Nonlinear problems related to the Thomas-Fermi equation

Journal of Evolution Equations

Volumn 3, Issue 4, 2003, Pages 673-770

  Bénilan, Philippe ^a   Brezis, Haǐm ^b,c  

^a UNIVERSITÉ DE FRANCHE COMTÉ   (France)

^b SORBONNE UNIVERSITÉ   (France)

^c RUTGERS UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0141692156     PISSN: 14243199     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00028-003-0117-8     Document Type: Article

References (88)

1. ANCONA, A., Une propriété d'invariance des ensembles absorbants par perturbation d'un opérateur elliptique, Comm. PDE 4 (1979), 321-337.
2. BAMBERGER, A., Étude de deux équations nonlinéaires avec une masse de Dirac au second membre, Rapport 13 du Centre de Mathématiques Appliquées de l'École Polytechnique, Oct. 1976.
3. BARAS, P. and PIERRE, M., [1], Singularités éliminables pour des équations semi-linéaires, Ann. Inst. Fourier (Grenoble) 34 (1984), 185-206; [2], Problèmes paraboliques semi-linéaires avec données mesures, Applicable Anal. 18 (1984), 111-149.
4. BARAS, P. and PIERRE, M., [1], Singularités éliminables pour des équations semi-linéaires, Ann. Inst. Fourier (Grenoble) 34 (1984), 185-206; [2], Problèmes paraboliques semi-linéaires avec données mesures, Applicable Anal. 18 (1984), 111-149.
5. BARENBLATT, G. I., "Scaling, self-similarity, and intermediate asymptotics". Cambridge Texts in Applied Mathematics, 14. Cambridge University Press, Cambridge, 1996.
6. BARENBLATT, G. I. and SIVASHINSKI, G. I., Self-similar solutions of the second kind in nonlinear filtration, Prikl. Mat. Meh. 33 (1969), 861-870 (Russian); translated as J. Appl. Math. Mech. 33 (1969), 836-845.
7. BARENBLATT, G. I. and SIVASHINSKI, G. I., Self-similar solutions of the second kind in nonlinear filtration, Prikl. Mat. Meh. 33 (1969), 861-870 (Russian); translated as J. Appl. Math. Mech. 33 (1969), 836-845.
8. BAXTER, J. R., Inequalities for potentials of particle systems, Illinois J. Math. 24 (1980), 645-652.
9. BENGURIA, R., The von Weizsäcker and exchange corrections in the Thomas-Fermi theory, Ph.D. Dissertation, Princeton Univ. 1979.
10. BENGURIA, R. and YÁÑEZ, J., Variational principle for the chemical potential in the Thomas-Fermi model, J. Phys. A 31 (1998), 585-593.
11. BENGURIA, R., BREZIS, H. and LIEB, E., The Thomas-Fermi-von Weizsäcker theory of atoms and molecules, Comm. Math. Phys. 79 (1981), 167-180.
12. N), Ann. Sc. Norm. Sup. Pisa, Serie IV, 2 (1975), 523-555.
13. BÉNILAN, PH., GOLSTEIN, G. and GOLDSTEIN, J., A nonlinear elliptic system arising in electron density theory, Comm. PDE 17 (1992), 2907-2917.
14. BOCCARDO, L., DALL'AGLIO, A., GALLOUËT, T. and ORSINA, L., Nonlinear parabolic equations with measure data, J. Funct. Anal. 147 (1976), 237-258.
15. BOCCARDO, L. and GALLOUËT, T., [1], Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989), 149-169; [2] Nonlinear elliptic equations with right-hand side measures, Comm. PDE 17 (1992), 641-655.
16. BOCCARDO, L. and GALLOUËT, T., [1], Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989), 149-169; [2] Nonlinear elliptic equations with right-hand side measures, Comm. PDE 17 (1992), 641-655.
17. BOCCARDO, L., GALLOUËT, T. and ORSINA, L., [1], Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré, Anal. Non Linéaire 13 (1996), 539-551; [2], Existence and nonexistence of solutions for some nonlinear elliptic equations, J. Anal. Math. 73 (1997), 203-223.
18. BOCCARDO, L., GALLOUËT, T. and ORSINA, L., [1], Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré, Anal. Non Linéaire 13 (1996), 539-551; [2], Existence and nonexistence of solutions for some nonlinear elliptic equations, J. Anal. Math. 73 (1997), 203-223.
