Abstract
We investigate here density questions. For real-valued Sobolev spaces, \(C^\infty (\overline{\varOmega }; \mathbb R)\) is dense in \(W^{s,p}(\varOmega ; \mathbb R)\), for any \(s>0\) and \(1\le p<\infty \). This need not be true for the Sobolev spaces \(W^{s,p}(\varOmega ; {\mathscr {N}})\), where \(\mathscr {N}\) is a manifold. In particular, this is not always the case when \({\mathscr {N}}={\mathbb S}^1\). We present the optimal conditions on s and p so that \(C^{\infty }(\overline{\varOmega }; {\mathbb S}^1)\) is dense in \(W^{s,p}(\varOmega ; {\mathbb S}^1)\).
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Brezis, H., Mironescu, P. (2021). Density. In: Sobolev Maps to the Circle. Progress in Nonlinear Differential Equations and Their Applications, vol 96. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-0716-1512-6_10
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DOI: https://doi.org/10.1007/978-1-0716-1512-6_10
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Publisher Name: Birkhäuser, New York, NY
Print ISBN: 978-1-0716-1510-2
Online ISBN: 978-1-0716-1512-6
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