An Algebra of Boundary Value Problems Not Requiring Shapiro–Lopatinskij Conditions

Abstract

We construct an algebra of pseudo-differential boundary value problems that contains the classical Shapiro Lopatinskij elliptic problems as well as all differential elliptic problems of Dirac type with APS boundary conditions, together with their parametrices. Global pseudo-differential projections on the boundary are used to define ellipticity and to show the Fredholm property in suitable scales of spaces. Academic PressKey Words: boundary value problems with APS conditions; pseudo-differential operators on manifolds with boundary; ellipticity and parametrix constructions.

Boundary value problems in edge representation

“…Let us also note that when the obstruction (1.24) is non-vanishing, there are pseudo-differential (boundary or edge) calculi with global projection conditions, cf. [32,34], which generalise those of Atiyah, Patodi and Singer [4].…”

“…The operators are realised in weighted edge Sobolev spaces. Ellipticity refers to additional conditions of trace and potential type along the edge which exist when a topological obstruction on the operator vanishes; this is an analogue of [2] for edge problems, see also [32,34]. We show (Theorem 1.3.2) that vanishing of the topological obstruction for the existence of elliptic edge conditions is independent of the weight.…”

Boundary value problems with the transmission property

“…For elliptic operators there is also another kind of ellipticity of boundary conditions, known in special cases, as conditions of Atiyah-Patodi-Singer type ("APS-conditions"), and in general as global projection conditions. While not every elliptic operator on a C ∞ manifold X with boundary admits ShapiroLopatinskij elliptic boundary conditions, there are always global projection conditions (when X is compact), see [21] where both concepts are unified to an operator algebra, containing also Boutet de Monvel's calculus. Let L µ tr (X; E, F ) for E, F ∈ Vect(X) denote the set of all operators A = r For convenience we assume that Y is compact.…”

“…The following theorem was first formulated in the case of differential operators in the paper [2] by Atiyah and Bott, and then for pseudo-differential operators with the transmission property at the boundary in [4] by Boutet de Monvel, cf. also [21]. An analogue for edge operators may be found in [17], cf.…”

Fredholm realizations of elliptic symbols on manifolds with boundary

“…Finally, we also mention [29] where an extension of the transmission algebra is decribed so as to allow arbitrary elliptic operators to be quantized as Fredholm operators. This approach is extended in [30] to remove the transmission condition on the principal symbol and in [31] to allow for a non-trivial fibration at the boundary.…”