19. BREAZNA, A., GOLDSTEIN, G. and GOLDSTEIN, J., Parameter dependence in Thomas-Fermi theory, Comm. Appl. Anal. 5 (2001), 421-432.
20. N without conditions at infinity, Appl. Math. Optim. 12 (1984), 271-282.
21. N without conditions at infinity, Appl. Math. Optim. 12 (1984), 271-282.
22. N without conditions at infinity, Appl. Math. Optim. 12 (1984), 271-282.
23. N without conditions at infinity, Appl. Math. Optim. 12 (1984), 271-282.
24. N without conditions at infinity, Appl. Math. Optim. 12 (1984), 271-282.
25. N without conditions at infinity, Appl. Math. Optim. 12 (1984), 271-282.
26. N without conditions at infinity, Appl. Math. Optim. 12 (1984), 271-282.
27. N without conditions at infinity, Appl. Math. Optim. 12 (1984), 271-282.
28. BREZIS, H. and FRIEDMAN, A., Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. 62 (1983), 73-97.
29. BREZIS, H. and LIEB, E., Long range atomic potentials in Thomas-Fermi theory, Comm. Math. Phys. 65 (1979), 231-246.
30. BREZIS, H., MARCUS, M. and PONCE, A. C., Nonlinear elliptic equations with measures revisited (to appear).
31. BREZIS, H. and NIRENBERG, L., Removable singularities for nonlinear elliptic equations, Topol. Methods Nonlinear Anal. 9 (1997), 201-219.
32. BREZIS, H. and OSWALD, L., Singular solutions for some semilinear elliptic equations, Arch. Rat. Mech. Anal. 99 (1987), 249-259.
33. BREZIS, H., PELETIER, L. and TERMAN, D., A very singular solution of the heat equation with absorption, Arch. Rat. Mech. Anal. 95 (1986), 185-209.
34. BREZIS, H. and PONCE, A. C., Remarks on the strong maximum principle, Diff. Int. Equations 16 (2003), 1-12.
35. BREZIS, H. and SERFATY, S., A variational formulation for the two-sided obstacle problem with measure data, Commun. Contemp. Math. 4 (2002), 357-374.
36. 1, J. Math. Soc. Japan 25 (1973), 565-590.
37. BREZIS, H. and VÉRON, L., Removable singularities of some nonlinear elliptic equations, Arch. Rat. Mech. Anal. 75 (1980), 1-6.
38. BROWDER, F., Nonlinear elliptic boundary value problems, Bull. Amer. Math. Soc. 69 (1963), 862-874.
39. CAFFARELLI, L. and FRIEDMAN, A., The free boundary in the Thomas-Fermi model, J. Diff. Eq. 32 (1979), 335-356.
40. DALL'AGLIO, P. and DAL MASO, G., Some properties of the solutions of obstacle problems with measure data, Ricerche Mat. suppl. 48 (1999), 99-116.
41. DUNFORD, N. and SCHWARTZ, J. T., "Linear operators", Part I, Wiley Interscience, 1958.
42. DYNKIN, E., "Diffusions, superdiffusions and partial differential equations", American Mathematical Society Colloquium Publications, 50. American Mathematical Society, Providence, RI, 2002.
43. EKELAND, I., On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353.
44. FRIEDMAN, A. and KAMIN, S., The asymptotic behavior of gas in an n-dimensional porous medium, Trans. Amer. Math. Soc. 262 (1980), 551-563.
45. FUKUSHIMA, M., SATO, K. and TANIGUCHI, S., On the closable part of pre-Dirichlet forms and the fine supports of underlying measures, Osaka Math. J. 28 (1991), 517-535.
46. 1, Boll. Un. Mat. Ital. 4 (1985), 123-131.
47. 1, Boll. Un. Mat. Ital. 4 (1985), 123-131.
48. 1, Boll. Un. Mat. Ital. 4 (1985), 123-131.
49. GMIRA, A. and VÉRON, L., Boundary singularities of solutions of nonlinear elliptic equations, Duke J. Math. 64 (1991), 271-324.
50. GOLDSTEIN, G., GOLDSTEIN, J. and JIA, W., Thomas-Fermi theory with magnetic fields and the Fermi-Amaldi correction, Diff. Int. Equations 8 (1995), 1305-1316.
51. GOLDSTEIN, J. and RIEDER, G., [1], A rigorous Thomas-Fermi theory for atomic systems, J. Math. Phys 28 (1987), 1198-1202; [2], Spin polarized Thomas-Fermi theory, J. Math. Phys 29 (1988), 709-716; [3], Thomas-Fermi theory with an external magnetic field, J. Math. Phys. 32 (1991), 2901-2917.
