Metodi di simmetrizzazione nelle equazioni alle derivate parziali
Trombetti, Guido
Bollettino dell'Unione Matematica Italiana, Tome 3-A (2000), p. 601-634 / Harvested from Biblioteca Digitale Italiana di Matematica

A survey of the fundamental ideas which are the base of the socalled symmetrization method; a priori estimates in partial differential equations.

Publié le : 2000-10-01
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     author = {Guido Trombetti},
     title = {Metodi di simmetrizzazione nelle equazioni alle derivate parziali},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {3-A},
     year = {2000},
     pages = {601-634},
     zbl = {0963.35006},
     mrnumber = {1801615},
     language = {it},
     url = {http://dml.mathdoc.fr/item/BUMI_2000_8_3B_3_601_0}
}
Trombetti, Guido. Metodi di simmetrizzazione nelle equazioni alle derivate parziali. Bollettino dell'Unione Matematica Italiana, Tome 3-A (2000) pp. 601-634. http://gdmltest.u-ga.fr/item/BUMI_2000_8_3B_3_601_0/

[1] Adams, D. R., A sharp inequality of J. Moser for higher order derivatives, Ann. of Math., 128, 1988. | MR 960950 | Zbl 0672.31008

[2] Aguilar, J. A.-Peral, I., An a priori estimate for the N-Laplacian, C. R. Acad. Sci. Paris, Sér. I Math., t. 319, 1994. | MR 1288396 | Zbl 0804.35044

[3] Alberico, A.-Ferone, V., Regularity properties of solutions of elliptic equations in R2 in limit cases, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (9) 6, 1995. | MR 1382708 | Zbl 0860.35015

[4] Almgren, F. J. Jr-Lieb, E. H., Symmetric decreasing rearrangement is sometimes continous, J. Amer. Math. Soc., vol. 2, no. 4, 1989. | MR 1002633 | Zbl 0688.46014

[5] Almgren, F. J. Jr-Lieb, E. H., Symmetric decreasing rearrangement can be discontinuous, Bull. Amer. Math. Soc. (N. S.), 20, no. 2, 1989. | MR 968686 | Zbl 0692.46028

[6] Alvino, A., Formule di maggiorazione e regolarizzazione per soluzioni di equazioni ellittiche del secondo ordine in un caso limite, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 62, 1977. | Zbl 0371.35009

[7] Alvino, A., Sulla diseguaglianza di Sobolev in spazi di Lorentz, Boll. U.M.I. (5), 14-A, 1977. | MR 438106 | Zbl 0352.46020

[8] Alvino, A., Un caso limite della diseguaglianza di Sobolev in spazi di Lorentz, Rend. Accad. Sci. Fis. Mat. Napoli, 44, 1977. | Zbl 0412.46024

[9] Alvino, A.-Boccardo, L.-Ferone, V.-Orsina, L.-Trombetti, G., in preparazione.

[10] Alvino, A.-Díaz, J. I.-Lions, P. L.-Trombetti, G., Équations elliptiques et symétrisation de Steiner, C. R. Acad. Sci. Paris, Sér. I Math., 314, 1992. | MR 1168527 | Zbl 0795.35022

[11] Alvino, A.-Díaz, J. I.-Lions, P. L.-Trombetti, G., Elliptic equations and Steiner symmetrization, Comm. Pure Appl. Math., 49 (3), 1996. | MR 1374171 | Zbl 0856.35034

[12] Alvino, A.-Ferone, V.-Lions, P. L.-Trombetti, G., Convex symmetrization and applications, Ann. Inst. H. Poincaré - Analyse Nonlinéaire, 14, 1997. | MR 1441395 | Zbl 0877.35040

[13] Alvino, A.-Ferone, V.-Trombetti, G., Moser-type inequalities in Lorentz spaces, Potential Analysis, 5, 1996. | MR 1389498 | Zbl 0856.46020

[14] Alvino, A.-Ferone, V.-Trombetti, G., A priori estimates for a class of non uniformly elliptic equations, Atti Sem. Mat. Fis. Univ. Modena, 46-suppl., 1998. | MR 1645729 | Zbl 0911.35025

[15] Alvino, A.-Ferone, V.-Trombetti, G., On the properties of some nonlinear eigenvalues, SIAM J. Math. Anal., 29, 1998. | MR 1616519 | Zbl 0908.35094

[16] Alvino, A.-Ferone, V.-Trombetti, G., Estimates for the gradient of nonlinear elliptic equations with L1 data, in corso di stampa su Ann. Mat. Pura Appl.. | Zbl pre02216904

[17] Alvino, A.-Lions, P. L.-Matarasso, S.-Trombetti, G., Comparison results for solutions to elliptic problems via symmetrization, Ann. Inst. H. Poincaré - Analyse Nonlinéaire, 16, 2, 1999. | MR 1674768 | Zbl 0924.35038

[18] Alvino, A.-Lions, P. L.-Trombetti, G., Comparaison des solutions d'équations paraboliques et elliptiques par symétrisation. Une méthode nouvelle, C. R. Acad. Sci. Paris, Sér. I Math., 303, no. 20, 1986. | MR 877184 | Zbl 0617.35008

[19] Alvino, A.-Lions, P. L.-Trombetti, G., On optimization problems with prescribed rearrangements, Nonlinear Analysis T.M.A, 13, 1989. | MR 979040 | Zbl 0678.49003

[20] Alvino, A.-Lions, P. L.-Trombetti, G., Comparison results for elliptic and parabolic equations via Schwartz symmetrization, Ann. Inst. H. Poincaré - Analysis Nonlinéaire, 7, 1990. | MR 1051227 | Zbl 0703.35007

