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Regularity for Variational Inequalities — A Survey of Results

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From Convexity to Nonconvexity

Part of the book series: Nonconvex Optimization and Its Applications ((NOIA,volume 55))

Abstract

We give an overview of the methods to prove regularity results for variational inequalities emphasizing application to elasticity.

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References

  1. Archipova, A.A., and N.N. Uralzeva: On the regularity of the solution of a problem with two-sided obstacles on the boundary (Russian). Vestnik Leningr. Univ., Ser. 1, 1986, No. 1, 3–10.

    Google Scholar 

  2. Baiocchi, C., and A. Capelo: Variational and quasivariational inequalities. Wiley 1984.

    Google Scholar 

  3. Biroli, M.: De Giorgi-Nash-Moser results for a variational inequality. Boll. UMI (5)16-A(1979), 598–605.

    MathSciNet  Google Scholar 

  4. Brézis, H.: Problèmes unilateraux. J. Math. Pures et appl. 51(1972), 1–168.

    Google Scholar 

  5. Brézis, H., and G. Stampacchia.: Sur la régularité de la solutions d’inéquations elliptiques. Bull. Soc. Math. France. 96(1968),153–180.

    MathSciNet  MATH  Google Scholar 

  6. Ciarlet, P.: Mathematical elasticity. North Holland, Amsterdam 1983.

    Google Scholar 

  7. Duvaut, G., and J.L. Lions: Les inéquations en mathématique et en physique. Dunod, Paris 1972.

    Google Scholar 

  8. Fichera, G: Problemi elastostatici con vincoli unilaterali: il problema di Sig-norini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei Mem. Cl. Fis. Mat. Nat. Sez. Ia 7(8)VII (1963/64), 91–140.

    Google Scholar 

  9. Fichera, G.: Existence theorems in elasticity.

    Google Scholar 

  10. Boundary value problems of elasticity with unilateral constraints. In: Handhuch der Physik (S. Fliigge, ed.), Vol. 6a/2, Springer-Verlag, Berlin 1972.

    Google Scholar 

  11. Frehse, J.: On Signorini’s problem and variational problems with thin obstacles. Ann. Sc. Norm. Sup. Pisa 4(1977), 343–362.

    MathSciNet  MATH  Google Scholar 

  12. Frehse, J.: On the smoothness of variational inequalities with obstacles. In: Partial differential equations, p. 87–128. Banach Centre Publications. Warsaw 1983.

    Google Scholar 

  13. Frehse, J. and U. Mosco: Variational inequalities with one sided irregular obstacles. Manuscripta math. 28(1979), 219–234.

    Article  MathSciNet  MATH  Google Scholar 

  14. Friedman, A.: Variational principles and free-boundary problems. Wiley 1982.

    MATH  Google Scholar 

  15. Fuchs, M.: Hölder continuity of the gradient for degenerate variational inequalities. Nonlin. Anal. 15(1990), 85–100.

    Article  MATH  Google Scholar 

  16. Gerhard, C.: Regularity of solutions of nonlinear variational inequalities. Arch. Rat. Mech. Anal. 52(1973), 389–393.

    Article  Google Scholar 

  17. Giaquinta, M.: Remarks on the regularity of weak solutions of some variational inequalities. Math.Z. 177(1981), 15–31.

    Article  MathSciNet  MATH  Google Scholar 

  18. Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems. Princeton University Press. Princeton 1983.

    MATH  Google Scholar 

  19. Hartman, P., and G. Stampacchia: On some non-linear elliptic differential functional equations. Acta Mat. 115(1966), 271–310.

    Article  MathSciNet  MATH  Google Scholar 

  20. Hildebrandt, S., and K.-O. Widman: Variational inequalities for vector-valued functions. J. reine und angew. Mathematik 309(1979), 181–220.

