Manifolds of mappings for continuum mechanics

Michor, Peter W.

doi:10.48550/arxiv.1909.00445

Cited by 1 publication

(2 citation statements)

References 63 publications

(122 reference statements)

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“…, (44), and (43). As a consequence, d(x, y(j)) ≤ d(x, x n )+d(x n , y(j)) < 2 −m (recalling (43)) and thus q(γ(y(j)) − γ(x)) ≤ ε if x n ∈ supp(h j ).…”

Section: Proof Of Proposition 114 and Related Resultsmentioning

confidence: 93%

“…We obtain a variant of Dugundji's extension operators by a simple construction, which can dispense with the complicated geometry pervading the work [14]. In the simplified approach, we can establish continuity with respect to the compactopen topology under mild hypotheses: Extension operators also play a role in the approach to manifolds of mappings pursued in [43].…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

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Smoothing operators for vector-valued functions and extension operators

Glöckner¹

2020

Preprint

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Let Ω ⊆ R d be open and ℓ ∈ N0 ∪ {∞}. Given a locally convex topological vector space F , endow C ℓ (Ω, F ) with the compact-open C ℓtopology. For ℓ < ∞, we describe a sequence (Sn) n∈N of continuous linear operators Sn :Moreover, we study the existence of continuous linear right inverses for restriction maps C ℓ (R d , F ) → C ℓ (R, F ), γ → γ|R for subsets R ⊆ R d with dense interior. As an application, we construct continuous linear right inverses for restriction operators between spaces of sections in vector bundles in many situations, and smooth local right inverses for restriction operators between manifolds of mappings. We also obtain smoothing results for sections in fibre bundles.12 As functions f with domain C ℓ (R, F ) are a frequent topic in infinite-dimensional calculus, we now denote the elements of C ℓ (R, F ) by γ (rather than f ).13 Let P := lim ←−C n (R, F ) ⊆ n∈N 0 C n (R, F ) be the standard projective limit. One readily verifies that the linear map φ : C ∞ (R, F ) → P , γ → (γ) n∈N 0 is bijective, continuous, and that φ −1 is continuous.

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“…, (44), and (43). As a consequence, d(x, y(j)) ≤ d(x, x n )+d(x n , y(j)) < 2 −m (recalling (43)) and thus q(γ(y(j)) − γ(x)) ≤ ε if x n ∈ supp(h j ).…”

Section: Proof Of Proposition 114 and Related Resultsmentioning

confidence: 93%

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%