[21] Alvino, A.-Lions, P. L.-Trombetti, G., Comparison results for elliptic and parabolic equations via Symmetrization: a new approach, Diff. Int. Eq., 4, 1991. | MR 1079609 | Zbl 0735.35003

[22] Alvino, A.-Matarasso, S.-Trombetti, G., Variational inequalities and rearrangements, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (9), 3, 1992. | MR 1203167 | Zbl 0803.35060

[23] Alvino, A.-Trombetti, G., Sulle migliori costanti di maggiorazione per una classe di equazioni ellittiche degeneri, Ricerche di Matematica, 27, 1978. | MR 527431 | Zbl 0403.35027

[24] Alvino, A.-Trombetti, G., Equazioni ellittiche con termini di ordine inferiore e riordinamenti, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 66, 1979. | MR 606083 | Zbl 0455.35052

[25] Alvino, A.-Trombetti, G., Sulle migliori costanti di maggiorazione per una classe di equazioni ellittiche degeneri e non, Ricerche di Matematica, 30, 1981. | MR 657553 | Zbl 0489.35013

[26] Alvino, A.-Trombetti, G., A lower bound for the first eigenvalue of an elliptic operator, Math. Anal. Appl., 94, 1983. | MR 706368 | Zbl 0525.35063

[27] Alvino, A.-Trombetti, G., Isoperimetric inequalities connected with torsion problem and capacity, Boll. U.M.I. (6), 4-B, 1985. | MR 831290

[28] Andrievskiĭ, V.-Hansen, W.-Nadirashvili, N., Isoperimetric inequalities for capacities in the plane, Math. Ann., 292, no. 2, 1992. | MR 1149031 | Zbl 0739.31001

[29] Aronsson, G., An integral inequality and plastic torsion, Arch. Rational Mech. Anal., 7, 1989. | MR 540220 | Zbl 0415.49005

[30] Ashbaugh, M. S.-Benguria, R. D., Proof of the Payne-Pólya-Weinberger conjecture, Bull. Amer. Math. Soc. (N. S.), 25, no. 1, 1991. | MR 1085824 | Zbl 0736.35075

[31] Ashbaugh, M. S.-Benguria, R. D., A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions, Ann. of Math., (2) 135, no. 3, 1992. | MR 1166646 | Zbl 0757.35052

[32] Ashbaugh, M. S.-Benguria, R. D., Isoperimetric bounds for higher eigenvalue ratios for the n-dimensional fixed membrane problem, Proc. Royal Soc. Edinburgh, 123-A, no. 6, 1993. | MR 1263898 | Zbl 0819.35104

[33] Ashbaugh, M. S.-Benguria, R. D., On Rayleigh's conjecture for the clamped plate and its generalization to three dimensions, Duke Math. J., 78, no. 1, 1995. | MR 1328749 | Zbl 0833.35035

[34] Ashbaugh, M. S.-Laugesen, R. S., Fundamental tones and buckling loads of clamped planes, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 23, no. 2, 1996. | MR 1433428 | Zbl 0891.73028

[35] Aubin, T., Problèmes isopèrimetriques et espaces de Sobolev, J. Diff. Geometry, 11, 1976. | MR 448404 | Zbl 0371.46011

[36] Aubin, T., Meilleures constantes dans le théorème d'inclusion de Sobolev et un théorème de Fredholm non linéaire pour la transformation conforme de la courbure scalaire, J. Funct. Anal., 32, 1979. | MR 534672 | Zbl 0411.46019

[37] Baernstein, A. Ii, A unified approach to symmetrization, Partial differential equations of elliptic type (A. Alvino, E. Fabes & G. Talenti eds.), Symposia Math., 35, Cambridge University Press, 1998. | Zbl 0830.35005

[38] Baernstein, A. Ii-Taylor, B. A., Spherical rearrangements, subharmonic functions, and *-functions in n-space, Duke Math. J., 43, no. 2, 1976. | MR 402083 | Zbl 0331.31002

[39] Bagby, R. J., Maximal functions and rearrangements: some new proofs, Ind. Univ. Math. J., vol. 32, no. 6, 1983. | MR 721570 | Zbl 0534.42004

[40] Bandle, C., Inégalités isopérimétriques pour des membranes vibrantes, C. R. Acad. Sci. Paris, Sér. A-B, 269, 1969. | MR 252873 | Zbl 0188.58101

[41] Bandle, C., Extensions of an inequality by Pólya and Schiffer for vibrating membranes, Pacific J. Math., 42, 1972. | MR 315563 | Zbl 0217.13301

[42] Bandle, C., Isoperimetric inequality for some eigenvalues of an inhomogeneous, free membrane, SIAM J. Appl. Math., 22, 1972. | MR 313648 | Zbl 0237.35069

[43] Bandle, C., A geometrical isoperimetric inequality and applications to problems of mathematical physics, Comment. Math. Helv., 49, 1974. | MR 358100 | Zbl 0292.52007

[44] Bandle, C., Isoperimetric inequalities for a class of nonlinear parabolic equations, Z.A.M.P., 27, no. 3, 1976. | MR 415067 | Zbl 0324.35051

[45] Bandle, C., On symmetrizations in parabolic equations, J. Analyse Math., 30, 1976. | MR 442477 | Zbl 0331.35036

[46] Bandle, C., Isoperimetric inequalities and applications, Monographs and Studies in Mathematics, 7. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1980. | MR 572958 | Zbl 0436.35063

[47] Bandle, C.-Giarrusso, E., Boundary blow up for semilinear elliptic equations with nonlinear gradient terms, Adv. Differential Equations, 1, no. 1, 1996. | MR 1357958 | Zbl 0840.35034

