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An Introduction to Differential Geometry with Applications to Elasticity

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Reference

  • Adams, R.A. (1975): Sobolev Spaces, Academic Press, New York.

    MATH  Google Scholar 

  • Agmon, S., Douglis, A., Nirenberg, L. (1964): Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17, 35–92.

    MathSciNet  MATH  Google Scholar 

  • Akian, J.L. (2003): A simple proof of the ellipticity of Koiter's model, Analysis and Applications 1, 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  • Alexandrescu, O. (1994): Théorème d'existence pour le modèle bidimensionnel de coque non linéaire de W.T. Koiter, C.R. Acad. Sci. Paris, Sér. I, 319, 899–902.

    MathSciNet  MATH  Google Scholar 

  • Amrouche, C., Girault, V. (1994): Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czech. Math. J. 44, 109–140.

    MathSciNet  MATH  Google Scholar 

  • Anicic, S., Le Dret, H., Raoult, A. (2005): The infinitesimal rigid displacement lemma in Lipschitz coordinates and application to shells with minimal regularity, Math. Methods Appl. Sci. 27, 1283–1299.

    Google Scholar 

  • Antman, S.S. (1976): Ordinary differential equations of non-linear elasticity I: Foundations of the theories of non-linearly elastic rods and shells, Arch. Rational Mech. Anal. 61, 307–351.

    MathSciNet  MATH  Google Scholar 

  • Antman, S.S. (1995): Nonlinear Problems of Elasticity, Springer-Verlag, Berlin (Second Edition: 2004).

    MATH  Google Scholar 

  • Ball, J.M. (1977): Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63, 337–403.

    MATH  Google Scholar 

  • Bamberger, Y. (1981): Mécanique de l'Ingénieur, Volume II, Hermann, Paris.

    Google Scholar 

  • Berger, M. (2003): A Panoramic View of Riemannian Geometry, Springer, Berlin.

    MATH  Google Scholar 

  • Berger M., Gostiaux, B. (1987): Géomémtrie Différentielle: Variétés, Courbes et Surfaces, Presses Universitaires de France, Paris.

    Google Scholar 

  • Bernadou, M. (1994): Méthodes d'Eléments Finis pour les Coques Minces, Masson, Paris (English translation: Finite Element Methods for Thin Shell Problems, John Wiley, New York, 1995).

  • Bernadou, M., Ciarlet, P.G. (1976): Sur l'ellipticité du modèle linéaire de coques de W.T. Koiter, in Computing Methods in Applied Sciences and Engineering (R. Glowinski and J.L. Lions, Editors), pp. 89–136, Lecture Notes in Economics and Mathematical Systems, 134, Springer-Verlag, Heidelberg.

  • Bernadou, M., Ciarlet, P.G., Miara, B. (1994): Existence theorems for twodimensional linear shell theories, J. Elasticity 34, 111–138.

    Article  MathSciNet  MATH  Google Scholar 

  • Blouza, A., Le Dret, H. (1999): Existence and uniqueness for the linear Koiter model for shells with little regularity, Quart. Appl. Math. 57, 317–337.

    MathSciNet  MATH  Google Scholar 

  • Bolley, P., Camus, J. (1976): Régularité pour une classe de problèmes aux limites elliptiques dégénérés variationnels, C.R. Acad. Sci. Paris, Sér. A, 282, 45–47.

    MathSciNet  MATH  Google Scholar 

  • Bonnet, O. (1848): Mémoire sur la théorie générale des surfaces, Journal de l'Ecole Polytechnique 19, 1–146.

    Google Scholar 

  • Boothby, W.M. (1975): An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, New York.

    MATH  Google Scholar 

  • Borchers, W., Sohr, H. (1990): On the equations rot v = g and div u = f with zero boundary conditions, Hokkaido Math. J. 19, 67–87.

    MathSciNet  MATH  Google Scholar 

  • Brezis, H. (1983): Analyse Fonctionnelle, Théorie et Applications, Masson, Paris.

  • Caillerie, D., Sanchez-Palencia, E. (1995): Elastic thin shells: asymptotic theory in the anisotropic and heterogeneous cases, Math. Models Methods Appl. Sci. 5, 473–496.

