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Archive for Rational Mechanics and Analysis

, Volume 233, Issue 1, pp 87–166 | Cite as

Cauchy Fluxes and Gauss–Green Formulas for Divergence-Measure Fields Over General Open Sets

  • Gui-Qiang G. ChenEmail author
  • Giovanni E. Comi
  • Monica Torres
Article
  • 47 Downloads

Abstract

We establish the interior and exterior Gauss–Green formulas for divergence-measure fields in Lp over general open sets, motivated by the rigorous mathematical formulation of the physical principle of balance law via the Cauchy flux in the axiomatic foundation, for continuum mechanics allowing discontinuities and singularities. The method, based on a distance function, allows us to give a representation of the interior (resp. exterior) normal trace of the field on the boundary of any given open set as the limit of classical normal traces over the boundaries of interior (resp. exterior) smooth approximations of the open set. In the particular case of open sets with a continuous boundary, the approximating smooth sets can explicitly be characterized by using a regularized distance. We also show that any open set with Lipschitz boundary has a regular Lipschitz deformable boundary from the interior. In addition, some new product rules for divergence-measure fields and suitable scalar functions are presented, and the connection between these product rules and the representation of the normal trace of the field as a Radon measure is explored. With these formulas to hand, we introduce the notion of Cauchy fluxes as functionals defined on the boundaries of general bounded open sets for the rigorous mathematical formulation of the physical principle of balance law, and show that the Cauchy fluxes can be represented by corresponding divergence-measure fields.

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Acknowledgements

The research of Gui-Qiang G. Chen was supported in part by the UK Engineering and Physical Sciences Research Council Award EP/E035027/1 and EP/L015811/1, and the Royal Society-Wolfson Research Merit Award (UK). The research of Giovanni E. Comi was supported in part by the PRIN2015 MIUR Project “Calcolo delle Variazioni”. The research of Monica Torres was supported in part by the Simons Foundation Award No. 524190 and by the National Science Foundation Award 1813695.

