, Volume 63, Issue 2, pp 249–276 | Cite as

Explicit div-curl inequalities in bounded and unbounded domains of \({{\mathbb {R}}}^3\)

  • Tahar Zamene Boulmezaoud
  • Keltoum Kaliche
  • Nabil Kerdid


We give a constructive proof of some functional inequalities related to the div and curl operators in bounded and unbounded domains of \({{\mathbb {R}}}^3\). Our new innovation consists in giving explicit constants in several geometric configurations. These inequalities are of a first use in solving div-curl systems and vector potential problems arising in physics.


Div-curl systems Div-curl inequalities Vector potentials Weighted inequalities Unbounded domains 

Mathematics Subject Classification

35Q60 35Q61 35B45 35F45 



The authors would like to thank the anonymous reviewer for careful and thorough reading of this manuscript, and for helpful and valuable comments, which significantly contributed to improving the quality of this publication.

The third author, N. Kerdid, gratefully acknowledge the support of the National Plan for Science, Technology and Information (Maarifah), King Abdulaziz City for Science and Technology, KSA, award number 12-MAT2996-08.


  1. 1.
    Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21(9), 823–864 (1998)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Amrouche, C., Girault, V., Giroire, J.: Weighted Sobolev spaces for Laplace’s equation in \({ R}^n\). J. Math. Pures Appl. (9) 73(6), 579–606 (1994)MathSciNetMATHGoogle Scholar
  3. 3.
    Amrouche, C., Girault, V., Giroire, J.: Dirichlet and Neumann exterior problems for the \(n\)-dimensional Laplace operator: an approach in weighted Sobolev spaces. J. Math. Pures Appl. (9) 76(1), 55–81 (1997)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Amrouche, Ch., Seloula, N.-H.: \(L^p\)-theory for vector potentials and Sobolev’s inequalities for vector fields: application to the Stokes equations with pressure boundary conditions. Math. Models Methods Appl. Sci. 23(1), 37–92 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Arar, N., Boulmezaoud, T.Z.: Eigenfunctions of a weighted Laplace operator in the whole space. J. Math. Anal. Appl. 400(1), 161–173 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Auchmuty, G.: Divergence \(L^2\)-coercivity inequalities. Numer. Funct. Anal. Optim. 27(5–6), 499–515 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Boulmezaoud, T.Z.: On the existence of non-linear Beltrami fields. Comptes Rendus de l’Académie des Sciences. Série I. Mathématiques 328(5), 442 (1999)MathSciNetMATHGoogle Scholar
  8. 8.
    Boulmezaoud, T.Z.: Vector potentials in the half-space of \({\mathbb{R}}^3\). C. R. Acad. Sci. Paris Sér. I Math. 332(8), 711–716 (2001)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Boulmezaoud, T.Z.: On the Stokes system and on the biharmonic equation in the half-space: an approach via weighted Sobolev spaces. Math. Methods Appl. Sci. 25(5), 373–398 (2002)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Boulmezaoud, T.Z.: On the Laplace operator and on the vector potential problems in the half-space: an approach using weighted spaces. Math. Methods Appl. Sci. 26(8), 633–669 (2003)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Boulmezaoud, T.Z.: Inverted finite elements: a new method for solving elliptic problems in unbounded domains. M2AN. Math. Model. Numer. Anal. 39(1), 109–145 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Boulmezaoud, T.Z.: Étude des champs de Beltrami dans des domaines de R3 bornés et non-bornés et applications en astrophysique. Ph.D. Thesis, Université Pierre et Marie Curie (Paris VI), Paris., Janvier (1999)Google Scholar
  13. 13.
    Boulmezaoud, T.Z., Maday, Y., Amari, T.: On the linear force-free fields in bounded and unbounded three-dimensional domains. M2AN Math. Modell. Numer. Anal. 33(2), 359–393 (1999)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Problems in Analysis (Papers dedicated to Salomon Bochner, 1969), pp. 195–199. Princeton University Press, Princeton (1970)Google Scholar
