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

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.KeywordsDifferential OperatorVector BundleCommutation RelationPseudo Differential OperatorCoerciveness ConditionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.