Skip to main content
SpringerLink
Log in

Nonlinear problems related to the Thomas-Fermi equation

  • Chapter
Nonlinear Evolution Equations and Related Topics

Abstract

Most of the results in this work were obtained over the period 1975-77 and were announced at various meetings (see e.g. items [3], [4], [5] under Brezis [16]). This paper has a rather unusual history. Around 1972 I became interested in nonlinear elliptic equations of the form

$$ - \Delta u + |u{|^{{p - 1}}}u = f in a domain \Omega \subset {\mathbb{R}^{N}}, $$
((P.1))

with zero Dirichlet condition, where 0 < p < ∞ and f ∈ L 1. The motivation came from the study of the porous medium equation

$$ \frac{{\partial \upsilon }}{{\partial t}} - \Delta (|\upsilon {|^{{m - 1}}}\upsilon ) = 0, $$
((P.2))

with 0 < m < ∞.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. ANCONA, A.Une propriété d’invariance des ensembles absorbants par perturbation d’un opérateur elliptiqueComm. Pde4(1979), 321–337.

    Article  MathSciNet  MATH  Google Scholar 

  2. BAMBERGER, A.Étude de deux équations nonlinéaires avec une masse de Dirac au second membreRapport 13 du Centre de Mathématiques Appliquées de l’École Polytechnique, Oct. 1976.

    Google Scholar 

  3. Babas, P. and Pierre, M., [1]Singularités éliminables pour des équations semi-linéairesAnn. Inst. Fourier (Grenoble)34 (1984)185–206; [2]Problèmes paraboliques semi-linéaires avec données mesuresApplicable Anal.18(1984), 111–149.

    Article  MathSciNet  Google Scholar 

  4. Barenblatt, G. I.“Scaling self-similarity and intermediate asymptotics”.Cambridge Texts in Applied Mathematics, 14. Cambridge University Press, Cambridge, 1996.

    Google Scholar 

  5. Barenblatt, G. I. and SIVASHINSKI, G.I.Self-similar solutions of the second kind in nonlinear filtrationPrikl. Mat. Meh.33(1969), 861–870 (Russian); translated as J. Appl. Math. Mech.33(1969), 836–845.

    Article  MathSciNet  MATH  Google Scholar 

  6. Baxter, J. R.Inequalities for potentials of particle systemsIllinois J. Math.24(1980), 645–652.

    MathSciNet  MATH  Google Scholar 

  7. Benguria, R.The von Weizsäcker and exchange corrections in the Thomas-Fermi theoryPh.D. Dissertation, Princeton Univ. 1979.

    Google Scholar 

  8. Benguria, R. and YÁNEz, J.Variational principle for the chemical potential in the Thomas-Fermi modelJ. Phys. A31(1998), 585–593.

    Article  MathSciNet  MATH  Google Scholar 

  9. Benguria, R., Brezis, H. and Lieb, E.The Thomas-Fermi-von Weizsäcker theory of atoms and moleculesComm. Math. Phys. 79 (1981), 167–180.

    Article  MathSciNet  MATH  Google Scholar 

  10. BÉnilan, PH., Brezis, H. and Crandall, M., Asemilinear equation in L i (1R N )Ann. Sc. Norm. Sup. Pisa, Serie IV, 2 (1975), 523–555.

    Google Scholar 

  11. BÉnilan, PH., Golstein, G. and Goldstein, J., Anonlinear elliptic system arising in electron density theory Comm. Pde/7(1992), 2907–2917.

    Google Scholar 

  12. Boccardo, L., Dall’Aglio, A., Gallouet, T. and Orsina, L.Nonlinear parabolic equations with measure dataJ. Funct. Anal.147(1976), 237–258.

    MathSciNet  Google Scholar 

  13. Boccardo, L. and Gallouet, T., [1]Nonlinear elliptic and parabolic equations involving measure dataJ. Funct. Anal87(1989), 149–169; [2]Nonlinear elliptic equations with right-hand side measuresComm. Pde17(1992), 641–655.

    MathSciNet  MATH  Google Scholar 

  14. Boccardo, L., Gallouet, T. and Orsina, L., [1]Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure dataAnn. Inst. H. Poincaré, Anal. Non Linéaire 13 (1996), 539–551; [2]Existence and nonexistence of solutions for some nonlinear elliptic equationsJ. Anal. Math.73(1997), 203–223.

    Article  MathSciNet  MATH  Google Scholar 

  15. Breazna, A., Goldstein, G. and Goldstein, J.Parameter dependence in Thomas-Fermi theoryComm. Appl. Anal.5(2001), 421–432.

