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# Partial Differential Equations VIII: Overdetermined Systems Dissipative Singular Schroedinger Operator Index Theory - Encyclopaedia of Mathematical Sciences 65 (Paperback)

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Paperback 261 Pages / Published: 14/04/2012
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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.

Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
ISBN: 9783642489464
Number of pages: 261
Weight: 416 g
Dimensions: 235 x 155 x 14 mm
Edition: Softcover reprint of the original 1st ed. 199

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