52. GOLDSTEIN, J. and RIEDER, G., [1], A rigorous Thomas-Fermi theory for atomic systems, J. Math. Phys 28 (1987), 1198-1202; [2], Spin polarized Thomas-Fermi theory, J. Math. Phys 29 (1988), 709-716; [3], Thomas-Fermi theory with an external magnetic field, J. Math. Phys. 32 (1991), 2901-2917.
53. GOLDSTEIN, J. and RIEDER, G., [1], A rigorous Thomas-Fermi theory for atomic systems, J. Math. Phys 28 (1987), 1198-1202; [2], Spin polarized Thomas-Fermi theory, J. Math. Phys 29 (1988), 709-716; [3], Thomas-Fermi theory with an external magnetic field, J. Math. Phys. 32 (1991), 2901-2917.
54. HILLE, E., Some aspects of the Thomas-Fermi equation, J. Anal. Math. 23 (1970), 147-170.
55. KAMENOMOSTSKAIA (KAMIN), S. L., [1], Equation of the elastoplastic mode of filtration, Prikl. 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 equation, Israel J. Math. 14 (1973), 76-87.
56. KAMENOMOSTSKAIA (KAMIN), S. L., [1], Equation of the elastoplastic mode of filtration, Prikl. 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 equation, Israel J. Math. 14 (1973), 76-87.
57. KAMENOMOSTSKAIA (KAMIN), S. L., [1], Equation of the elastoplastic mode of filtration, Prikl. 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 equation, Israel J. Math. 14 (1973), 76-87.
58. KAMIN, S. and PELETIER, L. A., Singular solutions of the heat equation with absorption, Proc. Amer. Math. Soc. 95 (1985), 145-158.
59. KAMIN, S., PELETIER, L. A. and VÁZQUEZ, J. L., [1], On the Barenblatt equation of elastoplastic filtration, Indiana Univ. Math. J. 40 (1991), 1333-1362; [2], Classification of singular solutions of a nonlinear heat equation, Duke Math. J. 58 (1989), 601-615.
60. KAMIN, S., PELETIER, L. A. and VÁZQUEZ, J. L., [1], On the Barenblatt equation of elastoplastic filtration, Indiana Univ. Math. J. 40 (1991), 1333-1362; [2], Classification of singular solutions of a nonlinear heat equation, Duke Math. J. 58 (1989), 601-615.
61. KATO, T., Schrödinger operators with singular potentials, Israel J. Math. 13 (1972), 135-148.
62. LABUTIN, D., Wiener regularity for large solutions of nonlinear equations, Arkiv för Math, (to appear).
63. 2 in a planar domain, Comm. Pure Appl. Math. 50 (1997), 69-103.
64. 2 in a planar domain, Comm. Pure Appl. Math. 50 (1997), 69-103.
65. LERAY, J. and LIONS, J. L., Quelques résultats de Visik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-Browder, Bull. Soc. Math. Fr. 93 (1965), 97-107.
66. LIEB, E. H., [1], The stability of matter, Rev. Mod. Phys. 48 (1976), 553-569; [2], Thomas-Fermi and related theories of atoms and molecules, Rev. 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.
67. LIEB, E. H., [1], The stability of matter, Rev. Mod. Phys. 48 (1976), 553-569; [2], Thomas-Fermi and related theories of atoms and molecules, Rev. 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.
68. LIEB, E. H., [1], The stability of matter, Rev. Mod. Phys. 48 (1976), 553-569; [2], Thomas-Fermi and related theories of atoms and molecules, Rev. 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.
69. LIEB, E. H. and SIMON, B., The Thomas-Fermi theory of atoms, molecules and solids, Advances in Math. 23 (1977), 22-116.
70. LOEWNER, C. and NIRENBERG, L., Partial differential equations invariant under conformal or projective transformations, in "Contributions to Analysis", Acad. Press, 1974, 245-272.
71. MARCUS, M. and VÉRON, L., [1], The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case, Arch. Rat. Mech. Anal. 144 (1998), 201-231 ; [2], The boundary trace of positive solutions of semilinear elliptic equations: the supercritical case, J. Math. Pures Appl. 77 (1998), 481-524; [3], Removable singularities and boundary traces, J. Math. Pures Appl. 80 (2001), 879-900; [4], Capacitary estimates of solutions of a class of nonlinear elliptic equations, C. R. Acad. Sc. Paris 336 (2003), 913-918.