    MathSciNet  MATH  Google Scholar 

  16. REZIS,H.,[1]MonotonicitymethodsinHilbertspacesandsomeapplicationstononlinearpartialdiiferential equationsin“Contributions to nonlinear functional analysis”(E. Zarantonello ed.), Acad. Press 1971, 101–156;[2] “Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert”North-Holland, Amsterdam, 1973;[3] Nonlinear problems related to the Thomas-Fermi equationin “Contemporary developments in continuum mechanics and partial differential equations” (Proc. Internat. Sympos., Rio de Janeiro, 1977), (de la Penha and Medeiros ed.), North-Holland, 1978, pp. 81–89; [4], Afree boundary problem in quantum mechanics. - Thomas-Fermi equationin “Free boundary problems vol II, (Proc. Sympos. Pavia, 1979), Ist. Naz. Alta Mat., Rome, 1980, pp. 85-91; [5]Some variational problems of the Thomas-Fermi typein ”Variational inequalities and complementary problems: theory and applications“, (Proc. Internat. School, Erice, 1978) (R. W. Cottle, F. Giannessi and J.-L. Lions ed.), Wiley, 1980, 53-73; [6]Problèmes elliptiques et paraboliques non linéaires avec données mesuresSeminaire Goulaouic-Meyer-Schwartz, 1981-82, XX.1—XX.12;[7] Nonlinear elliptic equations involving measuresin ”Contributions to Nonlinear Partial Differential Equations“ (Madrid, 1981), (C. Bardos, A. Damlamian, J. I. Diaz and J. Hernandez ed.), Pitman, 1983, 82-89; [8]Semilinear equations in R N without conditions at infinityAppl. Math. Optim.12(1984), 271–282.

    Google Scholar 

  17. Brezis, H. and Friedman, A.Nonlinear parabolic equations involving measures as initial conditionsJ. Math. Pures Appl. 62 (1983), 73–97.

    MathSciNet  MATH  Google Scholar 

  18. Brezis, H. and Lieb, E.Long range atomic potentials in Thomas-Fermi theoryComm. Math. Phys.65(1979), 231–246.

    Article  MathSciNet  MATH  Google Scholar 

  19. Brezis, H., Marcus, M. and Ponce, A. C.Nonlinear elliptic equations with measures revisited(to appear).

    Google Scholar 

  20. Brezis, H. and Nirenberg, L.Removable singularities for nonlinear elliptic equationsTopol. Methods Nonlinear Anal.9(1997), 201–219.

    MathSciNet  MATH  Google Scholar 

  21. Brezis, H. and Oswald, L.Singular solutions for some semilinear elliptic equationsArch. Rat. Mech. Anal. 99 (1987), 249–259.

    MathSciNet  MATH  Google Scholar 

  22. Brezis, H., Peletier, L. and TÈrman, D., Avery singular solution of the heat equation with absorptionArch. Rat. Mech. Anal.95(1986), 185–209.

    Article  MATH  Google Scholar 

  23. Brezis, H. and Ponce, A. C.Remarks on the strong maximum principleDiff. Int. Equations16(2003), 1–12.

    MathSciNet  MATH  Google Scholar 

  24. Brezis, H. and SERFATY, S.A variational formulation for the two-sided obstacle problem with measure dataCommun. Contemp. Math. 4 (2002), 357–374.

    MathSciNet  MATH  Google Scholar 

  25. Brezis, H. and Strauss, W.Semilinear second-order elliptic equations inL1J. Math. Soc. Japan25(1973), 565–590.

    Article  MathSciNet  MATH  Google Scholar 

  26. Brezis, H. and Veron, L.Removable singularities of some nonlinear elliptic equationsArch. Rat. Mech. Anal.75(1980), 1–6.

    MathSciNet  MATH  Google Scholar 

  27. Browder, F.Nonlinear elliptic boundary value problemsBull. Amer. Math. Soc. 69 (1963), 862–874.

    Article  MathSciNet  MATH  Google Scholar 

  28. Caffarelli, L. and Friedman, A.The free boundary in the Thomas-Fermi modelJ. Diff. Eq.32(1979), 335–356.

    Article  MathSciNet  MATH  Google Scholar 

  29. Dall’Aglio, P. and Dal Maso, G.Some properties of the solutions of obstacle problems with measure dataRicerche Mat. suppl.48(1999), 99–116.

    Google Scholar 

  30. Dunford, N. and Schwartz, J. T.`Linear operators“Part I, Wiley Interscience, 1958.

    Google Scholar 

  31. DYNKIN, E.“Diffusions superdiifusions and partial differential equations”American Mathematical Society Colloquium Publications50.American Mathematical Society, Providence, RI, 2002.