[48] Bandle, C.-Mossino, J., Rearrangement in variational inequalities, Ann. Mat. Pura Appl. (4), 138, 1984. | MR 779535 | Zbl 0563.49005

[49] Bandle, C.-Peletier, L. A., Best Sobolev constants and Emden equations for the critical exponent in S3, Math. Ann., 313, no. 1, 1999. | MR 1666821 | Zbl 0932.49023

[50] Beckner, W., Geometric inequalities in Fourier analysis, Essays on Fourier Analysis in honor of Elias M. Stein (Princeton NJ 1991), Princeton Math. Ser.42, Princeton University Press, Princeton, N. J.1995. | MR 1315541 | Zbl 0888.42006

[51] Bérard, P. P.-Meyer, D., Inégalités isopérimétriques et applications, Ann. Sci. École Norm. Sup., 4e série, t. 15, 1982. | MR 690651 | Zbl 0527.35020

[52] Betta, M. F., Estimates for solutions of nonlinear degenerate elliptic equations, Atti Sem. Mat. Fis. Univ. Modena, 45, 1997. | MR 1601934 | Zbl 0894.35018

[53] Betta, M. F.-Brock, F.-Mercaldo, A.-Posteraro, M. R., A weighted isoperimetric inequality and applications to symmetrization, in corso di stampa su J. Ineq. Appl.. | Zbl 1029.26018

[54] Betta, M. F.-Ferone, V.-Mercaldo, A., Regularity for solutions of nonlinear elliptic equations, Bull. Sci. Math., 118, 1994. | MR 1309088 | Zbl 0842.35014

[55] Betta, M. F.-Mercaldo, A., Comparison and regularity results for a nonlinear elliptic equation, Nonlinear Analalysis T.M.A., 20, 1993. | MR 1199064 | Zbl 0786.35008

[56] Betta, M. F.-Mercaldo, A., Geometric inequalities related to Steiner symmetrization, Diff. Int. Eq., 10, no. 3, 1997. | MR 1744857 | Zbl 0897.49024

[57] Bhattacharya, T.-Weitsman, A., Some estimates for the symmetrized first eigenfunction of the Laplacian, Potential Analysis, 9, 1998. | MR 1652553 | Zbl 0930.35044

[58] Bliss, G. A., An integral inequality, J. London Math. Soc., 5, 1930. | JFM 56.0434.02

[59] Boukrim, L., Inégalités isopérimétriques pour un probléme d'électrostatique, C. R. Acad. Sci. Paris, Sér. I Math., 318, no. 5, 1994. | MR 1267822 | Zbl 0806.31005

[60] Bramanti, M., Symmetrization in parabolic Neumann problems, Appl. Anal., vol. 40, no. 1, 1991. | MR 1121322 | Zbl 0693.35019

[61] Brandolini, B.-Monetti, V. M.-Randazzo, L., A comparison result for a class of semilinear elliptic equations, Rend. Accad. Sci. Fis. Mat. Napoli (4), 64, 1997. | MR 1635973 | Zbl 0944.35032

[62] Brascamp, H. J.-Lieb, E. H.-Luttinger, J. M., A general rearrangement inequality for multiple integrals, J. Funct. Anal., 17, 1974. | MR 346109 | Zbl 0286.26005

[63] Brenier, Y., Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 54, 1991. | MR 1100809 | Zbl 0738.46011

[64] Brezis, H., Laser beams and limiting case of Sobolev inequalities, Non-linear PDE's and their applications, Collège de France Seminar, vol. II (H. Brezis and J. L. Lions eds.), Res. Notes Math., no. 60, Pitman, London, 1982. | MR 652508 | Zbl 0489.35040

[65] Brezis, H.-Wainger, S., A note on limiting cases of Sobolev embeddings and convolution inequalities, Communications in P.D.E., 5, 1980. | MR 579997 | Zbl 0437.35071

[66] Brock, F., Continuous Steiner-symmetrization, Math. Nachr., 172, 1995. | MR 1330619 | Zbl 0886.49010

[67] Brothers, J.-Ziemer, W. P., Minimal rearrangements of Sobolev functions, J. Reine Angew. Math., 384, 1988. | MR 929981 | Zbl 0633.46030

[68] Buonocore, P., Isoperimetric inequalities in the torsion problem for multiply connected domains, Z.A.M.P., 36, no. 1, 1985. | MR 785926 | Zbl 0559.73010

[69] Buonocore, P., Some isoperimetric inequalities in a special case of the problem of torsional creep, Appl. Anal., vol. 27, no. 1-3, 1988. | MR 922773 | Zbl 0617.35037

[70] Burton, G. R., Rearrangements of functions, maximization of convex functionals, and vortex rings, Math. Ann., 276, no. 2, 1987. | MR 870963 | Zbl 0592.35049

[71] Burton, G. R.-Mcleod, J. B., Maximization and minimization on classes of rearrangements, Proc. Royal Soc. Edinburgh, 119-A, no. 3-4, 1991. | MR 1135975 | Zbl 0736.49006

[72] Cheng, S. S., Isoperimetric eigenvalue problem of even order differential equations, Pacific J. Math., 99, no. 2, 1982. | MR 658061 | Zbl 0505.34018

[73] Chiacchio, F., An existence result for a nonlinear problem in a limit case, Le Matematiche (Catania), 53, 1998. | MR 1710763 | Zbl 1153.35321

[74] Chiti, G., Rearrangements of functions and convergence in Orlicz spaces, Appl. Anal., 9, 1979. | MR 536688 | Zbl 0424.46023

[75] Chiti, G., Norme di Orlicz delle soluzioni di una classe di equazioni ellittiche, Boll. U.M.I. (5), 16-A, 1979. | Zbl 0398.35028