72. MARCUS, M. and VÉRON, L., [1], The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case, Arch. Rat. Mech. Anal. 144 (1998), 201-231 ; [2], The boundary trace of positive solutions of semilinear elliptic equations: the supercritical case, J. Math. Pures Appl. 77 (1998), 481-524; [3], Removable singularities and boundary traces, J. Math. Pures Appl. 80 (2001), 879-900; [4], Capacitary estimates of solutions of a class of nonlinear elliptic equations, C. R. Acad. Sc. Paris 336 (2003), 913-918.
73. MARCUS, M. and VÉRON, L., [1], The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case, Arch. Rat. Mech. Anal. 144 (1998), 201-231 ; [2], The boundary trace of positive solutions of semilinear elliptic equations: the supercritical case, J. Math. Pures Appl. 77 (1998), 481-524; [3], Removable singularities and boundary traces, J. Math. Pures Appl. 80 (2001), 879-900; [4], Capacitary estimates of solutions of a class of nonlinear elliptic equations, C. R. Acad. Sc. Paris 336 (2003), 913-918.
74. MARCUS, M. and VÉRON, L., [1], The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case, Arch. Rat. Mech. Anal. 144 (1998), 201-231 ; [2], The boundary trace of positive solutions of semilinear elliptic equations: the supercritical case, J. Math. Pures Appl. 77 (1998), 481-524; [3], Removable singularities and boundary traces, J. Math. Pures Appl. 80 (2001), 879-900; [4], Capacitary estimates of solutions of a class of nonlinear elliptic equations, C. R. Acad. Sc. Paris 336 (2003), 913-918.
75. ORSINA, L. and PRIGNET, A., Nonexistence of solutions for some nonlinear elliptic equations involving measures, Proc. Royal Soc. Edinburgh 130 (2000), 561-592.
76. OSWALD, L., Isolated positive singularities for a nonlinear heat equation, Houston J. Math. 14 (1988), 543-572.
77. PATTLE, R. E., Diffusion from an instantaneous point source with a concentration-dependent coefficient, Quart. J. Mech. Appl. Math. 12 (1959), 407-409.
78. t - Δφ(u) = 0 with initial datum a measure, Nonlinear Anal. 6 (1982), 175-187.
79. RIEDER, G., Mathematical contributions to Thomas-Fermi theory, Houston J. Math. 16 (1990), 179-201
80. STAMPACCHIA, G., "Équations elliptiques du second ordre à coefficients discontinus", Presses de l'Université de Montréal, 1966.
81. TRUDINGER, N., Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa 27 (1973), 265-308.
82. 2 involving bounded measures, Proc. Roy. Soc. Edinburgh, 95A (1983), 181-202; [2], A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191-202.
83. 2 involving bounded measures, Proc. Roy. Soc. Edinburgh, 95A (1983), 181-202; [2], A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191-202.
84. N, Ann. Mat. Pura Appl. 127 (1981), 25-50; [4], "Singularities of solutions of second order quasilinear equations", Pitman Research Notes, vol. 353, Longman, 1996.
85. N, Ann. Mat. Pura Appl. 127 (1981), 25-50; [4], "Singularities of solutions of second order quasilinear equations", Pitman Research Notes, vol. 353, Longman, 1996.
86. N, Ann. Mat. Pura Appl. 127 (1981), 25-50; [4], "Singularities of solutions of second order quasilinear equations", Pitman Research Notes, vol. 353, Longman, 1996.
87. N, Ann. Mat. Pura Appl. 127 (1981), 25-50; [4], "Singularities of solutions of second order quasilinear equations", Pitman Research Notes, vol. 353, Longman, 1996.
88. ZEL'DOVICH, Y. B. and KOMPANEEC, A. S., On the theory of propagation of heat with the heat conductivity depending upon the temperature, Collection in honor of the seventieth birthday of academician A. F. Ioffe, pp. 61-71. Izdat. Akad. Nauk USSR, Moscow, 1950.