Differential geometry/Mathematical problems in mechanics
W2,p-estimates for surfaces in terms of their two fundamental forms
Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 85-91.

Let p>2. We show how the fundamental theorem of surface theory for surfaces of class Wloc2,p(ω) over a simply-connected open subset of R2 established in 2005 by S. Mardare can be extended to surfaces of class W2,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 W2,p(ω). Finally, we show that the mapping that defines in this fashion a surface of class W2,p(ω), unique up to proper isometries of E3, in terms of its two fundamental forms is locally Lipschitz-continuous.

Soit p>2. Nous montrons comment le théorème fondamental de la théorie des surfaces de classe Wloc2,p(ω) sur un ouvert simplement connexe ω de R2 établi par S. Mardare in 2005 peut être étendu à des surfaces de classe W2,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 W2,p(ω). Nous établissons enfin que l'application qui définit une surface de classe W2,p(ω) à une isométrie propre de E3 près en fonction de ses deux formes fondamentales est localement lipschitzienne.

Published online:
DOI: 10.1016/j.crma.2017.12.003
Ciarlet, Philippe G. 1; Mardare, Cristinel 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
     author = {Ciarlet, Philippe G. and Mardare, Cristinel},
     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},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2017.12.003/}
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Ciarlet, Philippe G.; Mardare, Cristinel. W2,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. http://www.numdam.org/articles/10.1016/j.crma.2017.12.003/

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

[3] Ciarlet, P.G.; Mardare, S. Nonlinear Korn inequalities in Rn and immersions in W2,p, p>n, considered as functions of their metric tensors in W1,p, J. Math. Pures Appl., Volume 105 (2016), pp. 873-906

[4] P.G. Ciarlet, C. Mardare, A surface in W2,p is a locally Lipschitz-continuous function of its fundamental forms in W1,P and Lp, p>2, in preparation.

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

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

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