  15. 15.
    Costabel, M.: A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains. Math. Methods Appl. Sci. 12(4), 365–368 (1990)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Desvillettes, L., Villani, C.: On a variant of Korn’s inequality arising in statistical mechanics. ESAIM Control Optim. Calc. Var. 8:603–619 (electronic) (2002) (A tribute to J. L. Lions)Google Scholar
  17. 17.
    Duvaut, G., Lions, J.-L.: Les inéquations en mécanique et en physique. Dunod, Paris. Travaux et Recherches Mathématiques, No. 21 (1972)Google Scholar
  18. 18.
    Filonov, N.: On an inequality for the eigenvalues of the Dirichlet and Neumann problems for the Laplace operator. Algebra i Analiz 16(2), 172–176 (2004)MathSciNetGoogle Scholar
  19. 19.
    Friedrichs, K.O.: Differential forms on Riemannian manifolds. Commun. Pure Appl. Math. 8, 551–590 (1955)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Gaffney, M.P.: The harmonic operator for exterior differential forms. Proc. Nat. Acad. Sci. USA 37, 48–50 (1951)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Girault, V.: The divergence, curl and Stokes operators in exterior domains of \({\bf R}^3\). In: Recent Developments in Theoretical Fluid Mechanics (Paseky, 1992), Pitman Res. Notes Math. Ser., vol. 291, pp. 34–77. Longman Scientific and Technical, Harlow (1993)Google Scholar
  22. 22.
    Girault, V.: The Stokes problem and vector potential operator in three-dimensional exterior domains: an approach in weighted Sobolev spaces. Differ. Integral Equ. 7(2), 535–570 (1994)MathSciNetMATHGoogle Scholar
  23. 23.
    Girault, V., Giroire, J., Sequeira, A.: Formulation variationnelle en fonction courant-tourbillon du problème de Stokes extérieur dans des espaces de Sobolev à poids. C. R. Acad. Sci. Paris Sér. I Math. 313(8), 499–502 (1991)MathSciNetMATHGoogle Scholar
  24. 24.
    Girault, V., Raviart, P.A.: Finite element methods for Navier–Stokes equations. Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)Google Scholar
  25. 25.
    Girault, V., Sequeira, A.: A well-posed problem for the exterior Stokes equations in two and three dimensions. Arch. Rational Mech. Anal. 114(4), 313–333 (1991)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Giroire, J.: Etude de quelques problèmes aux limites extérieurs et résolution par équations intégrales. Thèse de Doctorat d’Etat. Université Pierre et Marie Curie, Paris (1987)Google Scholar
  27. 27.
    Gobert, J.: Sur une inégalité de coercivité. J. Math. Anal. Appl. 36, 518–528 (1971)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Grisvard, P.: Elliptic problems in nonsmooth domains, vol. 24, Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston (1985)Google Scholar
  29. 29.
    Hanouzet, B.: Espaces de Sobolev avec poids application au problème de Dirichlet dans un demi espace. Rend. Sem. Mat. Univ. Padova 46, 227–272 (1971)MathSciNetMATHGoogle Scholar
  30. 30.
    Jochmann, F.: A compactness result for vector fields with divergence and curl in \(L^q(\Omega )\) involving mixed boundary conditions. Appl. Anal. 66(1–2), 189–203 (1997)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Kuhn, P., Pauly, D.: Regularity results for generalized electro-magnetic problems. Analysis (Munich) 30(3), 225–252 (2010)MathSciNetMATHGoogle Scholar
  32. 32.
    Mitrea, M.: Dirichlet integrals and Gaffney–Friedrichs inequalities in convex domains. Forum Math. 13(4), 531–567 (2001)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Monk, P.: Finite Element Methods for Maxwell’s Equations. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2003)CrossRefGoogle Scholar
  34. 34.
    Nédélec, J.-C: Acoustic and electromagnetic equations, vol. 144, Applied Mathematical Sciences. Springer, New York (2001) (Integral representations for harmonic problems)Google Scholar
  35. 35.
    Pauly, D.: Low frequency asymptotics for time-harmonic generalized Maxwell’s equations in nonsmooth exterior domains. Adv. Math. Sci. Appl. 16(2), 591–622 (2006)MathSciNetMATHGoogle Scholar
  36. 36.
