TY - JOUR
T1 - V-shaped fronts around an obstacle
AU - Guo, Hongjun
AU - Monobe, Harunori
N1 - Funding Information:
Research partially supported by National Science Foundation (grant no. DMS-1514752). The authors are grateful to the anonymous referees for interesting comments which led to an improvement of the article. 1 The obstacle K is called star-shaped if either K = ∅ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K=\emptyset $$\end{document} or there is x in the interior Int ( K ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {Int}(K)$$\end{document} of K such that x + t ( y - x ) ∈ Int ( K ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x+t(y-x)\in \mathrm {Int}(K)$$\end{document} for all y ∈ ∂ K \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y\in \partial K$$\end{document} and t ∈ [ 0 , 1 ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in [0,1)$$\end{document} . 2 The obstacle K is called directionally convex with respect to a hyperplane H = { x ∈ R N : x · e = a } \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=\{x\in {\mathbb {R}}^N: x\cdot e=a\}$$\end{document} , with e ∈ S N - 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e\in \mathbb {S}^{N-1}$$\end{document} and a ∈ R \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\in {\mathbb {R}}$$\end{document} , if for every line Σ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varSigma $$\end{document} parallel to e , the set K ∩ Σ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\cap \varSigma $$\end{document} is either a single line segment or empty and if K ∩ H \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\cap H$$\end{document} is equal to the orthogonal projection of K onto H .
Funding Information:
Research partially supported by National Science Foundation (grant no. DMS-1514752). The authors are grateful to the anonymous referees for interesting comments which led to an improvement of the article.
Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2021/2
Y1 - 2021/2
N2 - In this paper, we investigate V-shaped fronts around an obstacle K. We first prove that there exist solutions emanating from any homogeneous transition front including V-shaped front for exterior domains Ω= RN\ K. By providing the complete propagation of the V-shaped front, we prove that the V-shaped front can recover after passing the obstacle.
AB - In this paper, we investigate V-shaped fronts around an obstacle K. We first prove that there exist solutions emanating from any homogeneous transition front including V-shaped front for exterior domains Ω= RN\ K. By providing the complete propagation of the V-shaped front, we prove that the V-shaped front can recover after passing the obstacle.
UR - http://www.scopus.com/inward/record.url?scp=85076841612&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85076841612&partnerID=8YFLogxK
U2 - 10.1007/s00208-019-01944-y
DO - 10.1007/s00208-019-01944-y
M3 - Article
AN - SCOPUS:85076841612
SN - 0025-5831
VL - 379
SP - 661
EP - 689
JO - Mathematische Annalen
JF - Mathematische Annalen
IS - 1-2
ER -