Injectivity in second-gradient nonlinear elasticity

Daniel Campbell

doi:10.1007/s00526-025-03086-3

Injectivity in second-gradient nonlinear elasticity

2025-09-16

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Sebastian Schwarzacher *

DOI：10.1007/s00526-025-03086-3

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Abstract

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We study injectivity for models of Nonlinear Elasticity that involve the second gradient. We assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb{R}}<^>n$$\end{document} is a domain, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in W<^>{2,q}(\Omega ,{\mathbb{R}}<^>n)$$\end{document} satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|J_f|<^>{-a}\in L<^>1$$\end{document} and that f equals a given homeomorphism on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega$$\end{document}. Under suitable conditions on q and a we show that f must be a homeomorphism. As a main new tool we find an optimal condition for a and q that imply that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}<^>{n-1}(\{J_f=0\})=0$$\end{document} and hence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_f$$\end{document} cannot change sign. We further specify in dependence of q and a the maximal Hausdorff dimension d of the critical set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{J_f=0\}$$\end{document}. The sharpness of our conditions for d is demonstrated by constructing respective counterexamples.

Keywords:

SOBOLEV FUNCTIONS

NEGATIVE POWER

INVERTIBILITY

DEFORMATIONS

REGULARITY

Injectivity in second-gradient nonlinear elasticity

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