Advertisement

Singular Domains

  • Gabriel Acosta
  • Ricardo G. Durán
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

In view of the previous chapters it is a natural question whether similar results can be obtained for more general domains. With this goal, we consider generalizations of Korn p and div p using weighted Sobolev norms.

References

  1. 1.
    Acosta, G., Durán, R.G., Lombardi, A.L.: Weighted Poincaré and Korn inequalities for Hölder α domains. Math. Methods Appl. Sci. 29 (4), 387–400 (2006). DOI 10.1002/mma.680. URL http://dx.doi.org/10.1002/mma.680 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Acosta, G., Durán, R.G., López Garcia, F.: Korn inequality and divergence operator: counterexamples and optimality of weighted estimates. Proc. Am. Math. Soc. 141 (1), 217–232 (2013). DOI 10.1090/S0002-9939-2012-11408-X. URL http://dx.doi.org/10.1090/S0002-9939-2012-11408-X MathSciNetCrossRefzbMATHGoogle Scholar
  3. 4.
    Acosta, G., Ojea, I.: Korns inequalities for generalized external cusps. Math. Methods Appl. Sci. 39 (17), 4935–4950 (2016). DOI 10.1002/mma.3170. URL http://dx.doi.org/10.1002/mma.3170 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 5.
    Adams, R.A., Fournier, J.J.F.: Sobolev spaces, Pure and Applied Mathematics (Amsterdam), vol. 140, second edn. Elsevier/Academic Press, Amsterdam (2003)Google Scholar
  5. 6.
    Agmon, S.: Lectures on elliptic boundary value problems. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. Van Nostrand Mathematical Studies, No. 2. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London (1965)Google Scholar
  6. 8.
    Andreou, E., Dassios, G., Polyzos, D.: Korn’s constant for a spherical shell. Quart. Appl. Math. 46 (3), 583–591 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 12.
    Boas, H.P., Straube, E.J.: Integral inequalities of Hardy and Poincaré type. Proc. Amer. Math. Soc. 103 (1), 172–176 (1988). DOI 10.2307/2047547. URL http://dx.doi.org/10.2307/2047547 MathSciNetzbMATHGoogle Scholar
  8. 13.
    Boffi, D., Brezzi, F., Demkowicz, L.F., Durán, R.G., Falk, R.S., Fortin, M.: Mixed finite elements, compatibility conditions, and applications, Lecture Notes in Mathematics, vol. 1939. Springer-Verlag, Berlin; Fondazione C.I.M.E., Florence (2008). DOI 10.1007/978-3-540-78319-0. URL http://dx.doi.org/10.1007/978-3-540-78319-0. Lectures given at the C.I.M.E. Summer School held in Cetraro, June 26–July 1, 2006, Edited by Boffi and Lucia Gastaldi
  9. 17.
    Buckley, S.M., Koskela, P.: New Poincaré inequalities from old. Ann. Acad. Sci. Fenn. Math. 23 (1), 251–260 (1998)MathSciNetzbMATHGoogle Scholar
  10. 18.
    Calderón, A.P.: Lebesgue spaces of differentiable functions and distributions. In: Proc. Sympos. Pure Math., Vol. IV, pp. 33–49. American Mathematical Society, Providence, R.I. (1961)Google Scholar
  11. 24.
    Ciarlet, P.G.: Mathematical elasticity. Vol. I, Studies in Mathematics and its Applications, vol. 20. North-Holland Publishing Co., Amsterdam (1988). Three-dimensional elasticityGoogle Scholar
  12. 25.
    Ciarlet, P.G.: On Korn’s inequality. Chin. Ann. Math. Ser. B 31 (5), 607–618 (2010). DOI 10.1007/s11401-010-0606-3. URL http://dx.doi.org/10.1007/s11401-010-0606-3 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 26.
    Ciarlet, P.G., Ciarlet Jr., P.: Another approach to linearized elasticity and a new proof of Korn’s inequality. Math. Models Methods Appl. Sci. 15 (2), 259–271 (2005). DOI 10.1142/S0218202505000352. URL http://dx.doi.org/10.1142/S0218202505000352 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 34.
