W2,p-estimates for surfaces in terms of their two fundamental forms

Philippe G. Ciarlet; Cristinel Mardare

doi:10.1016/j.crma.2017.12.003

Differential geometry/Mathematical problems in mechanics

W^2,p-estimates for surfaces in terms of their two fundamental forms
[Estimations dans W^2,p pour des surfaces à partir de leurs deux formes fondamentales]

Présenté par : Philippe G. Ciarlet

Philippe G. Ciarlet ¹ ; Cristinel Mardare ²

¹ Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
² Sorbonne Universités, Université Pierre-et-Marie-Curie, Laboratoire Jacques-Louis-Lions, 75005 Paris, France

Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 85-91.

Résumé
Abstract

Soit $p > 2$ . Nous montrons comment le théorème fondamental de la théorie des surfaces de classe $W_{loc}^{2, p} (ω)$ sur un ouvert simplement connexe ω de $R^{2}$ établi par S. Mardare in 2005 peut être étendu à des surfaces de classe $W^{2, p} (ω)$ lorsque ω est de plus borné et de frontière lipschitzienne. Ensuite, nous établissons une inégalité de Korn non linéaire pour des surfaces de classe $W^{2, p} (ω)$ . Nous établissons enfin que l'application qui définit une surface de classe $W^{2, p} (ω)$ à une isométrie propre de $E^{3}$ près en fonction de ses deux formes fondamentales est localement lipschitzienne.

Let $p > 2$ . We show how the fundamental theorem of surface theory for surfaces of class $W_{loc}^{2, p} (ω)$ over a simply-connected open subset of $R^{2}$ established in 2005 by S. Mardare can be extended to surfaces of class $W^{2, p} (ω)$ when ω is in addition bounded and has a Lipschitz-continuous boundary. Then we establish a nonlinear Korn inequality for surfaces of class $W^{2, p} (ω)$ . Finally, we show that the mapping that defines in this fashion a surface of class $W^{2, p} (ω)$ , unique up to proper isometries of $E^{3}$ , in terms of its two fundamental forms is locally Lipschitz-continuous.

Reçu le : 2017-12-07
Accepté le : 2017-12-07
Publié le : 2017-12-19

DOI : 10.1016/j.crma.2017.12.003

Affiliations des auteurs :

Philippe G. Ciarlet ¹ ; Cristinel Mardare ²

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     author = {Philippe G. Ciarlet and Cristinel Mardare},
     title = {\protect\emph{W}\protect\textsuperscript{2,\protect\emph{p}}-estimates for surfaces in terms of their two fundamental forms},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {85--91},
     publisher = {Elsevier},
     volume = {356},
     number = {1},
     year = {2018},
     doi = {10.1016/j.crma.2017.12.003},
     language = {en},
}

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Philippe G. Ciarlet; Cristinel Mardare. W^2,p-estimates for surfaces in terms of their two fundamental forms. Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 85-91. doi : 10.1016/j.crma.2017.12.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.12.003/

Bibliographie
Cité par

[1] M.P. do Carmo Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, 1976

[2] P.G. Ciarlet Linear and Nonlinear Functional Analysis with Applications, SIAM, Philadelphia, 2013

[3] P.G. Ciarlet; S. Mardare Nonlinear Korn inequalities in $R^{n}$ and immersions in $W^{2, p}$ , $p > n$ , considered as functions of their metric tensors in $W^{1, p}$ , J. Math. Pures Appl., Volume 105 (2016), pp. 873-906

[4] P.G. Ciarlet, C. Mardare, A surface in $W^{2, p}$ is a locally Lipschitz-continuous function of its fundamental forms in $W^{1, P}$ and $L^{p}$ , $p > 2$ , in preparation.

[5] W. Klingenberg Eine Vorlesung über Differentialgeometrie, Springer, Berlin, 1973 (English translation: A Course in Differential Geometry, Springer, Berlin, 1978)

[6] S. Mardare On Pfaff systems with $L^{p}$ coefficients and their applications in differential geometry, J. Math. Pures Appl., Volume 84 (2005), pp. 1659-1692

[7] S. Mardare On systems of first order linear partial differential equations with $L^{p}$ coefficients, Adv. Differ. Equ., Volume 12 (2007), pp. 301-360

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