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.
| Serija: | Encyclopaedia of Mathematical Sciences |
| Leidėjas: | Springer Berlin Heidelberg |
| Išleidimo metai: | 2012 |
| Knygos puslapių skaičius: | 272 |
| ISBN-10: | 364248946X |
| ISBN-13: | 9783642489464 |
| Formatas: | Knyga minkštu viršeliu |
| Kalba: | Anglų |
| Žanras: | Algebraic geometry |
Parašykite atsiliepimą apie „Partial Differential Equations VIII: Overdetermined Systems Dissipative Singular Schrödinger Operator Index Theory“
