Continuity of a surface in Fréchet spaces

Philippe G. Ciarlet; Maria Malin; Cristinel Mardare

doi:10.1016/j.crma.2019.10.010

Mathematical problems in mechanics/Differential geometry

Continuity of a surface in Fréchet spaces

Presented by: Philippe G. Ciarlet

Philippe G. Ciarlet ¹; Maria Malin ²; Cristinel Mardare ¹

¹ Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong
² Department of Mathematics, University of Craiova, Craiova, 200585, Romania

Comptes Rendus. Mathématique, Volume 357 (2019) no. 11-12, pp. 917-921.

Abstract
Résumé

We establish the continuity of a surface as a function of its first two fundamental forms for several Fréchet topologies, which include in particular those of the space $W_{loc}^{1, p}$ for the first fundamental form and of the space $L_{loc}^{p}$ for the second fundamental form, for any $p > 2$ .

On établit la continuité d'une surface en fonction de ses deux premières formes fondamentales pour plusieurs topologies de Fréchet, qui incluent en particulier celles de l'espace $W_{loc}^{1, p}$ pour la première forme et de l'espace $L_{loc}^{p}$ pour la deuxième forme, où $p > 2$ .

Received: 2019-10-18
Accepted: 2019-10-18
Published online: 2019-11-01

DOI: 10.1016/j.crma.2019.10.010

Author's affiliations:

Philippe G. Ciarlet ¹; Maria Malin ²; Cristinel Mardare ¹

¹ Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong
² Department of Mathematics, University of Craiova, Craiova, 200585, Romania

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     title = {Continuity of a surface in {Fr\'echet} spaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {917--921},
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Philippe G. Ciarlet; Maria Malin; Cristinel Mardare. Continuity of a surface in Fréchet spaces. Comptes Rendus. Mathématique, Volume 357 (2019) no. 11-12, pp. 917-921. doi : 10.1016/j.crma.2019.10.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.10.010/

References
Cited by

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[2] P.G. Ciarlet Continuity of a surface as a function of its two fundamental forms, J. Math. Pures Appl., Volume 82 (2002), pp. 253-274

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

[4] P.G. Ciarlet; C. Mardare Recovery of a surface with boundary and its continuity as a function of its two fundamental forms, Anal. Appl., Volume 3 (2005), pp. 99-117

[5] 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

[6] 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 $L^{p}$ , $p > 2$ , J. Math. Pures Appl., Volume 124 (2019), pp. 300-318

[7] P.G. Ciarlet, M. Malin, C. Mardare, Continuity in Fréchet topologies of a surface as a function of its fundamental forms, in preparation.

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[9] W. Kühnel Differential Geometry: Curves – Surfaces – Manifolds, American Mathematical Society, Providence, 2002

[10] 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

[11] W. Rudin Functional Analysis, McGraw-Hill, New York, 1973

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