    MathSciNet  MATH  Google Scholar 

  • Cartan, E. (1927): Sur la possibilité de plonger un espace riemannien donné dans un espace euclidien, Annales de la Société Polonaise de Mathématiques 6, 1–7.

    MATH  Google Scholar 

  • do Carmo, M.P. (1976): Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs.

    MATH  Google Scholar 

  • do Carmo, M.P. (1994): Differential Forms and Applications, Universitext, Springer-Verlag, Berlin (English translation of: Formas Diferenciais e Apli¸cões, Instituto da Matematica, Pura e Aplicada, Rio de Janeiro, 1971).

  • Chen, W., Jost, J. (2002): A Riemannian version of Korn's inequality, Calculus of Variations 14, 517–530.

    MathSciNet  MATH  Google Scholar 

  • Choquet-Bruhat, de Witt-Morette and Dillard-Bleick (1977): Analysis, Manifolds and Physics, North-Holland, Amsterdam (Revised Edition: 1982).

    MATH  Google Scholar 

  • Ciarlet, P.G. (1982): Introduction àl'Analyse Numérique Matricielle et àl'Optimisation, Masson, Paris (English translation: Introduction to Numerical Linear Algebra and Optimisation, Cambridge University Press, Cambridge, 1989).

  • Ciarlet, P.G. (1988): Mathematical Elasticity, Volume I: Three-Dimensional Elasticity, North-Holland, Amsterdam.

    MATH  Google Scholar 

  • Ciarlet, P.G. (1993): Modèles bi-dimensionnels de coques: Analyse asymptotique et théorèmes d'existence, in Boundary Value Problems for Partial Differential Equations and Applications (J.L. Lions and C. Baiocchi, Editors), pp. 61–80, Masson, Paris.

  • Ciarlet, P.G. (1997): Mathematical Elasticity, Volume II: Theory of Plates, North-Holland, Amsterdam.

    Google Scholar 

  • Ciarlet, P.G. (2000a): Mathematical Elasticity, Volume III: Theory of Shells, North-Holland, Amsterdam.

    Google Scholar 

  • Ciarlet, P.G. (2000b): Un modèle bi-dimensionnel non linéaire de coque analogue àcelui de W.T. Koiter, C.R. Acad. Sci. Paris, Sér. I, 331, 405–410.

    MathSciNet  MATH  Google Scholar 

  • Ciarlet, P.G. (2003): The continuity of a surface as a function of its two fundamental forms, J. Math. Pures Appl. 82, 253–274.

    MathSciNet  MATH  Google Scholar 

  • Ciarlet, P.G., Destuynder, P. (1979): A justification of the two-dimensional plate model, J. Mécanique 18, 315–344.

    MathSciNet  MATH  Google Scholar 

  • Ciarlet, P.G., Larsonneur, F. (2001): On the recovery of a surface with prescribed first and second fundamental forms, J. Math. Pures Appl. 81, 167–185.

    MathSciNet  Google Scholar 

  • Ciarlet, P.G., Laurent, F. (2003): Continuity of a deformation as a function of its Cauchy-Green tensor, Arch. Rational Mech. Anal. 167, 255–269.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Ciarlet, P.G., Lods, V. (1996a): On the ellipticity of linear membrane shell equations, J. Math. Pures Appl. 75, 107–124.

    MathSciNet  MATH  Google Scholar 

  • Ciarlet, P.G., Lods, V. (1996b): Asymptotic analysis of linearly elastic shells. I. Justification of membrane shell equations, Arch. Rational Mech. Anal. 136, 119–161.

    MathSciNet  ADS  MATH  Google Scholar 

  • Ciarlet, P.G., Lods, V. (1996c): Asymptotic analysis of linearly elastic shells. III. Justification of Koiter's shell equations, Arch. Rational Mech. Anal. 136, 191–200.

    MathSciNet  ADS  MATH  Google Scholar 

  • Ciarlet, P.G., Lods, V. (1996d): Asymptotic analysis of linearly elastic shells: “Generalized membrane shells”, J. Elasticity 43, 147–188.

