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Rearrangements of functions and partial differential equations

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Nonlinear Diffusion Problems

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References

  1. T. Aubin, Problèmes isopérimetriques et espaces de Sobolev (J. Diff. Geometry, 11, 1976).

    Google Scholar 

  2. H.J. Brascamp-E.H. Lieb-J.M. Luttinger, A general rearrangement inequality for multiple integrals (J. Functional Anal. 17, 1974).

    Google Scholar 

  3. G. Chiti, Rearrangements of functions and convergence in Orlicz spaces (Appl. Anal. 9, 1979).

    Google Scholar 

  4. G. Faber, Beweis dass unter allen homogenen membranen von gleicher fläche und gleicher spannung die kreisförmige den tiefsten grundton gibt (Sitzungsber. Bayer. Akad. Wiss., Math.-Naturwiss.Kl., 1923).

    Google Scholar 

  5. A. Friedman-B.McLeod, Strict inequalities for integrals of decreasingly rearranged functions (to appear).

    Google Scholar 

  6. A.M. Garsia-E. Rodemich, Monotonicity of certain functionals under rearrangements (Ann. Inst. Fourier Grenoble 24, 1974).

    Google Scholar 

  7. Hardy-Littlewood-Polya, Inequalities (Cambridge Univ. Press, 1964).

    Google Scholar 

  8. K. Hildén, Symmetrization of functions in Sobolev spaces and the isoperimetric inequality (Manuscripta Math. 18, 1976).

    Google Scholar 

  9. B. Kawohl, Rearrangements and convexity of level sets in PDE (Lecture Notes in Math. 1150, Springer Verlag 1985).

    Google Scholar 

  10. E. Krahn, Ãœber eine von Rayleigh formulierte Minimaleigenschaft des Kreises (Math. Ann. 94, 1924).

    Google Scholar 

  11. E.H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation (Studies in Appl. Math. 57, 1977).

    Google Scholar 

  12. E.H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities (Ann. of Math. 118, 1983).

    Google Scholar 

  13. G. Polya-G.Szegö, Isoperimetric inequalities in mathematical physics (Princeton Univ. Press, 1951).

    Google Scholar 

  14. F. Riesz, Sur une inegalité intégrale (J. London Math. Soc. 5, 1930).

    Google Scholar 

  15. C. Somigliana, Sulle funzioni reali d'una variabile, Considerazioni sulle funzioni ordinate (Rendiconti R. Accademia dei Lincei, vol. 8, 1899).

    Google Scholar 

  16. E. Sperner, Zur Symmetrisierung für Funktionen auf Sphären (Math. Z. 134, 1973).

    Google Scholar 

  17. E. Sperner, Symmetrisierung für Funktionen mehrerer reeller Variablen (Manuscripta Math. 11, 1974).

    Google Scholar 

  18. W. Spiegel, Ãœber die Symmetrisierung stetiger Funktionen im euklidischen Raum (Archiv der Math. 24, 1973).

    Google Scholar 

  19. G. Talenti, Best constant in Sobolev inequality (Ann. Mat. Pura Appl. 110, 1976).

    Google Scholar 

  20. A. Alvino, Formule di maggiorazione e regolarizzazione per soluzioni di equazioni ellittiche del secondo ordine in un caso limite Rend. Acc. Naz. Lincei, ser.8, vol. 62, 1977).

    Google Scholar 

  21. A. Alvino-G. Trombetti, Equazioni ellittiche con termini di ordine inferiore e riordinamenti (Pend. Acc. Naz. Lincei, ser.8, vol. 66, 1979).

    Google Scholar 

  22. A. Alvino-G. Trombetti, Sulle migliori costanti di maggiorazione per una classe di equazioni ellittiche degeneri (Ricerche di Mat. 27, 1979).

    Google Scholar 

  23. A. Alvino-G. Trombetti, Su una classe di equazioni ellittiche non lineari degeneri (Ricerche di Mat. 29, 1980).

    Google Scholar 

  24. A. Alvino-G. Trombetti, Sulle migliori costanti di maggiorazione per una classe di equazioni ellittiche degeneri e non (Ricerche di Mat. 30, 1981).

    Google Scholar 

  25. A. Alvino-G. Trombetti, A lower bound for the first eigenvalue of an elliptic operator (J. Math. Analysis and Applications 94, 1983).

    Google Scholar 

  26. A. Alvino-P.L. Lions-G. Trombetti, A remark on comparison results via symmetrization (in corso di stampa).

    Google Scholar 

  27. C. Bandle, Bounds for the solutions of boundary value problems (J. Math. Anal. Appl. 54, 1976).

    Google Scholar 

  28. C. Bandle, On symmetrizations in parabolic equations (J. Analyse Math. 30, 1976).

    Google Scholar 

  29. C. Bandle, Estimates for the Green's functions of elliptic operators (SIAM J. Math. Anal. 9, 1978).

    Google Scholar 

  30. C. Bandle-J. Mossino, Application du réarrangement à une inéquation variationnelle (C.R.A.S. Paris, t. 296, Série I, 1983).

