Parameter dependence for a class of ordinary differential equations with measurable right-hand side

Klose, Daniela; Schuricht, Friedemann

doi:10.1002/mana.200710188

Cited by 2 publications

(3 citation statements)

References 5 publications

(7 reference statements)

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“…Remark 3.5 follows from theorem 3.2 in [19] (see also [27]) and the fact that is C 1 near S (3.5), which in view of (3.24) implies that…”

Section: T/; Nt/; T/; T/mentioning

confidence: 84%

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Euler‐lagrange equation and regularity for flat minimizers of the Willmore functional

Hornung

2010

Comm Pure Appl Math

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Let S R 2 be a bounded domain with boundary of class C 1 , and let g ij D ı ij denote the flat metric on R 2 . Let u be a minimizer of the Willmore functional within a subclass (defined by prescribing boundary conditions on parts of @S) of all W 2;2 isometric immersions of the Riemannian manifold .S; g/ into R 3 . In this article we derive the Euler-Lagrange equation and study the regularity properties for such u. Our main regularity result is that minimizers u are C 3 away from a certain singular set † and C 1 away from a larger singular set † [ † 0 . We obtain a geometric characterization of these singular sets, and we derive the scaling of u and its derivatives near † 0 .Our main motivation to study this problem comes from nonlinear elasticity: On isometric immersions, the Willmore functional agrees with Kirchhoff's energy functional for thin elastic plates.

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“…Remark 3.5 follows from theorem 3.2 in [19] (see also [27]) and the fact that is C 1 near S (3.5), which in view of (3.24) implies that…”

Section: T/; Nt/; T/; T/mentioning

confidence: 84%

“…The quantities P r, P , P , P Y , and P are defined similarly. The existence of P is not covered by Remark 3.5, but it follows from theorem 3.2 in [19] or, more directly, from the fact that the difference quotients 1 " Y OE" converge weakin W 1;1 to the (unique) solution of the linear ODE system…”

Section: Variational Derivative Of the Energy Functionalmentioning

confidence: 95%

Euler‐lagrange equation and regularity for flat minimizers of the Willmore functional

Hornung

2010

Comm Pure Appl Math

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“…For previous work concerning differential equations in finite-dimensional (or Banach) spaces with measurable right hand sides, see e.g. [44], [62], [67], [68] and the references therein.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Measurable regularity properties of infinite-dimensional Lie groups

Glöckner¹

2016

Preprint

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Let G be a Banach-Lie group with Lie algebra g, and p ∈ [1, ∞]. Then the space AC L p ([0, 1], g) of absolutely continuous functions γ :with Lie algebra AC L p ([0, 1], g). We show that each γ ∈ L p ([0, 1], g) has a left evolution Evol(γ) ∈ AC L p ([0, 1], G) 0 , and that the map Evol : L p ([0, 1], g) → AC L p ([0, 1], G) 0 is smooth. Similar results are obtained for important classes of Fréchet-Lie groups and more general Lie groups, notably for diffeomorphism groups of paracompact finitedimensional smooth manifolds and gauge groups of principal bundles with Banach structure groups. The measurable regularity properties considered imply validity of the Trotter product formula and the commutator formula.

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Parameter dependence for a class of ordinary differential equations with measurable right-hand side

Cited by 2 publications

References 5 publications

Euler‐lagrange equation and regularity for flat minimizers of the Willmore functional

Euler‐lagrange equation and regularity for flat minimizers of the Willmore functional

Measurable regularity properties of infinite-dimensional Lie groups

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