On the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundary

Ginoux, Nicolas; Murro, Simone

doi:10.48550/arxiv.2007.02544

Cited by 3 publications

(11 citation statements)

References 62 publications

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“…By setting suitably such a function, we shall show that the resulting Møller operator is actually a unitary map between the spaces of initial data endowed with a naturally defined positive scalar product. Our goal is achieved with the help of [52,66].…”

Section: Møller Operators For Symmetric Weakly-hyperbolic Systemsmentioning

confidence: 99%

“…As shown in [52,Section 7.2], the class of symmetric positive systems includes not only hyperbolic PDEs, but also parabolic PDEs, e.g. any diffusion-reaction system with linear reaction term.…”

Section: Hyperbolic Friedrichs Systems Of Constant Characteristicmentioning

confidence: 99%

“…Roughly speaking, Equation (2.2) provides a sufficient and necessary condition to ensure C kregularity for the solution of the Cauchy problem (2.3) once Cauchy data are given on Σ t 0 . We recall one of the main results of [52], see [52, Theorem 1.2]: Theorem 2.12 (Smooth solutions for hyperbolic Friedrichs systems). Let M be a globally hyperbolic manifold with timelike boundary and let S be a hyperbolic Friedrichs system of constant characteristic.…”

Section: Admissible Boundary Conditionsmentioning

confidence: 99%

“…For the class of globally hyperbolic manifolds with timelike boundary, the Cauchy problem was investigated by two of us. In particular, we showed in [52] that the Cauchy problem for any symmetric hyperbolic system coupled with an admissible boundary condition is well-posed and the unique solution propagates with at most the speed of light. As a byproduct the existence of a causal propagator is guaranteed.…”

Section: Introductionmentioning

confidence: 99%

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Møller operators and Hadamard states for Dirac fields with MIT boundary conditions

Drago¹,

Ginoux²,

Murro³

2021

Preprint

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The aim of this paper is to prove the existence of Hadamard states for Dirac fields coupled with MIT boundary conditions on any globally hyperbolic manifold with timelike boundary. This is achieved by introducing a geometric Møller operator which implements a unitary isomorphism between the spaces of L 2 -initial data of particular symmetric systems we call weakly-hyperbolic and which are coupled with admissible boundary conditions. In particular, we show that for Dirac fields with MIT boundary conditions, this isomorphism can be lifted to a * -isomorphism between the algebras of Dirac fields and that any Hadamard state can be pulled back along this * -isomorphism preserving the singular structure of its two-point distribution.

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Section: Møller Operators For Symmetric Weakly-hyperbolic Systemsmentioning

confidence: 99%

Section: Hyperbolic Friedrichs Systems Of Constant Characteristicmentioning

confidence: 99%

Section: Admissible Boundary Conditionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Møller operators and Hadamard states for Dirac fields with MIT boundary conditions

Drago¹,

Ginoux²,

Murro³

2021

Preprint

Self Cite

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“…[28,33,44,54], the Cauchy problem for the Dirac operator has been investigated only with local boundary conditions, see e.g. [46,47]. On the contrary we are here considering boundary conditions which show non-local features (cf.…”

Section: Introductionmentioning

confidence: 99%

The Cauchy problem of the Lorentzian Dirac operator with APS boundary conditions

Drago¹,

Große²,

Murro³

2021

Preprint

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We consider the classical Dirac operator on globally hyperbolic manifolds with timelike boundary and show well-posedness of the Cauchy initial-boundary value problem coupled to APS-boundary conditions. This is achieved by deriving suitable energy estimates, which play a fundamental role in establishing uniqueness and existence of weak solutions. Finally, by introducing suitable mollifier operators, we study the differentiability of the solutions. For obtaining smoothness we need additional technical conditions.

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Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes

Dappiaggi,

Marta

2021

Preprint

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We consider the Klein-Gordon operator on an n-dimensional asymptotically anti-de Sitter spacetime (M, g) together with arbitrary boundary conditions encoded by a self-adjoint pseudodifferential operator on ∂M of order up to 2. Using techniques from b-calculus and a propagation of singularities theorem, we prove that there exist advanced and retarded fundamental solutions, characterizing in addition their structural and microlocal properties. We apply this result to the problem of constructing Hadamard two-point distributions. These are bi-distributions which are weak bi-solutions of the underlying equations of motion with a prescribed form of their wavefront set and whose anti-symmetric part is proportional to the difference between the advanced and the retarded fundamental solutions. In particular, under a suitable restriction of the class of admissible boundary conditions and setting to zero the mass, we prove their existence extending to the case under scrutiny a deformation argument which is typically used on globally hyperbolic spacetimes with empty boundary.

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On the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundary

Cited by 3 publications

References 62 publications

Møller operators and Hadamard states for Dirac fields with MIT boundary conditions

Møller operators and Hadamard states for Dirac fields with MIT boundary conditions

The Cauchy problem of the Lorentzian Dirac operator with APS boundary conditions

Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes

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