    Google Scholar 

  31. C. Bandle-R.P. Sperb-I. Stakgold, The single, steady-state irreversible reaction (in corso di stampa).

    Google Scholar 

  32. P. Buonocore, Sulla simmetrizzazione in equazioni paraboliche degeneri (Boll. U.M.I., (6) 3-B, 1984).

    Google Scholar 

  33. G. Chiti, Norme di Orlicz delle soluzioni di una classe di equazioni ellittiche (Boll. U.M.I. (5) 16-A, 1979).

    Google Scholar 

  34. G. Chiti, An isoperimetric inequality for the eigenfunctions of linear second order elliptic operators (Boll. U.M.I. (6) A-1, 1982).

    Google Scholar 

  35. G. Chiti, A reverse Hölder inequality for the eigenfunctions of linear second order elliptic equations (Z.A.M.P. 33, 1982).

    Google Scholar 

  36. P.S. Crooke-R.P. Sperb, Isoperimetric inequalities in a class of nonlinear eigenvalue problems (SIAM J. Math. Anal. 9, 1978).

    Google Scholar 

  37. E. Giarrusso-D. Nunziante, Symmetrization in a class of first-order Hamilton-Jacobi equations (Nonlinear Analysis T.M.A. 8, 1984).

    Google Scholar 

  38. E. Giarrusso-D. Nunziante, Comparison theorems for a class of first-order Hamilton-Jacobi equations (in corso di stampa).

    Google Scholar 

  39. B. Kawohl, On the isoperimetric nature of a rearrangement inequality and its consequences for some variational problems (LCDS Report 84-4, Providence 1984).

    Google Scholar 

  40. P. Laurence-E.W. Stredulinsky, A new approach to queer differential equations (to appear in Comm. Pure Appl. Math.)

    Google Scholar 

  41. P.L. Lions, Quelques remarques sur la symétrisation de Schwarz (Nonlinear partial differential equations and their applications, Collège de France Seminars, vol. 1, Pitman 1981).

    Google Scholar 

  42. C. Maderna, Optimal problems for a certain class of nonlinear Dirichlet problems (Boll. U.M.I., Suppl. 1, 1980).

    Google Scholar 

  43. C. Maderna, On level sets of Poisson integrals in disks and sectors (Boll. U.M.I. (6) ser. C-2, 1983).

    Google Scholar 

  44. C. Maderna-S. Salsa, Symmetrization in Neumann problems (Appl. Anal. 9, 1979).

    Google Scholar 

  45. C. Maderna-S. Salsa, A priori bounds in nonlinear Neumann problems (Boll. U.M.I. (5) 16-B, 1979).

    Google Scholar 

  46. C. Maderna-S. Salsa, Sharp estimates for solutions to a certain type of singular elliptic boundary value problem in two dimensions (Appl. Anal. 12, 1981).

    Google Scholar 

  47. C. Maderna-S. Salsa, Some special properties of solutions to obstacle problems (Rend. Sem. Mat. Univ. Padova 71, 1984).

    Google Scholar 

  48. J. Mossino, A priori estimates for a model of Grad-Mercier type in plasma confinement (Appl. Anal. 13, 1982).

    Google Scholar 

  49. J. Mossino, A generalization of the Payne-Rayner isoperimetric inequality (Boll. U.M.I. (6) A-2, 1983).

    Google Scholar 

  50. J. Mossino, Inégalités isopérimetriques et applications en physique (Herman, 1984).

    Google Scholar 

  51. J. Mossino-J.M. Rakotoson, Isoperimetric inequalities in parabolic equations (to appear).

    Google Scholar 

  52. J. Mossino-R. Temam, Directional derivative of the increasing rearrangement mapping and application to a queer differential equation in plasma physics (Duke Math. J. 48, 1981).

    Google Scholar 

  53. R.P. Sperb, Maximum principles and their applications (Academic Press, 1981).

    Google Scholar 

  54. E. Sperner, Spherical symmetrization and eigenvalue estimates (Math. Z. 176, 1981).

    Google Scholar 

  55. G. Talenti, Elliptic equations and rearrangements (Ann. Scuola Norm. Sup. Pisa (4) 3, 1976).

    Google Scholar 

  56. G. Talenti, Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces (Ann. Mat. Pura Appl. (4) 120, 1979).

    Google Scholar 

  57. G. Talenti, Some estimates for solutions to Monge-Ampère equations in dimension two (Ann. Scuola Norm. Sup. Pisa (4) 8, 1981).

    Google Scholar 

  58. G. Talenti, On the first eigenvalue of the clamped plate (Ann. Mat. Pura Appl. (4) 129, 1981).

    Google Scholar 

  59. G. Talenti, A note on the Gauss curvature of harmonic and minimal surfaces (Pacific J. Math. 101, 1982).

    Google Scholar 

  60. G. Talenti, Linear elliptic p.d.e.'s: level sets, rearrangements and a priori estimates of solutions (to appear).

    Google Scholar 

  61. J.L. Vazquez, Symétrisation pour ut = Δϕ(u) et applications (C.R.A.S.P., t. 295, 1982).

    Google Scholar 

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Antonio Fasano Mario Primicerio

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© 1986 Springer-Verlag

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Talenti, G. (1986). Rearrangements of functions and partial differential equations. In: Fasano, A., Primicerio, M. (eds) Nonlinear Diffusion Problems. Lecture Notes in Mathematics, vol 1224. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072690

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  • DOI: https://doi.org/10.1007/BFb0072690

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