    Google Scholar 

  32. EKELAND, I.On the variational principleJ. Math. Anal. Appl.47(1974), 324–353.

    Article  MathSciNet  MATH  Google Scholar 

  33. Friedman, A. and Kamin, S.The asymptotic behavior of gas in an n-dimensional porous mediumTrans. Amer. Math. Soc.262(1980), 551–563.

    MATH  Google Scholar 

  34. Fukushima, M., Sato, K. and TANIGUCHI, S.On the closable part of pre-Dirichlet forms and the fine supports of underlying measuresOsaka Math. J.28(1991), 517–535.

    MathSciNet  MATH  Google Scholar 

  35. Gallouet, TII. and Morel, J. M., PIOn some properties of the solution of the Thomas-Fermi problemNonlinear Anal. 7 (1983), 971–979; [2]Resolution of a semilinear equation in L IProc. Roy. Soc. Edinburgh96A(1984), 275-288; [3]On some semilinear problems in L IBoll. Un. Mat. Ital. 4 (1985), 123–131.

    MathSciNet  MATH  Google Scholar 

  36. Gmira, A. and Veron, L.Boundary singularities of solutions of nonlinear elliptic equationsDuke J. Math.64(1991), 271–324.

    MathSciNet  MATH  Google Scholar 

  37. Goldstein, G., Goldstein, J. and JIA, W.Thomas-Fermi theory with magnetic fields and the FermiAmaldi correctionDiff. Int. Equations8(1995), 1305–1316.

    MATH  Google Scholar 

  38. Goldstein, J. and Rieder, G., [1], Arigorous Thomas-Fermi theory for atomic systemsJ. Math. Phys28(1987), 1198–1202; [2]Spin polarized Thomas-Fermi theoryJ. Math. Phys29(1988), 709–716; [3]Thomas-Fermi theory with an external magnetic fieldJ. Math. Phys.32(1991), 2901–2917.

    MathSciNet  Google Scholar 

  39. HILLE, E.Some aspects of the Thomas-Fermi equationJ. Anal. Math.23(1970), 147–170.

    Article  MathSciNet  MATH  Google Scholar 

  40. KAMENOMOSTSKATA (Kamin), S. L., [1]Equation of the elastoplastic mode of filtrationPrikl. Mat. Meh.33(1969), 1076–1084 (Russian); translated as J. Appl. Math. Mech.33(1969) 1042–1049; [2]The asymptotic behavior of the solution of the filtration equationIsrael J. Math.14(1973), 76–87.

    Google Scholar 

  41. Kamin, S. and Peletier, L. A.Singular solutions of the heat equation with absorptionProc. Amer. Math. Soc. 95 (1985), 145–158.

    MathSciNet  Google Scholar 

  42. Kamin, S., Peletier, L. A. and Vazquez, J. L., [1]On the Barenblatt equation of elastoplastic filtrationIndiana Univ. Math. J.40(1991), 1333–1362; [2]Classification of singular solutions of a nonlinear heat equationDuke Math. J.58(1989), 601–615.

    MathSciNet  MATH  Google Scholar 

  43. KATO, T.Schrödinger operators with singular potentialsIsrael J. Math. 13 (1972), 135–148.

    Google Scholar 

  44. LABUTIN, D.Wiener regularity for large solutions of nonlinear equationsArkiv för Math. (to appear).

    Google Scholar 

  45. Legall, J. F., [1]The Brownian snake and solutions ofAu= u 2 in a domainProbab. Theory Related Fields102(1995), 393-432; [2], Aprobabilistic Poisson representation for positive solutions of Au = u 2 in a planar domainComm. Pure Appl. Math.50(1997), 69–103.

    MathSciNet  Google Scholar 

  46. Leray, J. and Lions, J. L.Quelques résultats de Visik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-BrowderBull. Soc. Math, Fr.93(1965), 97–107.

    MathSciNet  Google Scholar 

  47. Lieb, E. H., [1]The stability of matterRev. Mod. Phys.48(1976), 553–569; [2]Thomas-Fermi and related theories of atoms and moleculesRev. Mod. Phys.53(1981), 603–641; [3]“The stability of matter: from atoms to stars”Selecta of E. Lieb (W. Thirring ed.) Springer, second edition, 1997.

    Google Scholar 

  48. Lieb, E. H. and Simon, B.The Thomas-Fermi theory of atoms molecules and solidsAdvances in Math.23(1977), 22–116.

    Article  MathSciNet  Google Scholar 

  49. Loewner, C. and Nirenberg, L.Partial differential equations invariant under conformal or pmjective transformationsin “Contributions to Analysis”, Acad. Press, 1974, 245–272.