References

  1. 1.
    Ambrosio, L., Crippa, G., Maniglia, S.: Traces and fine properties of a \(BD\) class of vector fields and applications. Ann. Fac. Sci. Toulouse Math. (6), 14(4), 527–561, 2005.  https://doi.org/10.5802/afst.1102
  2. 2.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. The Clarendon Press, Oxford University Press: New York, 2000.  https://doi.org/10.1017/S0024609301309281
  3. 3.
    Ambrosio, L., Tilli, P.: Topics on Analysis in Metric Spaces. Oxford University Press, Oxford (2004). ISBN 9780198529385zbMATHGoogle Scholar
  4. 4.
    Anzellotti, G.: Pairings between measures and bounded functions and compensated compactness. Annali di Matematica Pura ed Applicata 135(1), 293–318 (1983).  https://doi.org/10.1007/bf01781073 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Anzellotti, G.: Traces of bounded vector-fields and the divergence theorem. Preprint, 1983Google Scholar
  6. 6.
    Auscher, P., Russ, E., Tchamitchian, P.: Hardy Sobolev spaces on strongly Lipschitz domains of \({\mathbb{R}}^{n}\). J. Funct. Anal. 218(1), 54–109 (2005).  https://doi.org/10.1016/j.jfa.2004.06.005 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ball, J.M., Zarnescu, A.: Partial-regularity and smooth topology preserving approximations of rough domains, Calc. Var. & PDEs. 56(1), Art. 13, 32 pp., 2017.  https://doi.org/10.1007/s00526-016-1092-6
  8. 8.
    Burago, Y.D., Maz'ya, V.G.: Potential Theory and Function Theory for Irregular Regions. Translated from Russian. Seminars in Mathematics, V. A. Steklov Mathematical Institute, Leningrad, Vol. 3. Consultants Bureau: New York, 1969. ASIN: B01KK6TXLKGoogle Scholar
  9. 9.
    Cauchy, A.L.: Recherches sur l'équilibre et le mouvement intérieur des corps solides ou fluides, élastiques or non élastiques. Bull. Soc. Philomathique 10(2), 9–13 (1823).  https://doi.org/10.1017/CBO9780511702518.038 Google Scholar
  10. 10.
    Cauchy, A.L.: Da la pression ou tension dans un corps solide. Exercises de Matehématiques 2(2), 42–56 (1827)Google Scholar
  11. 11.
    Chen, G.-Q., Frid, H.: Divergence-measure fields and hyperbolic conservation laws. Arch. Ration. Mech. Anal. 147(2), 89–118 (1999).  https://doi.org/10.1007/s002050050146 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chen, G.-Q., Frid, H.: Extended divergence-measure fields and the Euler equations for gas dynamics. Commun. Math. Phys. 236(2), 251–280 (2003).  https://doi.org/10.1007/s00220-003-0823-7 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chen, G.-Q., Torres, M.: Divergence-measure fields, sets of finite perimeter, and conservation laws. Arch. Ration. Mech. Anal. 175(2), 245–267 (2005).  https://doi.org/10.1007/s00205-004-0346-1 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chen, G.-Q., Torres, M., Ziemer, W.P.: Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws. Commun. Pure Appl. Math. 62(2), 242–304 (2009).  https://doi.org/10.1002/cpa.20262 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Comi, G.E., Magnani, V.: The Gauss–Green theorem in stratified groups. Preprint, 2018. arXiv:1806.04011
  16. 16.
    Comi, G.E., Payne, K.R.: On locally essentially bounded divergence measure fields and sets of locally finite perimeter. Adv. Calc. Var. 2018 (to appear).  https://doi.org/10.1515/acv-2017-0001
  17. 17.
    Comi, G.E., Torres, M.: One sided approximations of sets of finite perimeter. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28(1), 181–190, 2017.  https://doi.org/10.4171/RLM/757
  18. 18.
    Crasta, G., De Cicco, V.: Anzellotti's pairing theory and the Gauss–Green theorem. Preprint, 2017. arXiv:1708.00792
  19. 19.
    Crasta, G., De Cicco, V.: An extension of the pairing theory between divergence-measure fields and BV functions. J. Funct. Anal. 2018 (to appear).  https://doi.org/10.1016/j.jfa.2018.06.007
  20. 20.
    Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, 4th Ed., Springer, Berlin, 2016.  https://doi.org/10.1007/978-3-662-49451-6
  21. 21.
    Daners, D.: Domain perturbation for linear and semilinear boundary value problems. In: Handbook of Differential Equations—Stationary Partial Differential Equations, pp. 1–81. Elsevier, 2008.  https://doi.org/10.1016/S1874-5733(08)80018-6
  22. 22.
