1960
DOI: 10.1002/cpa.3160130307
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Local boundary conditions for dissipative symmetric linear differential operators

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Cited by 357 publications
(277 citation statements)
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“…A well-posed IBVP for Einstein's vacuum equations was presented in [13]. This work, which is based on a tetrad formulation, recasts the evolution equations into a first-order symmetric hyperbolic quasilinear form with maximally dissipative boundary conditions [14,15], for which (local in time) well posedness is guaranteed [16]. There has been a substantial effort to obtain wellposed formulations for the more commonly used metric formulations of gravity using similar mathematical techniques (see [3,[17][18][19] for partial results).…”
Section: Introductionmentioning
confidence: 99%
“…A well-posed IBVP for Einstein's vacuum equations was presented in [13]. This work, which is based on a tetrad formulation, recasts the evolution equations into a first-order symmetric hyperbolic quasilinear form with maximally dissipative boundary conditions [14,15], for which (local in time) well posedness is guaranteed [16]. There has been a substantial effort to obtain wellposed formulations for the more commonly used metric formulations of gravity using similar mathematical techniques (see [3,[17][18][19] for partial results).…”
Section: Introductionmentioning
confidence: 99%
“…Remark 1. Essentially the same argument (see Lax and Phillips [3]) shows that weak equals strong for solutions of homogeneous boundary value problems with a boundary condition of the form Pw=0, where P is a bounded operator on L2(T) which commutes with translations. In this case the defining relation for weak solutions takes the form…”
Section: \Fyetpd(-2)mentioning
confidence: 89%
“…In studying 'weak equals strong' for boundary value problems for first-order linear systems of partial differential or pseudodifferential operators ( [1], [3], [4], [5], and others), one of the principal tools has been the use of tangential mollifiers as introduced in [3]. Their success usually depends on the fact that the system of equations can be reduced to an equivalent one of the same type for which the coefficient An of normal differentiation Dn commutes with a standard tangential mollifier.…”
Section: Introductionmentioning
confidence: 99%
“…For An invertible this was proved by Friedrichs [6] and a direct proof valid under the more restrictive constant multiplicity hypothesis was given by Lax and Phillips [9].…”
Section: Theorem 1 the Mapmentioning
confidence: 93%
“…First we suppose that L is symmetric positive, that is A j = A* for all x g 12, and there is a constant a > 0 so that (9) Z(x)s*±*l-^.Aj>aI j for all x G 12. Second, we suppose that N is maximal positive in the sense that (10) (A"(x)v,v) > 0 Vjce38,«eyV(x).…”
Section: Theorem 1 the Mapmentioning
confidence: 99%