A necessary condition for lower semicontinuity of line energies View Full Text


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Article Info

DATE

2017-01-05

AUTHORS

Pierre Bochard, Antonin Monteil

ABSTRACT

We are interested in some energy functionals concentrated on the discontinuity lines of divergence-free 2D vector fields valued in the circle S1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^1$$\end{document}. This kind of energy has been introduced first by Aviles and Giga (A mathematical problem related to the physical theory of liquid crystal configurations, 1987). They show in particular that, with the cubic cost function f(t)=t3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(t)=t^3$$\end{document}, this energy is lower semicontinuous. In this paper, we construct a counter-example which excludes the lower semicontinuity of line energies for cost functions of the form tp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^p$$\end{document} with 0 More... »

PAGES

8

References to SciGraph publications

  • 2011-07-08. Entropy method for line-energies in CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
  • 2010-04-20. Lower Bound for the Energy of Bloch Walls in Micromagnetics in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 2000-06-01. Singular Perturbation and the Energy of Folds in JOURNAL OF NONLINEAR SCIENCE
  • 2003-06. Structure of entropy solutions to the eikonal equation in JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
  • 1999-12. Line energies for gradient vector fields in the plane in CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
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    http://scigraph.springernature.com/pub.10.1007/s00526-016-1093-5

    DOI

    http://dx.doi.org/10.1007/s00526-016-1093-5

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    https://app.dimensions.ai/details/publication/pub.1050538640


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