    De Giorgi, E.: Su una teoria generale della misura \((r-1)\)-dimensionale in uno spazio ad \(r\) dimensioni. Ann. Mat. Pura Appl. 36(1), 191–213 (1954).  https://doi.org/10.1007/BF02412838 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    De Giorgi, E.: Frontiere orientate di misura minima. Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960–61. Editrice Tecnico Scientifica, Pisa, 1961Google Scholar
  24. 24.
    De Lellis, C., Otto, F., Westdickenberg, M.: Structure of entropy solutions for multi-dimensional scalar conservation laws. Arch. Ration. Mech. Anal. 170(2), 137–184 (2003).  https://doi.org/10.1007/s00205-003-0270-9 MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Degiovanni, M., Marzocchi, A., Musesti, A.: Cauchy fluxes associated with tensor fields having divergence measure. Arch. Ration. Mech. Anal. 147(3), 197–223 (1999).  https://doi.org/10.1007/s002050050149 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Doktor, P.: Approximation of domains with Lipschitzian boundary. Časopis pro Pěstováni Matematiky, 101(3), 237–255, 1976. ISSN: 0528-2195Google Scholar
  27. 27.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, FL (1992). ISBN 9780849371578zbMATHGoogle Scholar
  28. 28.
    Evans, L.C.: Partial Differential Equations. AMS, Providence, RI, 2010.  https://doi.org/10.1090/gsm/019
  29. 29.
    Federer, H.: The Gauss-Green theorem. Trans. Am. Math. Soc. 58, 44–76 (1945).  https://doi.org/10.2307/1990234 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Federer, H.: A note on the Gauss-Green theorem. Proc. Am. Math. Soc. 9, 447–451 (1958).  https://doi.org/10.2307/2033002 MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Frid, H.: Remarks on the theory of the divergence-measure fields. Quart. Appl. Math. 70(3), 579–596 (2012).  https://doi.org/10.1090/S0033-569X-2012-01311-5 MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Frid, H.: Divergence-measure fields on domain with Lipschitz boundary. In: Chen, G.-Q. Holden, H., Karlsen, K. (eds.) Hyperbolic Conservation Laws and Related Analysis with Applications, pp. 207–225. Springer, Heidelberg, 2014.  https://doi.org/10.1007/978-3-642-39007-4_10
  33. 33.
    Fuglede, B.: On a theorem of F. Riesz. Math. Scand. 3, 283–302 (1955).  https://doi.org/10.7146/math.scand.a-10448 CrossRefzbMATHGoogle Scholar
  34. 34.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin, 1983.  https://doi.org/10.1007/978-3-642-61798-0
  35. 35.
    Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Birkhäuser-Verlag, Basel, 1984.  https://doi.org/10.1007/978-1-4684-9486-0
  36. 36.
    Gurtin, M.E., Martins, L.C.: Cauchy's theorem in classical physics. Arch. Ration. Mech. Anal. 60(4), 305–324, 1975/76.  https://doi.org/10.1007/BF00248882
  37. 37.
    Hofmann, S., Mitrea, M., Taylor, M.: Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains. J. Geometric Anal. 17, 593–647 (2007).  https://doi.org/10.1007/BF02937431 MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Kawohl, B., Schuricht, F.: Dirichlet problems for the 1-Laplace operator, including the eigenvalue problem. Commun. Contemp. Math. 9(4), 515–543 (2007).  https://doi.org/10.1142/S0219199707002514 MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Lax, P.D.: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. CBMS-RCSM, SIAM, Philadelphia (1973)CrossRefzbMATHGoogle Scholar
  40. 40.
    Leonardi, G.P., Saracco G.: The prescribed mean curvature equation in weakly regular domains. NoDEA Nonlinear Differ. Equ. Appl. 25, 9, 29 pp., 2018.  https://doi.org/10.1007/s00030-018-0500-3
  41. 41.
    Lieberman, G. M.: Regularized distance and its applications. Pac. J. Math. 117(2), 329–352, 1985. ISSN: 0030-8730Google Scholar
  42. 42.
    Lieberman, G.M.: The conormal derivative problem for equations of variational type in nonsmooth domains. Trans. Am. Math. Soc. 330(1), 41–67 (1992).  https://doi.org/10.2307/2154153 MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Maggi, F.: Sets of Finite Perimeter and Geometric Variational Problems. Cambridge University Press, Cambridge, 2012.  https://doi.org/10.1017/CBO9781139108133
  44. 44.
    Maz'ya, V.G.: Sobolev Spaces with Applications to Elliptic Partial Differential Equations. Springer, Berlin, 2011.  https://doi.org/10.1007/978-3-642-15564-2
  45. 45.
    Nečas, J.: On domains of type \(\mathfrak N \it \) (Russian). Czech. Math. J. 12(2), 274–287, 1962. ISSN: 0011-4642Google Scholar