    MathSciNet  Google Scholar 

  21. Hörmander, L.: The analysis of linear partial differential operators, Vol. 3. Springer-Verlag New York, Berlin 1985.

    Google Scholar 

  22. Jentsch, L.: Über Wärmespannungen in Körpern mit stückweise konstanten Laméschen Elastizitätsmoduln. Schriftenreihe des Zentralinstitus für Mathematik und Mechanik. Akademieverlag, Berlin 1972.

    Google Scholar 

  23. Kinderlehrer, D., and G. Stampacchia: An introduction to variational inequalities and their applications. Academic Press, New York 1980.

    MATH  Google Scholar 

  24. Kinderlehrer, D.: The smothness of the solution of the boundary obstacle problem. J. Math. Pures Appl. 60(1981), 193–212.

    MathSciNet  MATH  Google Scholar 

  25. Kinderlehrer, D.: Remarks about Signorini’s problem in linear elasticity. Ann. Sc. Norm. Sup. Pisa (IV) 8(1981), 605–645.

    MathSciNet  MATH  Google Scholar 

  26. Kupradze, V.D.: Three dimensional mathematical theory of elasticity and thermoelasticity. Mir, Moscow 1976.

    Google Scholar 

  27. Lions, J.L.: Quelques méthodes de résolution des рroЬl èmes aux limites non linéaires. Dunod, Paris 1969.

    Google Scholar 

  28. Lions, J.L., and Stampacchia, G.: Variational inequalities. Comm. Pure Appl. Math. 20(1967). 439–519

    MathSciNet  Google Scholar 

  29. Nećas, J. and I. Hlavaćek: Mathematical theory of elastic and elastoplastic bodies. SNTL Prague and Elsevier, Amsterdam 1981.

    Google Scholar 

  30. Naniewicz, Z. and P.D. Panagiotopoulos: Mathematical theory of hemivariational inequalities and applications. Marcel Dekker, New York 1995.

    Google Scholar 

  31. Richardson, D.: The regularity of the solution of a variational inequality. Report No. 5, 1977, Institut Mittag-Leffler

    Google Scholar 

  32. Rodrigues, J.F.: Obstacle problems in mathematical physics. North Holland, Amsterdam 1987.

    MATH  Google Scholar 

  33. Schumann, R.: Regularity for Signorini’s problem in linear elasticity, Manuscripta math. 63(1989), 255–291.

    Article  MathSciNet  MATH  Google Scholar 

  34. Schumann, R.: A remark on a boundary contact problem in linear elasticity. Manuscripta math. 63(1989), 455–468.

    Article  MathSciNet  MATH  Google Scholar 

  35. Schumann, R.: Regularity for a variational inequality with a pseudodifferential operator of negative order. Zeitschrift für Analysis und ihre Anwendungen 15(1996), 357–375.

    MathSciNet  MATH  Google Scholar 

  36. Shamir, E.: Regularition of mixed second order elliptic problems. Israel Math. J. 6(1968), 150–168.

    MathSciNet  MATH  Google Scholar 

  37. Signorini, A.: Questioni di elasticità non linearizzata o semilinearizzata. Rend. di Matem, e delle sue appl. 18(1959).

    Google Scholar 

  38. Troianiello, G.M.: Elliptic differential equations and obstacle problems. Plenum, NewYork 1987.

    MATH  Google Scholar 

  39. Uralzeva, N.N.: On the regularity of solutions of variational inqualities (Russian). Uspechi mat. Nauk 42(1987)6, 151–174.

    Google Scholar 

  40. Zeidler, E.: Nonlinear functional analysis and applications, Vols. 1–4. Springer-Verlag New York, Berlin 1985–1992.

    Google Scholar 

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Schumann, R. (2001). Regularity for Variational Inequalities — A Survey of Results. In: Gilbert, R.P., Panagiotopoulos, P.D., Pardalos, P.M. (eds) From Convexity to Nonconvexity. Nonconvex Optimization and Its Applications, vol 55. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0287-2_20

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  • DOI: https://doi.org/10.1007/978-1-4613-0287-2_20

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7979-9

  • Online ISBN: 978-1-4613-0287-2

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