[76] G., A reverse Hölder inequality for the eigenfunctions of linear second order elliptic equations, Z.A.M.P., 33, 1982. | MR 652928 | Zbl 0508.35063

[77] Chong, K. M.-Rice, N. M., Equimeasurable rearrangements of functions, Queen's papers in pure and applied mathematics, n. 28, Queen's University, Ontario, 1971. | MR 372140 | Zbl 0275.46024

[78] Cianchi, A., Local minimizers and rearrangements, Appl. Math. Optim., 27, no. 3, 1993. | MR 1201624 | Zbl 0801.35005

[79] Cianchi, A., On the Lq norm of functions having equidistributed gradients, Nonlinear Anal., 26, no. 12, 1996. | MR 1386130 | Zbl 0884.49011

[80] Cianchi, A.-Schianchi, R., A priori sharp estimates for minimizers, Boll. U.M.I. (7), 7-B, no. 4, 1993. | MR 1255649 | Zbl 0833.49001

[81] Coron, J. M., The continuity of rearrangement in W1,pR, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 11, 1984. | MR 752580 | Zbl 0574.46021

[82] Crandall, M. G.-Tartar, L., Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc., 78, 1980. | MR 553381 | Zbl 0449.47059

[83] Crooke, P. S.-Sperb, R. P., Isoperimetric inequalities in a class of nonlinear eigenvalue problems, SIAM J. Math. Anal., 9, 1978. | MR 492835 | Zbl 0385.35006

[84] Crowe, J. A.-Zweibel, J. A., Rearrangements of functions, J. Funct. Anal., 66, 1986. | MR 839110 | Zbl 0612.46027

[85] De Giorgi, E., Su una teoria generale della misura r-1-dimensionale in uno spazio ad r dimensioni, Ann. Mat. Pura Appl. (4), 36, 1954. | Zbl 0055.28504

[86] Del Vecchio, T., Nonlinear elliptic equations with measure data, Potential Analysis, 4, no. 2, 1995. | MR 1323826 | Zbl 0815.35023

[87] Díaz, J. I., Applications of symmetric rearrangement to certain nonlinear elliptic equations with a free boundary, Nonlinear differential equations (Granada, 1984), Res. Notes in Math., 132, Pitman, Boston, Mass.-London, 1985. | MR 908901 | Zbl 0653.35027

[88] Díaz, J. I.-Mossino, J., Isoperimetric inequalities in the parabolic obstacle problems, J. Math. Pures Appl. (9), 71, no. 3, 1992. | MR 1172451 | Zbl 0679.49011

[89] Douglas, R. J., Rearrangements of functions on unbounded domains, Proc. Royal Soc. Edinburgh, 124-A, 1994. | MR 1298584 | Zbl 0818.49010

[90] Duff, G. F., Integral inequalities for equimeasurable rearrangements, Canad. J. Math., 22, 1970. | MR 257304 | Zbl 0194.36004

[91] Egnell, H.-Pacella, F.-Tricarico, M., Some remarks on Sobolev inequalities, Nonlinear Anal., 13, no. 6, 1989. | MR 998512 | Zbl 0687.46018

[92] Ehrhard, A., Inégalités isopérimetriques et intégrales de Dirichlet gaussiennes, Ann. Sci. École Norm. Sup., 17, 1984. | MR 760680 | Zbl 0546.49020

[93] Epperson, J. B., A class of monotone decreasing rearrangements, J. Math. Anal. Appl., 150, 1990. | MR 1059584 | Zbl 0714.28002

[94] Faber, G., 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. | JFM 49.0342.03

[95] Ferone, A.-Ferone, V.-Volpicelli, R., Moser-type inequalities for solutions of linear elliptic equations with lower order terms, Diff. Int. Eq., 10, 1997. | MR 1608021 | Zbl 0940.35068

[96] Ferone, A.-Volpicelli, R., Symmetrization in parabolic obstacle problems, Bull. Sci. Math., 120, 1996. | MR 1420971 | Zbl 0884.35080

[97] Ferone, A.-Volpicelli, R., Some relations between pseudo-rearrangement and relative rearrangement, in corso di stampa su Nonlinear Analysis T.M.A.. | Zbl 1008.28003

[98] Ferone, V., Symmetrization in a Neumann problem, Le Matematiche (Catania), 41, 1986. | MR 998687 | Zbl 0687.35029

[99] Ferone, V., Symmetrization results in electrostatic problems, Ricerche di Matematica, 37, 1988. | MR 1060493 | Zbl 0711.31005

[100] Ferone, V.-Mercaldo, A., A second order derivation formula for functions defined by integrals, C. R. Acad. Sci. Paris, Sér. I Math., 326, 1998. | MR 1649548 | Zbl 0946.49015

[101] Ferone, V.-Mercaldo, A., in preparazione.

[102] Ferone, V.-Murat, F., Quasilinear problems having quadratic growth in the gradient: an existence result when the source term is small, Équations aux dérivées partielles et applications, Gauthier - Villars, Éd. Sci. Méd. Elsevier, Paris, 1998. | MR 1648236 | Zbl 0917.35039

[103] Ferone, V.-Murat, F., Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small, in corso di stampa su Nonlinear Analysis T.M.A. | Zbl 1158.35358

[104] Ferone, V.-Posteraro, M. R., Maximization on classes of functions with fixed rearrangemet, Diff. Int. Eq., 4, no. 4, 1989. | Zbl 0734.49002

[105] Ferone, V.-Posteraro, M. R., On a class of quasilinear elliptic equations with quadratic growth in the gradient, Nonlinear Analysis T.M.A., 20, 1993. | MR 1214736 | Zbl 0801.35028