    Article  MathSciNet  MATH  Google Scholar 

  • Ciarlet, P.G., Lods, V., Miara, B. (1996): Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations, Arch. Rational Mech. Anal. 136, 163–190.

    MathSciNet  ADS  MATH  Google Scholar 

  • Ciarlet, P.G., Mardare, C. (2003): On rigid and infinitesimal rigid displacements in three-dimensional elasticity, Math. Models Methods Appl. Sci. 13, 1589–1598.

    Article  MathSciNet  MATH  Google Scholar 

  • Ciarlet, P.G., Mardare, C. (2004a): On rigid and infinitesimal rigid displacements in shell theory, J. Math. Pures Appl. 83, 1–15.

    MathSciNet  MATH  Google Scholar 

  • Ciarlet, P.G., Mardare, C. (2004b): Recovery of a manifold with boundary and its continuity as a function of its metric tensor, J. Math. Pures Appl. 83, 811–843.

    MathSciNet  MATH  Google Scholar 

  • Ciarlet, P.G., Mardare, C. (2004c): Continuity of a deformation in H1 as a function of its Cauchy-Green tensor in L1, J. Nonlinear Sci. 14, 415–427.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Ciarlet, P.G., Mardare, C. (2005): Recovery of a surface with boundary and its continuity as a function of its two fundamental forms, Analysis and Applications, 3, 99–117.

    Article  MathSciNet  MATH  Google Scholar 

  • Ciarlet, P.G., Mardare, S. (2001): On Korn's inequalities in curvilinear coordinates, Math. Models Methods Appl. Sci. 11, 1379–1391.

    Article  MathSciNet  MATH  Google Scholar 

  • Ciarlet, P.G., Miara, B. (1992): Justification of the two-dimensional equations of a linearly elastic shallow shell, Comm. Pure Appl. Math. 45, 327–360.

    MathSciNet  MATH  Google Scholar 

  • Ciarlet, P.G., Neèas, J. (1987): Injectivity and self-contact in nonlinear elasticity, Arch. Rational Mech. Anal. 19, 171–188.

    ADS  Google Scholar 

  • Ciarlet, P.G., Roquefort, A. (2001): Justification of a two-dimensional nonlinear shell model of Koiter's type, Chinese Ann. Math. 22B, 129–244.

    MathSciNet  Google Scholar 

  • Ciarlet, P.G., Sanchez-Palencia, E. (1996): An existence and uniqueness theorem for the two-dimensional linear membrane shell equations, J. Math. Pures Appl. 75, 51–67.

    MathSciNet  MATH  Google Scholar 

  • Dacorogna (1989): Direct Methods in the Calculus of Variations, Springer, Berlin.

    MATH  Google Scholar 

  • Dautray, R., Lions, J.L. (1984): Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, Tome 1, Masson, Paris.

  • Destuynder, P. (1980): Sur une Justification des Modèles de Plaques et de Coques par les Méthodes Asymptotiques, Doctoral Dissertation, Université Pierre et Marie Curie, Paris.

  • Destuynder, P. (1985): A classification of thin shell theories, Acta Applicandae Mathematicae 4, 15–63.

    MathSciNet  MATH  Google Scholar 

  • Duvaut, G., Lions, J.L. (1972): Les Inéquations en Mécanique et en Physique, Dunod, Paris (English translation: Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976).

  • Flanders, H. (1989): Differential Forms with Applications to the Physical Sciences, Dover, New York.

    MATH  Google Scholar 

  • Friesecke, G., James, R.D., Mora, M.G., Müller, S. (2003): Derivation of nonlinear bending theory for shells from three dimensional nonlinear elasticity by Gamma-convergence, C.R. Acad. Sci. Paris, Sér. I, 336, 697–702.

    MATH  Google Scholar 

  • Friesecke, G., James, R.D., Müller, S. (2002): A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity, Comm. Pure Appl. Math. 55, 1461–1506.