    Google Scholar 

  50. Marcus, M. and Veron, L., [1]The boundary trace of positive solutions of semilinear elliptic equations: the subcritical caseArch. Rat. Mech. Anal.144(1998), 201–231;[2]The boundary trace of positive solutions of semilinear elliptic equations: the supercritical caseJ. Math. Pures Appl. 77 (1998), 481–524;[3]Removable singularities and boundary tracesJ. Math. Pures Appl.80(2001), 879–900;[4]Capacitary estimates of solutions of a class of nonlinear elliptic equationsC. R. Acad. Sc. Paris336(2003), 913–918.

    MATH  Google Scholar 

  51. Orsina, L. and Prignet, A.Nonexistence of solutions for some nonlinear elliptic equations involving measuresProc. Royal Soc. Edinburgh130(2000), 561–592.

    MathSciNet  Google Scholar 

  52. Oswald, L.Isolated positive singularities fora nonlinear heat equationHouston J. Math.14(1988), 543–572.

    MathSciNet  MATH  Google Scholar 

  53. Pattle, R. E.Diffusion from an instantaneous point source with a concentration-dependent coefficientQuart. J. Mech. Appl. Math.12(1959), 407–409.

    Article  MathSciNet  MATH  Google Scholar 

  54. Pierre, M.Uniqueness of the solutions of u t -A(p(u) = 0 with initial datum a measureNonlinear Anal.6(1982), 175–187.

    Article  MathSciNet  MATH  Google Scholar 

  55. Rieder, G.Mathematical contributions to Thomas-Fermi theoryHouston J. Math.16(1990), 179–201.

    MathSciNet  Google Scholar 

  56. STAMPACCHIA, G.“Équations elliptiques du second ordre a coefficients discontinus”Presses de l’Université de Montréal, 1966.

    Google Scholar 

  57. Trudinger, N.Linear elliptic operators with measurable coefficientsAnn. Scuola Norm. Sup. Pisa27(1973), 265–308.

    MathSciNet  MATH  Google Scholar 

  58. VÁzquez, J. L., [1]On a semilinear equation in R2 involving bounded measuresProc. Roy. Soc. Edinburgh95A(1983), 181–202; [2], Astrong maximum principle for some quasilinear elliptic equationsAppl. Math. Optim.12(1984), 191–202.

    Article  MATH  Google Scholar 

  59. Veron, L., [1]Singularités éliminables d’équations nonlinéaires J.Differential Equations41(1981), 87–95; [2]Singular solutions of some nonlinear elliptic equationsNonlinear Anal.5(1981), 225–242; [3]Comportement asymptotique des solutions d’équations elliptiques semi-linéaires dansR’’, Ann. Mat. Pura Appl.127(1981), 25–50;[4] “Singularities of solutions of second order quasilinear equations”Pitman Research Notes, vol.353Longman, 1996.

    Google Scholar 

  60. Zel’Dovich, Y. B. and KOMPANEEC, A. S.On the theory of propagation of heat with the heat conductivity depending upon the temperatureCollection in honor of the seventieth birthday of academician A. F. Ioffe, pp. 61–71. Izdat. Akad. Nauk USSR, Moscow, 1950.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departement de Mathématiques, Université de Franche-comté, Besançon, 25030, Cedex

    Philippe Bénilan

  2. Analyse Numérique, Université P. et M. Curie, B. C. 187 4 Pl. Jussieu, Paris Cedex 05, 75252, France

    Haïm Brezis

  3. Dept. of Math. Hill Center, Busch Campus, Rutgers University, 110 Frelinghuysen RD, Piscataway, NJ, 08854, USA

    Haïm Brezis

Authors
  1. Philippe Bénilan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Haïm Brezis
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Applied Analysis, University of Ulm, 89069, Ulm, Germany

    Wolfgang Arendt

  2. Analyse Numérique, Université Pierre et Marie Curie, B.C. 187, 4, place Jussieu, 75252, Paris Cedex 05, France

    Haïm Brézis

  3. Department of Mathematics Hill Center, Busch Campus, Rutgers University, 110 Frelinghuysen RD, Piscataway, NJ, 08854, USA

    Haïm Brézis

  4. Campus de Ker Lann, Antenne de Bretagne de l’ENS Cachan, 35170, Bruz, France

    Michel Pierre

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer Basel AG

About this chapter

Cite this chapter

Bénilan, P., Brezis, H. (2004). Nonlinear problems related to the Thomas-Fermi equation. In: Arendt, W., Brézis, H., Pierre, M. (eds) Nonlinear Evolution Equations and Related Topics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7924-8_35

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-7924-8_35

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-7643-7107-4

  • Online ISBN: 978-3-0348-7924-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation