Injectivity in second-gradient nonlinear elasticity

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F25%3A10511031" target="_blank" >RIV/00216208:11320/25:10511031 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=mo_3iYHB.J" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=mo_3iYHB.J</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00526-025-03086-3" target="_blank" >10.1007/s00526-025-03086-3</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Injectivity in second-gradient nonlinear elasticity

Popis výsledku v původním jazyce

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}$$Omega subset {mathbb{R}}&lt;^&gt;n$$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}$$fin W&lt;^&gt;{2,q}(Omega ,{mathbb{R}}&lt;^&gt;n)$$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}$$|J_f|&lt;^&gt;{-a}in L&lt;^&gt;1$$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}$$mathcal {H}&lt;^&gt;{n-1}({J_f=0})=0$$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.

Název v anglickém jazyce

Injectivity in second-gradient nonlinear elasticity

Popis výsledku anglicky

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}$$Omega subset {mathbb{R}}&lt;^&gt;n$$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}$$fin W&lt;^&gt;{2,q}(Omega ,{mathbb{R}}&lt;^&gt;n)$$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}$$|J_f|&lt;^&gt;{-a}in L&lt;^&gt;1$$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}$$mathcal {H}&lt;^&gt;{n-1}({J_f=0})=0$$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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2025

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Calculus of Variations and Partial Differential Equations

ISSN

0944-2669

e-ISSN

1432-0835

Svazek periodika

64

Číslo periodika v rámci svazku

8

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

29

Strana od-do

256

Kód UT WoS článku

001572380100003

EID výsledku v databázi Scopus

2-s2.0-105016612526