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Linear Overdetermined Systems of Partial Differential Equations. Initial and Initial-Boundary Value Problems

  • Chapter
Partial Differential Equations VIII

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 65))

Abstract

Consider a linear partial differential operator A that maps a vector-valued function y = (y 1 , ...,y m ) into a vector-valued function f = (f 1 ,...,f l ) We assume at first that all the functions, as well as the coefficients of the differential operator, are defined in an open domain Ω in the n-dimensional Euclidean space ℝn, 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′ f = 0 is a necessary condition for the solvability of the system A y = f with an unknown vector-valued function y.

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Authors
  1. P. I. Dudnikov
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  2. S. N. Samborski
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Editors and Affiliations

  1. Department of Mathematics, Northeastern University, 02115, Boston, MA, USA

    M. A. Shubin

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Dudnikov, P.I., Samborski, S.N. (1996). Linear Overdetermined Systems of Partial Differential Equations. Initial and Initial-Boundary Value Problems. In: Shubin, M.A. (eds) Partial Differential Equations VIII. Encyclopaedia of Mathematical Sciences, vol 65. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-48944-0_1

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  • DOI: https://doi.org/10.1007/978-3-642-48944-0_1

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