    Pauly, D.: Generalized electro-magneto statics in nonsmooth exterior domains. Analysis (Munich) 27(4), 425–464 (2007)MathSciNetMATHGoogle Scholar
  37. 37.
    Pauly, D.: On polynomial and exponential decay of eigen-solutions to exterior boundary value problems for the generalized time-harmonic Maxwell system. Asymptot. Anal. 79(1–2), 133–160 (2012)MathSciNetMATHGoogle Scholar
  38. 38.
    Pauly, D.: On constants in Maxwell inequalities for bounded and convex domains. J. Math. Sci. (N.Y.) 210(6), 787–792 (2015)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Pauly, D.: On Maxwell’s and Poincaré’s constants. Discrete Contin. Dyn. Syst. Ser. S 8(3), 607–618 (2015)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Pauly, D., Repin, S.: Functional a posteriori error estimates for elliptic problems in exterior domains. J. Math. Sci. (N. Y.) 162(3), 393–406 (2009) (Problems in mathematical analysis. No. 42)Google Scholar
  41. 41.
    Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5(286–292), 1960 (1960)MATHGoogle Scholar
  42. 42.
    Pflüger, K.: Semilinear elliptic problems in unbounded domains; solutions in weighted sobolev spaces. preprint (1995)Google Scholar
  43. 43.
    Picard, R.: Randwertaufgaben in der verallgemeinerten Potentialtheorie. Math. Methods Appl. Sci. 3(2), 218–228 (1981)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Picard, R.: On the boundary value problems of electro- and magnetostatics. Proc. Roy. Soc. Edinburgh Sect. A 92(1–2), 165–174 (1982)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Picard, R.: An elementary proof for a compact imbedding result in generalized electromagnetic theory. Math. Z. 187(2), 151–164 (1984)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Picard, R.: On the low frequency asymptotics in electromagnetic theory. J. Reine Angew. Math. 354, 50–73 (1984)MathSciNetMATHGoogle Scholar
  47. 47.
    Picard, R.: Some decomposition theorems and their application to nonlinear potential theory and Hodge theory. Math. Methods Appl. Sci. 12(1), 35–52 (1990)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Picard, R.: On a selfadjoint realization of curl and some of its applications. Ricerche Mat. 47(1), 153–180 (1998)MathSciNetMATHGoogle Scholar
  49. 49.
    Picard, R., Weck, N., Witsch, K.-J.: Time-harmonic Maxwell equations in the exterior of perfectly conducting, irregular obstacles. Analysis (Munich) 21(3), 231–263 (2001)MathSciNetMATHGoogle Scholar
  50. 50.
    Pólya, G.: Remarks on the foregoing paper. J. Math. Phys. 31, 55–57 (1952)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Saranen, J.: On an inequality of Friedrichs. Math. Scand. 51(2), 310–322 (1983). (1982)MathSciNetMATHGoogle Scholar
  52. 52.
    Saranen, J., Witsch, K.J.: Exterior boundary value problems for elliptic equations. Ann. Acad. Sci. Fenn. Ser. A I Math. 8(1), 3–42 (1983)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Weber, C.: Regularity theorems for Maxwell’s equations. Math. Methods Appl. Sci. 3(4), 523–536 (1981)MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Weber, Ch.: A local compactness theorem for Maxwell’s equations. Math. Methods Appl. Sci. 2(1), 12–25 (1980)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Weck, N.: Maxwell’s boundary value problem on Riemannian manifolds with nonsmooth boundaries. J. Math. Anal. Appl. 46, 410–437 (1974)MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Witsch, K.J.: A remark on a compactness result in electromagnetic theory. Math. Methods Appl. Sci. 16(2), 123–129 (1993)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Università degli Studi di Ferrara 2016

Authors and Affiliations

  • Tahar Zamene Boulmezaoud
    • 1
  • Keltoum Kaliche
    • 1
  • Nabil Kerdid
    • 2
  1. 1.Laboratoire de Mathématiques de VersaillesUVSQ, CNRS, Université Paris-SaclayVersaillesFrance
  2. 2.College of Sciences, Department of Mathematics and Statistics, Al Imam Mohammad Ibn Saud University (IMSIU)RiyadhKingdom of Saudi Arabia

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