A ${\mathcal{C}}^k$-Seeley-Extension-Theorem for Bastiani's Differential Calculus

Abstract

We generalize a classical extension result by Seeley in the context of Bastiani's differential calculus to infinite dimensions. The construction follows Seeley's original approach, but is significantly more involved as not only $C^k$-maps (for $k\in \mathbb{N}\cup \{\infty \}$) on (subsets of) half spaces are extended, but also continuous extensions of their differentials to some given piece of boundary of the domains under consideration. A further feature of the generalization is that we construct families of extension operators (instead of only one single extension operator) that fulfill certain compatibility (and continuity) conditions. Various applications are discussed as well.

arXiv comment29 pages. Version as published in Canadian Journal of Mathematics

References in corpus (5)

• Differentiable mappings on products with different degrees of differentiability in the two factors
• Manifolds of mappings for continuum mechanics
• Weighted diffeomorphism groups of Banach spaces and weighted mapping groups
• Strong topologies for spaces of smooth maps with infinite-dimensional target
• Extending Whitney's extension theorem: nonlinear function spaces

Cited by in corpus (4)

• Manifolds of mappings on cartesian products
• Manifolds of continuous BV-functions and vector measure regularity of Banach-Lie groups
• Controllability and diffeomorphism groups on manifolds with boundary
• The Lax Equation and Weak Regularity of Asymptotic Estimate Lie Groups