[106] Ferone, V.-Posteraro, M. R.-Rakotoson, J.-M., L-estimates for nonlinear elliptic problems with p-growth in the gradient, J. Ineq. Appl., vol. 3, 1999. | MR 1733106 | Zbl 0928.35060

[107] Ferone, V.-Posteraro, M. R.-Volpicelli, R., An inequality concerning rearrangements of functions and Hamilton-Jacobi equations, Arch. Rational Mech. Anal., 125, 1993. | MR 1245072 | Zbl 0787.35020

[108] De Figueiredo, D. G.-Lions, P. L., On pairs of positive solutions for a class of semilinear elliptic problems, Ind. Univ. Math. J., vol. 34, 1985. | MR 794577 | Zbl 0587.35033

[109] Fleming, W. H.-Rishel, R., An integral formula for total gradient variation, Arch. Math., 11, 1960. | MR 114892 | Zbl 0094.26301

[110] Friedman, A.-Mcleod, J. B., Strict inequalities for integrals of decreasingly rearranged functions, Proc. Royal Soc. Edinburgh, 102-A, no. 3-4, 1986. | MR 852361 | Zbl 0601.49012

[111] Garsia, A. M.-Rodemich, E., Monotonicity of certain functionals under rearrangements, Ann. Inst. Fourier (Grenoble), 24, 1974. | MR 414802 | Zbl 0274.26006

[112] Giarrusso, E., Estimates for generalized subsolutions of first order Hamilton-Jacobi equations and rearrangements, Nonlinear Analysis T.M.A., 18, 1992. | MR 1138640 | Zbl 0758.35008

[113] Giarrusso, E.-Nunziante, D., Su un problema di autovalori per una classe di equazioni ellittiche degeneri, Le Matematiche (Catania), 36, 1981. | MR 762915

[114] Giarrusso, E.-Nunziante, D., Symmetrization in a class of first-order Hamilton-Jacobi equations, Nonlinear Analysis T.M.A., 8, 1984. | MR 739660 | Zbl 0543.35014

[115] Giarrusso, E.-Nunziante, D., Regularity theorems in limit cases for solutions of linear and nonlinear elliptic equations, Rend. Istit. Mat. Univ. Trieste, 20, 1988. | MR 1013097 | Zbl 0699.35035

[116] Giarrusso, E.-Trombetti, G., Estimates for solutions of elliptic equations in a limit case, Bull. Austral. Math. Soc., vol. 36, 1987. | MR 923824 | Zbl 0641.35019

[117] Grenon-Isselkou, N.-Mossino, J., Existence de solutions bornées pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris, 321, 1995. | MR 1340081 | Zbl 0837.35045

[118] Hansen, W.-Nadirashvili, N., Isoperimetric inequalities for capacities, Harmonic analysis and discrete potential theory (Frascati, 1991), Plenum, New York, 1992. | MR 1222458

[119] Hardy, G. H.-Littelwood, J. E.-Pólya G., G., Inequalities, Cambridge University Press, 1964.

[120] Hayman, W. K., Some bounds for principal frequency, Appl. Anal., vol. 7, 1978. | MR 492339 | Zbl 0383.35053

[121] Hersch, J., Une interpretation du principe de Thomson et son analogue pour la fréquence fondamentale d'une membrane. Application., C. R. Acad. Sci. Paris, t. 248, 1959. | MR 104892 | Zbl 0085.19302

[122] Hersch, J., Isoperimetric monotonicity: some properties and conjectures (connections between isoperimetric inequalities), SIAM Rev., 30, no. 4, 1988. | MR 967959 | Zbl 0669.35089

[123] Hildén, K., Symmetrization of functions in Sobolev spaces and the isoperimetric inequality, Manuscripta Math., 18, 1976. | MR 409773 | Zbl 0365.46031

[124] Kawohl, B., Rearrangements and convexity of level sets in PDE, Lecture Notes in Mathematics, 1150. Springer-Verlag, Berlin-New York, 1985. | MR 810619 | Zbl 0593.35002

[125] B., On a family of torsional creep problems, J. Reine Angew. Math., 410, 1990. | MR 1068797 | Zbl 0701.35015

[126] Keller, J. B., The shape of the strongest column, Arch. Rational Mech. Anal., 5, 1960. | MR 128160

[127] Kesavan, S., Some remarks on a result of Talenti, Ann. Scuola Norm. Sup. Pisa Cl.Sci. (4), 15, no. 3, 1988. | MR 1015803 | Zbl 0703.35047

[128] Kesavan, S., On a comparison theorem via symmetrization, Proc. Royal Soc.Edinburgh, 119-A, no. 1-2, 1991. | MR 1130603 | Zbl 0762.35005

[129] Kohler-Jobin, M. T., Démonstration de l'inégalité isopérimétrique Pλ2πj04/2, conjecturée par Polya et Szegö, C. R. Acad. Sc. Paris, t. 281, 1975. | MR 385691 | Zbl 0303.52003

[130] Kohler-Jobin, M. T., Sur la première fonction propre d'une membrane: une extension à N dimensions de l'inégalité isopérimétrique de Payne-Rayner, J.Math. Phys. Appl., 28, 1977. | MR 637226 | Zbl 0379.73071

[131] Kohler-Jobin, M. T., Une méthode de comparaison isopérimétrique de fonctionelles de domaines de la physique mathématique I. Première part: une démonstration de la conjecture isopérimétrique Pλ2πj04/2 de Polya et Szegö, J. Math. Phys. Appl., 29, 1978. | Zbl 0427.73056

[132] Krahn, E., Über eine von Rayleigh formulierte minimaleigenschaft des kreises, Math. Ann., 94, 1924. | JFM 51.0356.05

