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TheN-membranes problem

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Abstract

We study the equilibrium position ofN elastic membranes attached to rigid supports and submitted to the action of forces. They are constrained because they cannot pass through each other. As in the case of the obstacle problem, the solution fails to beC 2 and thus fails to be classical, so we provide some new regularity results in different larger spaces using an iterative penalization technique.

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References

  1. Agmon S, Douglis A, Nirenberg L (1959) Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions (I). Comm Pure Appl Math 12:623–727

    Google Scholar 

  2. Brezis H, Evans LC (1979) A variational inequality approach to the Bellman—Dirichlet equation for two elliptic operators. Arch Rat Mech Anal 71:1–13

    Google Scholar 

  3. Brezis H, Kinderlehrer D (1974) The smoothness of solutions to nonlinear variational inequalities. Indiana Univ Math J 23:831–844

    Google Scholar 

  4. Brezis H, Stampacchia G (1968) Sur la Regularite de la solution d'inequations elliptiques. Bull Soc Math France 96:152–180

    Google Scholar 

  5. Chipot M (1984) Variational Inequalities and Flow in Porous Media. Applied Mathematical Sciences 52, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo

    Google Scholar 

  6. Gilbarg D, Trudinger NS (1984) Elliptic Partial Differential Equations of Second Order (2nd ed) Springer Verlag, Berlin, Heidelberg, New York, Tokyo

    Google Scholar 

  7. Kinderlehrer D, Stampacchia G (1980) An Introduction to Variational Inequalities and Their Applications. Academic Press, New York

    Google Scholar 

  8. Lions JL (1969) Quelques Methodes de Resolution des Problemes aux Limites non Lineaires. Dunod-Gauthier-Villars, Paris

    Google Scholar 

  9. Lions PL (1981) Une inegalite pour les operateurs elliptiques du second ordre. Anal di Mat pure Appl 127:1–11

    Google Scholar 

  10. Stampacchia G (1965) Equations Elliptiques du Second Ordre a Coefficients Discontinus. Presse de l'universite de Montreal, Montreal

    Google Scholar 

  11. Vergara-Caffarelli G (1971) Regularita di un problema di disequazioni variazionali relativo a due membrane. Rend Acad Lincei 50:659–662

    Google Scholar 

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Authors and Affiliations

  1. Institute for Mathematics and its Applications, 514 Vincent Hall, 206 Church Street SE, 55455, Minneapolis, MN, USA

    M. Chipot & G. Vergara-Caffarelli

Authors
  1. M. Chipot
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  2. G. Vergara-Caffarelli
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Additional information

Communicated by D. Kinderlehrer

On leave from Departement de Mathematiques, Universite de Metz, Ile du Saulcy, 57045 Metz-Cedex, France

On leave from Dipartamento di Matematica, Universita di Pisa, 56100 Pisa, Italy

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Chipot, M., Vergara-Caffarelli, G. TheN-membranes problem. Appl Math Optim 13, 231–249 (1985). https://doi.org/10.1007/BF01442209

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  • DOI: https://doi.org/10.1007/BF01442209

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