Partial Differential Equations VIII: Overdetermined Systems Dissipative Singular Schrödinger Operator Index Theory (Encyclopaedia of Mathematical Sciences)

ISBN 10: 364248946X / ISBN 13: 9783642489464

Editore: Springer, 2012

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• Titolo: Partial Differential Equations VIII: Overdetermined Systems Dissipative Singular Schrödinger Operator Index Theory (Encyclopaedia of Mathematical Sciences)
• Editore: Springer
• Data di pubblicazione: 2012
• Lingua: Inglese
• ISBN 10: 364248946X
• ISBN 13: 9783642489464
• Rilegatura: Brossura
• Condizione: New

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Riassunto

Consider a linear partial differential operator A that maps a vector-valued function Y = (Yl,"" Ym) into a vector-valued function I = (h,···, II). We assume at first that all the functions, as well as the coefficients of the differen­ tial operator, are defined in an open domain Jl in the n-dimensional Euclidean n space IR , and that they are smooth (infinitely differentiable). A is called an overdetermined operator if there is a non-zero differential operator A' such that the composition A' A is the zero operator (and underdetermined if there is a non-zero operator A" such that AA" = 0). If A is overdetermined, then A'I = 0 is a necessary condition for the solvability of the system Ay = I with an unknown vector-valued function y. 3 A simple example in 1R is the operator grad, which maps a scalar func­ tion Y into the vector-valued function (8y/8x!, 8y/8x2, 8y/8x3)' A necessary solvability condition for the system grad y = I has the form curl I = O.

Dalla quarta di copertina

This volume of the EMS contains three articles, on linear overdetermined systems of partial differential equations, dissipative Schroedinger operators, and index theorems. Each article presents a comprehensive survey of its subject, discussing fundamental results such as the construction of compatibility operators and complexes for elliptic, parabolic and hyperbolic coercive problems, the method of functional models and the Atiyah-Singer index theorem and its generalisations. Both classical and recent results are explained in detail and illustrated by means of examples.