[133] Lieb, E. B., Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57, no. 2, 1977. | Zbl 0369.35022

[134] Lieb, E. H., Sharp constants in the Hardy-Littelwood-Sobolev and related inequalities, Ann. of Math., 118, 1983. | MR 717827 | Zbl 0527.42011

[135] Lions, P. L., Quelques remarques sur la symétrisation de Schwarz, Nonlinear partial differential equations and their applications, Collège de France Seminars, vol. 1, Pitman1981. | MR 631400 | Zbl 0467.35008

[136] Lions, P.-L.-Pacella, F.-Tricarico, M., Best constants in Sobolev inequalities for functions vanishing on some part of the boundary and related questions, Ind. Univ. Math. J., vol. 37, no. 2, 1988. | MR 963504 | Zbl 0631.46033

[137] Liskevich, V. A., Some limit cases in estimates for solutions of second order elliptic equations, Houston J. Math., vol. 19, no. 4, 1993. | MR 1251616 | Zbl 0797.35019

[138] Maderna, C.-Pagani, C. D.-Salsa, S., Quasilinear elliptic equations with quadratic growth in the gradient, J. Differential Equations, 97, 1992. | MR 1161311 | Zbl 0785.35039

[139] Maderna, C.-Salsa, S., Symmetrization in Neumann problems, Appl. Anal., vol. 9, 1979. | MR 553957 | Zbl 0422.35027

[140] Maderna, C.-Salsa, S., Dirichlet problem for elliptic equations with nonlinear first order terms: a comparison result, Ann. Mat. Pura Appl., 148, 1987. | MR 932768 | Zbl 0657.35053

[141] Makai, E., Bounds for the principal frequency of a membrane and the torsional rigidity of a beam, Acta Sci. Math. (Szeged), 20, 1959. | MR 105222 | Zbl 0089.42404

[142] Makai, E., A proof of Saint-Venant's theorem on torsional rigidity, Acta Math. Acad. Sci. Hungar., 17, 1966. | MR 202053 | Zbl 0151.36305

[143] Maz'Ja, V. G., On weak solutions of the Dirichlet and Neumann problems, Trudy Moskov. Mat. Obshch, 20, 1969; trad. ingl.: Trans. Moscow Math. Soc., 20, 1969. | Zbl 0226.35027

[144] Mercaldo, A., Boundedness of minimizers of degenerate functionals, Diff. Int. Eq., 9, 1996. | MR 1371706 | Zbl 0858.49006

[145] Mercaldo, A., A remark on comparison results for first-order Hamilton-Jacobi equations, Nonlinear Analysis T.M.A., 9, 1997. | Zbl 0885.35014

[146] Moser, J., A sharp form of an inequality by N. Trudinger, Ind. Univ. Math. J., vol. 20, 1971. | MR 301504 | Zbl 0203.43701

[147] Mossino, J., Inégalitée isopérimétriques et applications en physique, Travaux en Cours. Hermann, Paris, 1984. | MR 733257 | Zbl 0537.35002

[148] Mossino, J.-Rakotoson, J.-M., Isoperimetric inequalities in parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4) 13, no. 1, 1986. | MR 863635 | Zbl 0652.35053

[149] Mossino, J.-Temam, R., Directional derivative of the increasing rearrangement mapping and application to a queer differential equation in plasma physics, Duke Math. J., 48, no. 3, 1981. | MR 630581 | Zbl 0476.35031

[150] Mugelli, F.-Talenti, G., Sobolev inequalities in 2-D hyperbolic space: a borderline case, J. Ineq. Appl., vol. 2, no. 3, 1998. | MR 1671679 | Zbl 0935.26005

[151] Mugnai, D.-Talenti, G., Estimating the resolvent of elliptic second-order partial differential operators, Le Matematiche (Catania), 51 (1996), no. 2, 1997. | MR 1488072 | Zbl 1010.35034

[152] Nadirashvili, N., Rayleigh's conjecture on the principal frequency of the clamped plate, Arch. Rational Mech. Anal., 129, no. 1, 1995. | MR 1328469 | Zbl 0826.73035

[153] Oklander, E. T., Interpolación, espacios de Lorentz y teorema de Marchinkiewicz, Cursos y Seminarios de Matemática, Fasc. 20Universidad de Buenos Aires, Buenos Aires, 1965. | MR 217591 | Zbl 0143.35702

[154] Osserman, R., The isoperimetric inequality, Bull. Amer. Math. Soc., 84, no. 6, 1978. | MR 500557 | Zbl 0411.52006

[155] Pacella, F.-Tricarico, M., Symmetrization for a class of elliptic equations with mixed boundary conditions, Atti Sem. Mat. Fis. Univ. Modena, 34, no. 1, 1985/86. | MR 876139 | Zbl 0629.35032

[156] Payne, L. E., Inequalities for eigenvalues of membranes and plates, J. Rational Mech. Anal., 4, 1955. | MR 70834 | Zbl 0064.34802

[157] Payne, L. E., News isoperimetric inequalities for eigenvalues and other physical quantities, Comm. Pure Appl. Math., 9, 1956. | MR 81433 | Zbl 0074.31405

[158] Payne, L. E., Inequalities for eigenvalues of supported and free plates, Quart. Appl. Math., 16, 1958. | MR 96440 | Zbl 0084.20705

[159] Payne, L. E., Isoperimetric inequalities for eigenvalues and their applications, Autovalori e autosoluzioni, Centro Internazionale Matematico Estivo 20 ciclo, Chieti, 1962. | Zbl 0201.57403

[160] Payne, L. E., Some isoperimetric inequalities in the torsion problem for multiply connected regions, Studies in Mathematical Analysis and Related Topics: Essays in honor of Pólya G., Stanford, California, 1962. | MR 163472 | Zbl 0114.40702