    Duoandikoetxea, J.: Fourier analysis, Graduate Studies in Mathematics, vol. 29. American Mathematical Society, Providence, RI (2001). Translated and revised from the 1995 Spanish original by David Cruz-UribeGoogle Scholar
  15. 35.
    Duran, R., Muschietti, M.A., Russ, E., Tchamitchian, P.: Divergence operator and Poincaré inequalities on arbitrary bounded domains. Complex Var. Elliptic Equ. 55 (8–10), 795–816 (2010). DOI 10.1080/17476931003786659. URL http://dx.doi.org/10.1080/17476931003786659 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 38.
    Durán, R.G., Muschietti, M.A.: The Korn inequality for Jones domains. Electron. J. Differential Equations pp. No. 127, 10 pp. (electronic) (2004)Google Scholar
  17. 41.
    Fichera, G.: Existence theorems in linear and semi-linear elasticity. Z. Angew. Math. Mech. 54, T24–T36 (1974). Vorträge der Wissenschaftlichen Jahrestagung der Gesellschaft für Angewandte Mathematik und Mechanik (Munich, 1973)Google Scholar
  18. 42.
    Friedrichs, K.: On certain inequalities and characteristic value problems for analytic functions and for functions of two variables. Trans. Amer. Math. Soc. 41 (3), 321–364 (1937). DOI 10.2307/1989786. URL http://dx.doi.org/10.2307/1989786 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 43.
    Friedrichs, K.O.: On the boundary-value problems of the theory of elasticity and Korn’s inequality. Ann. of Math. (2) 48, 441–471 (1947)Google Scholar
  20. 44.
    Friesecke, G., James, R.D., Müller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math. 55 (11), 1461–1506 (2002). DOI 10.1002/cpa.10048. URL http://dx.doi.org/10.1002/cpa.10048 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 48.
    Geymonat, G., Gilardi, G.: Contre-exemples à l’inégalité de Korn et au lemme de Lions dans des domaines irréguliers. In: Équations aux dérivées partielles et applications, pp. 541–548. Gauthier-Villars, Éd. Sci. Méd. Elsevier, Paris (1998)Google Scholar
  22. 49.
    Gobert, J.: Une inégalité fondamentale de la théorie de l’élasticité. Bull. Soc. Roy. Sci. Liège 31, 182–191 (1962)MathSciNetzbMATHGoogle Scholar
  23. 53.
    Hlaváček, I., Nečas, J.: On inequalities of Korn’s type. I. Boundary-value problems for elliptic system of partial differential equations. Arch. Rational Mech. Anal. 36, 305–311 (1970)zbMATHGoogle Scholar
  24. 54.
    Hlaváček, I., Nečas, J.: On inequalities of Korn’s type. II. Applications to linear elasticity. Arch. Rational Mech. Anal. 36, 312–334 (1970)CrossRefzbMATHGoogle Scholar
  25. 55.
    Horgan, C.O.: On Korn’s inequality for incompressible media. SIAM J. Appl. Math. 28, 419–430 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 56.
    Horgan, C.O.: Korn’s inequalities and their applications in continuum mechanics. SIAM Rev. 37 (4), 491–511 (1995). DOI 10.1137/1037123. URL http://dx.doi.org/10.1137/1037123 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 57.
    Horgan, C.O., Payne, L.E.: On inequalities of Korn, Friedrichs and Babuška-Aziz. Arch. Rational Mech. Anal. 82 (2), 165–179 (1983). DOI 10.1007/BF00250935. URL http://dx.doi.org/10.1007/BF00250935 zbMATHGoogle Scholar
  28. 58.
    Jiang, R., Kauranen, A.: Korn inequality on irregular domains. J. Math. Anal. Appl. 423 (1), 41–59 (2015). DOI 10.1016/j.jmaa.2014.09.076. URL http://dx.doi.org/10.1016/j.jmaa.2014.09.076 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 59.