    Article  MathSciNet  MATH  Google Scholar 

  • Gallot, S., Hulin, D., Lafontaine, J. (2004): Riemannian Geometry, Third Edition, Springer, Berlin.

    MATH  Google Scholar 

  • Gauss, C.F. (1828): Disquisitiones generales circas superficies curvas, Commentationes societatis regiae scientiarum Gottingensis recentiores 6, Göttingen.

  • Germain, P. (1972): Mécanique des Milieux Continus, Tome I, Masson, Paris.

    Google Scholar 

  • Geymonat, G. (1965): Sui problemi ai limiti per i sistemi lineari ellittici, Ann. Mat. Pura Appl. 69, 207–284.

    MathSciNet  MATH  Google Scholar 

  • Geymonat, G., Gilardi, G. (1998): Contre-exemples à l'inégalité de Korn et au lemme de Lions dans des domaines irréguliers, in Equations aux Dérivées Partielles et Applications. Articles Dédiés à Jacques-Louis Lions, pp. 541–548, Gauthier-Villars, Paris.

  • Geymonat, G., Suquet, P. (1986): Functional spaces for Norton-Hoff materials, Math. Methods Appl. Sci. 8, 206–222.

    MathSciNet  MATH  ADS  Google Scholar 

  • Goldenveizer, A.L. (1963): Derivation of an approximate theory of shells by means of asymptotic integration of the equations of the theory of elasticity, Prikl. Mat. Mech. 27, 593–608.

    MathSciNet  MATH  Google Scholar 

  • Gurtin, M.E. (1981): Topics in Finite Elasticity, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia.

  • Hartman, P., Wintner, A. (1950): On the embedding problem in differential geometry, Am. J. Math. 72, 553–564.

    MathSciNet  MATH  Google Scholar 

  • Jacobowitz, H. (1982): The Gauss-Codazzi equations, Tensor, N.S. 39, 15–22.

  • Janet, M. (1926): Sur la possibilité de plonger un espace riemannien donné dans un espace euclidien, Annales de la Société Polonaise de Mathématiques 5, 38–43.

    Google Scholar 

  • John, F. (1961): Rotation and strain, Comm. Pure Appl. Math. 14, 391–413.

    MathSciNet  MATH  Google Scholar 

  • John, F. (1965): Estimates for the derivatives of the stresses in a thin shell and interior shell equations, Comm. Pure Appl. Math. 18, 235–267.

    MathSciNet  Google Scholar 

  • John, F. (1971): Refined interior equations for thin elastic shells, Comm. Pure Appl. Math. 18, 235–267.

    Google Scholar 

  • John, F. (1972): Bounds for deformations in terms of average strains, in Inequalities III (O. Shisha, Editor), pp. 129–144, Academic Press, New York.

  • Klingenberg, W. (1973): Eine Vorlesung über Differentialgeometrie, Springer-Verlag, Berlin (English translation: A Course in Differential Geometry, Springer-Verlag, Berlin, 1978).

  • Kohn, R.V. (1982): New integral estimates for deformations in terms of their nonlinear strains, Arch. Rational Mech. Anal. 78, 131–172.

    Article  MathSciNet  MATH  ADS  Google Scholar 

  • Koiter, W.T. (1966): On the nonlinear theory of thin elastic shells, Proc. Kon. Ned. Akad. Wetensch. B69, 1–54.

    MathSciNet  Google Scholar 

  • Koiter, W.T. (1970): On the foundations of the linear theory of thin elastic shells, Proc. Kon. Ned. Akad. Wetensch. B73, 169–195.

    MathSciNet  Google Scholar 

  • Kühnel, W. (2002): Differentialgeometrie, Fried. Vieweg and Sohn, Wiesbaden (English translation: Differential Geometry: Curves-Surfaces-Manifolds, American Mathematical Society, Providence, 2002).

  • Lebedev, L.P., Cloud, M.J. (2003): Tensor Analysis, World Scientific, Singapore.

    MATH  Google Scholar 

  • Le Dret, H. (2004): Well-posedness for Koiter and Naghdi shells with a G1-midsurface, Analysis and Applications 2, 365–388.