[161] Payne, L. E., Isoperimetric inequalities and their applications, SIAM Rev., 9, 1967. | MR 218975 | Zbl 0154.12602

[162] Payne, L. E.-Pólya, G.-Weinberger, H. F., On the ratio of consecutive eigenvalues, J. Math. and Phys., 35, 1956. | MR 84696 | Zbl 0073.08203

[163] Payne, L. E.-Rayner, M. E., An isoperimetric inequality for the first eigenfunction in the fixed membrane problem, Z.A.M.P., 23, 1972. | MR 313649 | Zbl 0241.73080

[164] Payne, L. E.-Rayner, M. E., Some isoperimetric norm bounbs for solutions of the Helmholtz equation, Z.A.M.P., 24, 1973. | MR 324202 | Zbl 0256.35023

[165] Payne, L. E.-Weinberger, H. F., Some isoperimetric inequalities for membrane frequencies and torsional rigidity, J. Math. Anal. Appl., 2, 1961. | MR 149735 | Zbl 0098.39201

[166] Payne, L. E.-Weistein, A., Capacity, virtual mass and generalized symmetrization, Pacific J. Math., 2, 1952. | MR 50738 | Zbl 0048.08102

[167] Păsić, M., Isoperimetric inequalities in quasilinear elliptic equations of Leray-Lions type, J. Math. Pures Appl. (9), 75, no. 4, 1996. | MR 1411156 | Zbl 0859.35028

[168] Poincaré, H., Figures d'equilibre d'une masse fluide, Naud, Paris, 1903. | JFM 34.0757.05

[169] Pólya, G., Torsional rigidity, principal frequency, eletrostatic capacity and symmetrization, Quart. Appl. Math., 6, 1948. | MR 26817 | Zbl 0037.25301

[170] Pólya, G., On the eigenvalues of vibrating membranes, Proc. London Math. Soc., 11, 1961. | MR 129219 | Zbl 0107.41805

[171] Pólya, G.-Szegö, G., Isoperimetric inequalities in mathematical physics, Ann. of Math. Studies, n. 27, Princeton University Press, 1951. | MR 43486 | Zbl 0044.38301

[172] Pólya, G.-Weinstein, A., On the torsional rigidity of multiply connected crosssections, Ann. of Math., 52, 1950. | MR 40159 | Zbl 0038.37502

[173] Poornima, S., An embedding theorem for the Sobolev space W1,1, Bull. Sci. Math. (2), 107, 1983. | MR 719267 | Zbl 0529.46025

[174] Posteraro, M. R., Estimates for solutions of nonlinear variational inequalities, Ann. Inst. H. Poincare - Anal. Nonlineaire, 12, 5, 1995. | MR 1353260 | Zbl 0874.49009

[175] Posteraro, M. R., On the solutions of the equation Δu=eu blowing up on the boundary, C. R. Acad. Sci. Paris, Ser. I Math, 332, 1996. | MR 1381781 | Zbl 0848.35037

[176] Protter, M. H., Lower bounds for the first eigenvalue of elliptic equations, Ann. of Math., 71, 1960. | MR 111923 | Zbl 0095.20004

[177] Rakotoson, J.-M., Rearrangement relatif dans les equations elliptiques quasi-lineaires avec un second membre distribution: application a un theoreme d'existence et de regularite, J. Differential Equations, 66, no. 3, 1987. | MR 876805 | Zbl 0652.35041

[178] Rakotoson, J.-M., Some properties of the relative rearrangement, J. Math. Anal. Appl., 135, no. 2, 1988. | MR 967224 | Zbl 0686.28003

[179] Rakotoson, J.-M., A differentiability result for the relative rearrangement, Diff. Int. Eq., 2, no. 3, 1989. | MR 983687 | Zbl 0772.35018

[180] Rakotoson, M., Relative rearrangement for highly nonlinear equations, Nonlinear Anal., 24, no. 4, 1995. | MR 1315691 | Zbl 0830.35036

[181] Rakotoson, J.-M.-Simon, B., Relative rearrangement on a measure space application to the regularity of weighted monotone rearrangement. I, II, Appl. Math. Letters, 6, no. 1, 1993. | MR 1347759 | Zbl 0781.49025

[182] Rakotoson, J.-M.-Temam, R., Relative rearrangement in quasilinear elliptic variational inequalities, Ind. Univ. Math. J., 36, no. 4, 1987. | MR 916743

[183] Rakotoson, J.-M.-Temam, R., A co-area formula with applications to monotone rearrangement and to regularity, Arch. Rational Mech. Anal., 109, no. 3, 1990. | MR 1025171 | Zbl 0735.49039

[184] Lord Rayleigh, The Theory of Sound, 2nd ed., Macmillan, London, 1894/96. | JFM 27.0701.05

[185] Ryff, J. V., Measure preserving transformations and rearrangements, J. Math. Anal. Appl., 31, 1970. | MR 419734 | Zbl 0214.13701

[186] De St. Venant, B., Memoire sur la torsion des prismes, Memor. pres. divers. savants Acad. Sci., 14, 1856.

