Fractional Sobolev metrics on spaces of immersed curves
Fractional Sobolev metrics on spaces of immersed curves
复制标题
浸没曲线空间上的分数 Sobolev 度量
DOI：
10.1007/s00526-018-1300-7
复制
发表时间：
2017
影响因子：
2.1
通讯作者：
B. Kolev
中科院分区：
文献类型：
--
作者：
Martin Bauer;Martins Bruveris;B. Kolev
Motivated by applications in the field of shape analysis, we study reparametrization invariant, fractional order Sobolev-type metrics on the space of smooth regular curves Imm(S1,Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {Imm}(\mathrm {S}^{1},\mathbb {R}^d)$$\end{document} and on its Sobolev completions Iq(S1,Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {I}}^{q}(\mathrm {S}^{1},{\mathbb {R}}^{d})$$\end{document}. We prove local well-posedness of the geodesic equations both on the Banach manifold Iq(S1,Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {I}}^{q}(\mathrm {S}^{1},{\mathbb {R}}^{d})$$\end{document} and on the Fréchet-manifold Imm(S1,Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {Imm}(\mathrm {S}^{1},\mathbb {R}^d)$$\end{document} provided the order of the metric is greater or equal to one. In addition we show that the Hs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^s$$\end{document}-metric induces a strong Riemannian metric on the Banach manifold Is(S1,Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {I}}^{s}(\mathrm {S}^{1},{\mathbb {R}}^{d})$$\end{document} of the same order s, provided s>32\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s>\frac{3}{2}$$\end{document}. These investigations can be also interpreted as a generalization of the analysis for right invariant metrics on the diffeomorphism group.
