IJPAM: Volume 13, No. 1 (2004)

PARAMETER-DEPENDENT BOUNDARY VALUE
PROBLEMS ON MANIFOLDS WITH EDGES

A. Oliaro$^1$, B.-W. Schulze$^2$
$^1$Department of Mathematics
University of Turin
Via Carlo Alberto 10, Turin, I-10123, ITALY
e-mail: oliaro@dm.unito.it
$^2$Institute of Mathematics
University of Potsdam
Neue Palais 10, P.O. Box 601553, D-14415, Potsdam, GERMANY
e-mail: schulze@math.uni-potsdam.de
url: https://www.math.uni-potsdam.de/$\sim$shulze


Abstract.As is known from Kondratyev work, boundary value problems for elliptic operators on a manifold with conical singularities and boundary are controlled by a principal symbolic hierachy, where the conormal symbols belong to the typical new components (compared with the smooth case) with interior and boundary symbols. A similar picture may be expected on manifolds with corners when the base of the cone itself is a manifold with conical or edge singularities. This is a natural situation in a number of applications, although with essential new difficulties. We investigate here corresponding conormal symbols in terms of a calculus of holomorphic parameter-dependent edge boundary value problems on the base. We show that a certain kernel cut-off procedure generates all holomorphic families of that kind, modulo smoothing elements, and we establish the algebra of such corner conormal symbols.

Received: February 27, 2004

AMS Subject Classification: 35S15, 35J70, 58J32

Key Words and Phrases: elliptic operator on a manifold, boundary value problem, manifolds with corners, conormal symbols, edge boundary value problems, algebra of corner conormal symbols

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 13
Issue: 1