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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Differential equations
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Crack Theory and Edge Singularities (Paperback, Softcover reprint of hardcover 1st ed. 2003)
Loot Price: R2,993
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Crack Theory and Edge Singularities (Paperback, Softcover reprint of hardcover 1st ed. 2003)
Series: Mathematics and Its Applications, 561
Expected to ship within 10 - 15 working days
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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."
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