $ x $ are chosen in a neighbourhood of a boundary point such that $ \{ x _ {n} = 0 \} $ is the equation of the boundary, $ x _ {n} \geq 0 $ is on the manifold itself; and $ \xi = ( \xi ^ \prime , \xi _ {n} ) $ are coordinates dual to the coordinates $ x = ( x ^ \prime , x _ {n} ) $.

References

[1]	G.I. Eskin, "Boundary value problems for elliptic pseudodifferential equations" , Amer. Math. Soc. (1986) (Translated from Russian)
[2]	L. Boutet de Monvel, "Boundary problems for pseudo-differential operators" Acta Math. , 126 : 1/2 (1971) pp. 11–51
[3]	S. Rempel, B.-W. Schulze, "Index theory of elliptic boundary problems" , Akademie Verlag (1982)
[4]	L.V. Hörmander, "The analysis of linear partial differential operators" , 3 , Springer (1985)
[5]	G. Grubb, "Functional calculus of pseudo-differential boundary problems" , Birkhäuser (1986)