    MathSciNet  MATH  Google Scholar 

  • Le Dret, H., Raoult, A. (1996): The membrane shell model in nonlinear elasticity: A variational asymptotic derivation, J. Nonlinear Sci. 6, 59–84.

    MathSciNet  MATH  ADS  Google Scholar 

  • Lions, J.L. (1961): Equations Différentielles Opérationnelles et Problèmes aux Limites, Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • Lions, J.L. (1969): Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris.

    MATH  Google Scholar 

  • Lions, J.L. (1973): Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal, Lecture Notes in Mathematics, Vol. 323, Springer-Verlag, Berlin.

  • Lions, J.L., Magenes, E. (1968): Problèmes aux Limites Non Homogènes et Applications, Vol. 1, Dunod, Paris (English translation: Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, Berlin, 1972).

  • Lods, V. and Miara, B. (1998): Nonlinearly elastic shell models. II. The flexural model, Arch. Rational Mech. Anal. 142, 355–374.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Magenes, E., Stampacchia, G. (1958): I problemi al contorno per le equazioni differenziali di tipo ellittico, Ann. Scuola Norm. Sup. Pisa 12, 247–358.

    MathSciNet  MATH  Google Scholar 

  • Mardare, C. (2003): On the recovery of a manifold with prescribed metric tensor, Analysis and Applications 1, 433–453.

    Article  MathSciNet  MATH  Google Scholar 

  • Mardare, S. (2003a): Inequality of Korn's type on compact surfaces without boundary, Chinese Annals Math. 24B, 191–204

    MathSciNet  Google Scholar 

  • Mardare, S. (2003b): The fundamental theorem of surface theory for surfaces with little regularity, J. Elasticity 73, 251–290.

    Article  MathSciNet  MATH  Google Scholar 

  • Mardare, S. (2004): On isometric immersions of a Riemannian space with little regularity, Analysis and Applications 2, 193–226.

    Article  MathSciNet  MATH  Google Scholar 

  • Mardare, S. (2005): On Pfaff systems with Lp coefficients and their applications in differential geometry, J. Math. Pures Appl., to appear.

  • Marsden, J.E., Hughes, T.J.R. (1983): Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs (Second Edition: 1999).

    MATH  Google Scholar 

  • Miara, B. (1998): Nonlinearly elastic shell models. I. The membrane model, Arch. Rational Mech. Anal. 142, 331–353.

    Article  MathSciNet  MATH  ADS  Google Scholar 

  • Miara, B., Sanchez-Palencia, E. (1996): Asymptotic analysis of linearly elastic shells, Asymptotic Anal. 12, 41–54.

    MathSciNet  MATH  Google Scholar 

  • Morrey, Jr., C.B. (1952): Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2, 25–53.

    MathSciNet  MATH  Google Scholar 

  • Naghdi, P.M. (1972): The theory of shells and plates, in Handbuch der Physik, Vol. VIa/2 (S. Flügge and C. Truesdell, Editors), pp. 425–640, Springer-Verlag, Berlin.

  • Nash, J. (1954): C 1 isometric imbeddings, Annals of Mathematics, 60, 383–396.

    MathSciNet  MATH  Google Scholar 

  • Neèas, J. (1967): Les Méthodes Directes en Théorie des Equations Elliptiques, Masson, Paris.

  • Nirenberg, L. (1974): Topics in Nonlinear Functional Analysis, Lecture Notes, Courant Institute, New York University (Second Edition: American Mathematical Society, Providence, 1994).

  • Reshetnyak, Y.G. (1967): Liouville's theory on conformal mappings under minimal regularity assumptions, Sibirskii Math. J. 8, 69–85.

    MATH  Google Scholar 

  • Reshetnyak, Y.G. (2003): Mappings of domains in Rn and their metric tensors, Siberian Math. J. 44, 332–345.

    Article  MathSciNet  MATH  Google Scholar 

  • Sanchez-Palencia, E. (1989a): Statique et dynamique des coques minces. I. Cas de flexion pure non inhibée, C.R. Acad. Sci. Paris, Sér. I, 309, 411–417.