    Jiang, R., Kauranen, A.: Korn’s inequality and John domains. Preprint (2015)Google Scholar
  30. 62.
    Jones, P.W.: Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147 (1–2), 71–88 (1981). DOI 10.1007/BF02392869. URL http://dx.doi.org/10.1007/BF02392869 MathSciNetCrossRefzbMATHGoogle Scholar
  31. 63.
    Kesavan, S.: On Poincaré’s and J. L. Lions’ lemmas. C. R. Math. Acad. Sci. Paris 340 (1), 27–30 (2005). DOI 10.1016/j.crma.2004.11.021. URL http://dx.doi.org/10.1016/j.crma.2004.11.021 MathSciNetCrossRefzbMATHGoogle Scholar
  32. 64.
    Kikuchi, N., Oden, J.T.: Contact problems in elasticity: a study of variational inequalities and finite element methods, SIAM Studies in Applied Mathematics, vol. 8. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1988). DOI 10.1137/1.9781611970845. URL http://dx.doi.org/10.1137/1.9781611970845
  33. 65.
    Kondratiev, V.A., Oleinik, O.A.: On Korn’s inequalities. C. R. Acad. Sci. Paris Sér. I Math. 308 (16), 483–487 (1989). DOI 10.1070/RM1989v044n06ABEH002297. URL http://dx.doi.org/10.1070/RM1989v044n06ABEH002297 MathSciNetzbMATHGoogle Scholar
  34. 66.
    Kondratiev, V.A., Oleinik, O.A.: Hardy’s and Korn’s type inequalities and their applications. Rend. Mat. Appl. (7) 10 (3), 641–666 (1990)Google Scholar
  35. 67.
    Korn, A.: Die eigenschwingungen eines elastichen korpers mit ruhender oberflache. Akad. der Wissensch Munich, Math-phys. Kl, Beritche 36 (0), 351–401 (1906)Google Scholar
  36. 69.
    Korn, A.: Ubereinige ungleichungen, welche in der theorie der elastischen und elektrischen schwingungen eine rolle spielen. Bulletin Internationale, Cracovie Akademie Umiejet, Classe de sciences mathematiques et naturelles (0), 705–724 (1909)Google Scholar
  37. 70.
    Kufner, A., Persson, L.E.: Weighted inequalities of Hardy type. World Scientific Publishing Co., Inc., River Edge, NJ (2003). DOI 10.1142/5129. URL http://dx.doi.org/10.1142/5129 CrossRefzbMATHGoogle Scholar
  38. 75.
    López Garcia, F.: A decomposition technique for integrable functions with applications to the divergence problem. J. Math. Anal. Appl. 418 (1), 79–99 (2014). DOI 10.1016/j.jmaa.2014.03.080. URL http://dx.doi.org/10.1016/j.jmaa.2014.03.080 MathSciNetCrossRefzbMATHGoogle Scholar
  39. 78.
    Martio, O.: Definitions for uniform domains. Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1), 197–205 (1980). DOI 10.5186/aasfm.1980.0517. URL http://dx.doi.org/10.5186/aasfm.1980.0517 MathSciNetCrossRefzbMATHGoogle Scholar
  40. 81.
    Nitsche, J.A.: On Korn’s second inequality. RAIRO Anal. Numér. 15 (3), 237–248 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 82.
    Payne, L.E., Weinberger, H.F.: On Korn’s inequality. Arch. Rational Mech. Anal. 8, 89–98 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 84.
    Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J. (1970)Google Scholar
  43. 86.
    Ting, T.W.: Generalized Korn’s inequalities. Tensor (N.S.) 25, 295–302 (1972). Commemoration volumes for Prof. Dr. Akitsugu Kawaguchi’s seventieth birthday, Vol. IIGoogle Scholar
  44. 87.
    Weck, N.: Local compactness for linear elasticity in irregular domains. Math. Methods Appl. Sci. 17 (2), 107–113 (1994). DOI 10.1002/mma.1670170204. URL http://dx.doi.org/10.1002/mma.1670170204 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Gabriel Acosta
    • 1
  • Ricardo G. Durán
    • 1
  1. 1.Department of Mathematics and IMASUniversity of Buenos Aires and CONICETBuenos AiresArgentina

Personalised recommendations