A theory of pseudo-differential boundary value problems with discontinuous conditions. I (Q1059228)

This paper deals with mixed boundary value problems and transmission problems for elliptic differential and pseudo-differential operators on an open set \(\Omega\) of \({\mathbb{R}}^ n\) in the case where the conditions have a jump along a submanifold Y of codimension 1 in the boundary X of \(\Omega\). These problems are considered as elements of a certain algebra and a symbolic calculus is developed. By means of additional conditions (of trace and potential type with respect to Y) a given mixed problem A is extended to an element B of the algebra and the Fredholm property of B is established, under an ellipticity condition, the parametrix is constructed and the index of B is expressed. In the second part the authors study the case of certain classes of differential operators of second and higher order with mixed boundary conditions where explicit index formulas are given.