    MathSciNet  MATH  Google Scholar 

  • Sanchez-Palencia, E. (1989b): Statique et dynamique des coques minces. II. Cas de flexion pure inhibée - Approximation membranaire, C.R. Acad. Sci. Paris, Sér. I, 309, 531–537.

    MathSciNet  MATH  Google Scholar 

  • Sanchez-Palencia, E. (1990): Passages à la limite de l'élasticité tri-dimensionnelle à la théorie asymptotique des coques minces, C.R. Acad. Sci. Paris, Sér. II, 311, 909–916.

    MathSciNet  MATH  Google Scholar 

  • Sanchez-Palencia, E. (1992): Asymptotic and spectral properties of a class of singular-stiff problems, J. Math. Pures Appl. 71, 379–406.

    MathSciNet  MATH  Google Scholar 

  • Sanchez-Hubert, J., Sanchez-Palencia, E. (1997): Coques Elastiques Minces: Propriétés Asymptotiques, Masson, Paris.

    MATH  Google Scholar 

  • Schwartz, L. (1966): Théorie des Distributions, Hermann, Paris.

    MATH  Google Scholar 

  • Schwartz, L. (1992): Analyse II: Calcul Différentiel et Equations Différentielles, Hermann, Paris.

    Google Scholar 

  • Simmonds, J.G. (1994): A Brief on Tensor Analysis, Second Edition, Springer-Verlag, Berlin.

    Google Scholar 

  • Slicaru, S. (1998): Quelques Résultats dans la Théorie des Coques Linéairement Elastiques à Surface Moyenne Uniformément Elliptique ou Compacte Sans Bord, Doctoral Dissertation, Université Pierre et Marie Curie, Paris.

  • Spivak, M. (1999): A Comprehensive Introduction to Differential Geometry, Volumes I to V, Third Edition, Publish or Perish, Berkeley.

  • Stein, E. (1970): Singular Integrals and Differentiability Properties of Functions, Princeton University Press.

  • Stoker, J.J. (1969): Differential Geometry, John Wiley, New York.

    MATH  Google Scholar 

  • Szczarba, R.H. (1970): On isometric immersions of Riemannian manifolds in Euclidean space, Boletim da Sociedade Brasileira de Matemática 1, 31–45.

    MathSciNet  MATH  Google Scholar 

  • Szopos, M. (2005): On the recovery and continuity of a submanifold with boundary, Analysis and Applications 3, 119–143.

    Article  MathSciNet  MATH  Google Scholar 

  • Tartar, L. (1978): Topics in Nonlinear Analysis, Publications Mathématiques d'Orsay No. 78.13, Université de Paris-Sud, Orsay.

  • Tenenblat, K. (1971): On isometric immersions of Riemannian manifolds, Boletim da Sociedade Brasileira de Matemática 2, 23–36.

    MathSciNet  MATH  Google Scholar 

  • Truesdell, C., Noll, W. (1965): The non-linear field theories of mechanics, in Handbuch der Physik, Vol. III/3 (S. Flügge, Editor), pp. 1–602, Springer-Verlag, Berlin.

  • Valent, T. (1988): Boundary Value Problems of Finite Elasticity, Springer Tracts in Natural Philosophy, Vol. 31, Springer-Verlag, Berlin.

  • Wang, C.C., Truesdell, C. (1973): Introduction to Rational Elasticity, Noordhoff, Groningen.

    MATH  Google Scholar 

  • Whitney, H. (1934): Analytic extensions of differentiable functions defined in closed sets, Trans. Am. Math. Soc. 36, 63–89.

    MathSciNet  MATH  Google Scholar 

  • Yosida, K. (1966): Functional Analysis, Springer-Verlag, Berlin.

    Google Scholar 

  • Zeidler, E. (1986): Nonlinear Functional Analysis and its Applications, Vol. I: Fixed-Point Theorems, Springer-Verlag, Berlin.

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    Philippe G. Ciarlet

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Ciarlet, P.G. An Introduction to Differential Geometry with Applications to Elasticity. J Elasticity 78, 1–215 (2005). https://doi.org/10.1007/s10659-005-4738-8

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