Boundary value problems for partial differential equations playa crucial
role in many areas of physics and the applied sciences. Interesting
phenomena are often connected with geometric singularities, for
instance, in mechanics. Elliptic operators in corresponding models are
then sin- gular or degenerate in a typical way. The necessary structures
for constructing solutions belong to a particularly beautiful and
ambitious part of the analysis. Cracks in a medium are described by
hypersurfaces with a boundary. Config- urations of that kind belong to
the category of spaces (manifolds) with geometric singularities, here
with edges. In recent years the analysis on such (in general,
stratified) spaces has become a mathematical structure theory with many
deep relations with geometry, topology, and mathematical physics. Key
words in this connection are operator algebras, index theory,
quantisation, and asymptotic analysis. Motivated by Lame's system with
two-sided boundary conditions on a crack we ask the structure of
solutions in weighted edge Sobolov spaces and subspaces with discrete
and continuous asymptotics. Answers are given for elliptic sys- tems in
general. We construct parametrices of corresponding edge boundary value
problems and obtain elliptic regularity in the respective scales of
weighted spaces. The original elliptic operators as well as their
parametrices belong to a block matrix algebra of pseudo-differential
edge problems with boundary and edge conditions, satisfying analogues of
the Shapiro-Lopatinskij condition from standard boundary value problems.
Operators are controlled by a hierarchy of principal symbols with
interior, boundary, and edge components.