  46. 46.
    Nečas, J.: Sur les équations différentielles aux dérivées partielles du type elliptique du deuxième ordre. Czech. Math. J. 14(1), 125–146, 1964. ISSN: 0011-4642Google Scholar
  47. 47.
    Noll, W.: The foundations of classical mechanics in the light of recent advances in continuum mechanics. In: Henkin, P., Suppes, L., Tarski, A. (eds.) The Axiomatic Method. With Special Reference to Geometry and Physics, Studies in Logic and the Foundations of Mathematics, pp. 266–281. North-Holland Publishing Co., Amsterdam (1959)CrossRefGoogle Scholar
  48. 48.
    Phuc, N.C., Torres, M.: Characterizations of the existence and removable singularities of divergence-measure vector fields. Indiana Univ. Math. J. 57(4), 1573–1597 (2008).  https://doi.org/10.1512/iumj.2008.57.3312 MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Phuc, N.C., Torres, M.: Characterizations of signed measures in the dual of \(BV\) and related isometric isomorphisms. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 17(1), 385–417, 2017. ISSN: 0391-173XGoogle Scholar
  50. 50.
    Scheven, C., Schmidt, T.: \({B}{V}\) supersolutions to equations of 1-Laplace and minimal surface type. J. Differ. Equ. 261(3), 1904–1932 (2016).  https://doi.org/10.1016/j.jde.2016.04.015 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Scheven, C., Schmidt, T.: An Anzellotti type pairing for divergence-measure fields and a notion of weakly super-\(1\)-harmonic functions. Preprint, 2017. arXiv:1701.02656
  52. 52.
    Scheven, C., Schmidt, T.: On the dual formulation of obstacle problems for the total variation and the area functional. Ann. Inst. Henri Poincaré, Anal. Non Lineaire, 2018 (to appear).  https://doi.org/10.1016/j.anihpc.2017.10.003
  53. 53.
    Schuricht, F.: A new mathematical foundation for contact interactions in continuum physics. Arch. Ration. Mech. Anal. 184, 495–551 (2007).  https://doi.org/10.1007/s00205-006-0032-6 MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Šilhavý, M.: The existence of the flux vector and the divergence theorem for general Cauchy fluxes. Arch. Ration. Mech. Anal. 90(3), 195–212 (1985).  https://doi.org/10.1007/BF00251730 MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Šilhavý, M.: Cauchy's stress theorem and tensor fields with divergences in \(L^p\). Arch. Ration. Mech. Anal. 116(3), 223–255 (1991).  https://doi.org/10.1007/BF00375122 CrossRefzbMATHGoogle Scholar
  56. 56.
    Šilhavý, M.: Divergence-measure fields and Cauchy's stress theorem. Rend. Sem. Mat Padova, 113, 15–45, 2005. ISSN: 0041-8994Google Scholar
  57. 57.
    Šilhavý, M.: The divergence theorem for divergence measure vectorfields on sets with fractal boundaries. Math. Mech. Solids 14, 445–455 (2009).  https://doi.org/10.1177/1081286507081960 MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
  59. 59.
    Verchota, G.C.: Layer Potentials and Boundary Value Problems for the Laplace Equation in Lipschitz Domains. PhD thesis, 1982Google Scholar
  60. 60.
    Verchota, G.C.: Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains. J. Funct. Anal. 59(3), 572–611 (1984).  https://doi.org/10.1016/0022-1236(84)90066-1 MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Verchota, G.C.: The biharmonic Neumann problem in Lipschitz domains. Acta Math. 194(2), 217–279 (2005).  https://doi.org/10.1007/BF02393222 MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Vol'pert, A.I.: Spaces \(BV\) and quasilinear equations (Russian). Mat. Sb. (N.S.), 73(115), 255–302, 1967Google Scholar
  63. 63.
    Vol'pert, A.I., Hudjaev, S.I.: Analysis in Classes of Discontinuous Functions and Equation of Mathematical Physics. Martinus Nijhoff Publishers, Dordrecht (1985). ISBN 90-247-3109-7Google Scholar
  64. 64.
    von Koch, H.: Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire. Archiv för Matemat., Astron. och Fys. 1, 681–702, 1904Google Scholar
  65. 65.
    Ziemer, W.P.: Cauchy flux and sets of finite perimeter. Arch. Ration. Mech. Anal. 84(3), 189–201 (1983).  https://doi.org/10.1007/BF00281518 MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK
  2. 2.Scuola Normale SuperiorePisaItaly
  3. 3.Department of MathematicsPurdue UniversityWest LafayetteUSA

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