On the regularity of differential forms satisfying mixed boundary conditions in a class of Lipschitz domains

Jakab, Tünde; Mitrea, Irina; Mitrea, Marius

doi:10.1512/iumj.2009.58.3678

Cited by 28 publications

(42 citation statements)

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“….q C 1/-forms. More precisely, Both latter properties of , that is, the approximation and the compactness property, hold, for example, if the boundary manifolds , t , n are Lipschitz and the boundary manifolds t , n are separated by a .N 2/-dimensional Riemannian and Lipschitz sub-manifold, the interface :D t \ n , see [23,24] for details and proofs. We note that…”

Section: A1 More General Operatorsmentioning

confidence: 99%

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On the Maxwell constants in 3D

Pauly

2014

Math Methods in App Sciences

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Section: A1 More General Operatorsmentioning

confidence: 99%

“…|| || In , it is proved that

D_{normalΓ@ @_{monospacet}}^{q} (@@Ω) \cap {normalΔ}_{normalΓ@ @_{n}}^{q} (@@Ω)

even embeds continuously to

H^{1 / 2, q} (normalΩ)

and hence compactly to L 2, q (Ω). We note that the compactness property is independent of ε , see, for example, .…”

mentioning

confidence: 99%

On the Maxwell constants in 3D

Pauly

2014

Math Methods in App Sciences

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“…for some c > 0 independent of h. Here, we recall,ũ (see (34)) is a good approximation of u h (see (35)). Let us now explain how we will construct the family ψ j j∈{1,...,n} in order to obtain (38) and (39). Then, we explain how the terms Σj ψ j · n e − 2 h f j∈{1,...,n} and ∇ũ, ψ j L 2 w j∈{1,...,n} appearing in (38) and (39) are computed.…”

Section: Sketch Of the Proofs Of Proposition 4 And Theoremmentioning

confidence: 99%

“…with mixed boundary conditions onΩ j , the recent results of [38] and [28] are used. The 1-form ψ j associated with z j is then defined using an eigenform v (1) h,j associated with the eigenvalue 0 of the operator L (1) f,h associated with mixed boundary conditions onΩ j :…”

Section: Sketch Of the Proofs Of Proposition 4 And Theoremmentioning

confidence: 99%

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Exit Event from a Metastable State and Eyring-Kramers Law for the Overdamped Langevin Dynamics

Lelièvre

Peutrec

Nectoux

2019

Springer Proceedings in Mathematics &Amp; Statistics

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In molecular dynamics, several algorithms have been designed over the past few years to accelerate the sampling of the exit event from a metastable domain, that is to say the time spent and the exit point from the domain. Some of them are based on the fact that the exit event from a metastable region is well approximated by a Markov jump process. In this work, we present recent results on the exit event from a metastable region for the overdamped Langevin dynamics obtained in [22,23,57]. These results aim in particular at justifying the use of a Markov jump process parametrized by the Eyring-Kramers law to model the exit event from a metastable region.The objective of this note is to give motivations (Section 1) and outlines of the proofs (Section 2) of results recently obtained in [22,23,57]. These results justify the use of the Eyring-Kramers formulas together with a kinetic Monte Carlo model to model the exit event from a metastable state for the overdamped Langevin dynamics. Such results are particularly useful to justify algorithms and models which use such formulas to build reduced description of the overdamped Langevin dynamics.

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On an Extension of the First Korn Inequality to Incompatible Tensor Fields on Domains of Arbitrary Dimensions

Neff

Pauly

Witsch

2014

Computational Methods in Applied Sciences

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For a bounded domain Ω in R N with Lipschitz boundary Γ = ∂Ω and a relatively open and non-empty 'admissible' subset Γ t of Γ, we prove the existence of a positive constant c such that inequalityholds for all tensor fields T ∈ • H(Curl; Γ t , Ω, R N ×N ), this is, for all square-integrable tensor fields T : Ω → R N ×N having a row-wise square-integrable rotation tensor field Curl T : Ω → R N ×N (N −1)/2 and vanishing row-wise tangential trace on Γ t .For compatible tensor fields T = ∇v with v ∈ H 1 (Ω, R N ) having vanishing tangential Neumann trace on Γ t the inequality (0.1) reduces to a non-standard variant of Korn's first inequality since Curl T = 0, while for skew-symmetric tensor fields T Poincaré's inequality is recovered.If Γ t = ∅, our estimate (0.1) still holds at least for simply connected Ω and for all tensor fields T ∈ H(Curl; Ω, R N ×N ) which are L 2 (Ω, R N ×N )-perpendicular to so(N ), i.e., to all skew-symmetric constant tensors.

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On the regularity of differential forms satisfying mixed boundary conditions in a class of Lipschitz domains

Cited by 28 publications

References 0 publications

On the Maxwell constants in 3D

On the Maxwell constants in 3D

Exit Event from a Metastable State and Eyring-Kramers Law for the Overdamped Langevin Dynamics

On an Extension of the First Korn Inequality to Incompatible Tensor Fields on Domains of Arbitrary Dimensions

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