[187] Schulz, F.-Vera De Serio, V., Symmetrization with respect to a measure, Trans. Amer. Math. Soc., 337, 1993. | MR 1088477 | Zbl 0844.49025

[188] Schumann, W., On isoperimetric inequalities in plasticity, Quart. Appl. Math., 16, 1958. | MR 103658 | Zbl 0086.39104

[189] Schwarz, B., Bounds for the principal frequency of the nonhomogeneous membrane and for the generalized Dirichlet integral, Pacific J. Math., 7, 1957. | MR 92918 | Zbl 0081.31703

[190] Sperner, E., Zur symmetrisierung für funktionen auf sphären, Math. Z., 134, 1973. | MR 340558 | Zbl 0283.26015

[191] Sperner, E., Symmetrisierung für funktionen mehrerer reeller variabel, Manuscripta Math., 11, 1974. | MR 328000 | Zbl 0268.26011

[192] Sperner, E., Spherical symmetrization and eigenvalue estimates, Math. Z., 176, 1976. | MR 606173 | Zbl 0464.49009

[193] Spiegel, W., Über die symmetrisierung stetiger funktionen im euklidishen raum, Archiv. der Math., 24, 1973. | MR 412365 | Zbl 0274.52011

[194] Steiner, J., Einfache Beweise der isoperimetrischen Hauptsätze, Ges. Werke II, Berlin, 1882. | Zbl 018.0600cj

[195] Szegö, G., Über einige neue Extremalaufgaben der Potential-theorie, Math. Z., 31, 1930. | JFM 56.1069.03

[196] Szegö, G., Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal., 3, 1954. | MR 61749 | Zbl 0055.08802

[197] Tahraoui, R., Symmetrization inequalities, Nonlinear Anal., 27, no. 8, 1996. | MR 1404592 | Zbl 0879.35013

[198] Talenti, G., Best constant in Sobolev inequality, Ann. Mat. Pura Appl., (4) 110, 1976. | MR 463908 | Zbl 0353.46018

[199] Talenti, G., Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4) 3, no. 4, 1976. | MR 601601 | Zbl 0341.35031

[200] Talenti, G., Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl. (4), 120, 1979. | MR 551065 | Zbl 0419.35041

[201] Talenti, G., Some estimates for solutions to Monge-Ampère equations in dimension two, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 8, 1981. | MR 623935 | Zbl 0467.35044

[202] Talenti, G., On the first eigenvalue of the clamped plate, Ann. Mat. Pura Appl. (4), 129, 1981. | MR 648335 | Zbl 0475.73050

[203] Talenti, G., Linear elliptic p.d.e.'s: level sets, rearrangements and a priori estimates of solutions, Boll. U.M.I. (6), 4-B, no. 3, 1985. | MR 831299 | Zbl 0602.35025

[204] Talenti, G., Rearrangements of functions and partial differential equations, Nonlinear diffusion problems (Montecatini Terme, 1985), Lect. Notes in Math., 1224, Springer, Berlin-New York, 1986. | MR 877989 | Zbl 0607.65077

[205] Talenti, G., Assembling a rearrangement, Arch. Rational Mech. Anal., 98, no. 4, 1987. | MR 872748 | Zbl 0619.35113

[206] Talenti, G., Some inequalities of Sobolev type on two-dimensional spheres, General inequalities, 5 (Oberwolfach, 1986), Internat. Schriftenreihe Numer. Math., 80, Birkhäuser, Basel-Boston, MA, 1987. | MR 1018163 | Zbl 0652.26020

[207] Talenti, G., Boundness of minimizers, Hokkaido Math. J., 19, no. 2, 1990. | MR 1059170 | Zbl 0723.58015

[208] Talenti, G., Rearrangements and PDE, Inequalities (Birmingham, 1987), Lecture Notes in Pure and Appl. Math., 129, Dekker, New York, 1991. | MR 1112579 | Zbl 0733.35013

[209] Talenti, G., On isoperimetric theorems of mathematical physics, Handbook of convex geometry, vol. A, B, North-Holland, Amsterdam, 1993. | MR 1243005 | Zbl 0804.35005

[210] Talenti, G., The standard isoperimetric theorem, Handbook of convex geometry, vol. A, B, North-Holland, Amsterdam, 1993. | MR 1242977 | Zbl 0799.51015

[211] Tonelli, L., Sur un probleme de Lord Rayleigh, Monatsh. Math. Phys., 37, 1930. | JFM 56.1080.03 | MR 1549794

[212] Toupin, R., St. Venant's principle, Arch. Rational Mech. Anal., 18, 1965. | MR 172506 | Zbl 0203.26803

[213] Transirico, M., Symmetrization in a nonlinear degenerate parabolic problem, Ann. Mat. Pura Appl. (4), 149, 1987. | MR 932792 | Zbl 0655.35048

[214] Trombetti, C., Existence and regularity for a class of non uniformly elliptic equations in two dimension, in corso di stampa su Diff. Int. Eq.. | Zbl 0980.35054

[215] Trombetti, G.-Vazquez, G. L., Symmetrization results for elliptic equations with lower order terms, Ann. Fac. Sci. Toulouse, 7, 1985. | Zbl 0617.35036

[216] Trudinger, N. S., On embeddings into Orlicz spaces and some applications, J. Math. Mech., vol. 17, 1967. | MR 216286 | Zbl 0163.36402

[217] Trudinger, N. S., On new isoperimetric inequalities and symmetrization, J. Reine Angew. Math., 488, 1997. | MR 1465371 | Zbl 0883.52006

[218] Tso, K., On symmetrization and Hessian equations, J. Analyse Math., 52, 1989. | MR 981497 | Zbl 0675.35040

[219] Vazquez, J. L., Symétrization pour ut=Δϕu et applications, C. R. Acad. Sci. Paris, t. 295, 1982 e C. R. Acad. Sci. Paris, t. 296, 1983. | MR 676365 | Zbl 0501.35015

[220] Voas, C.-Yaniro, D., Symmetrization and optimal control for elliptic equations, Proc. Amer. Math. Soc., 99, no. 3, 1987. | MR 875390 | Zbl 0609.49016

[221] Voas, C.-Yaniro, D., Elliptic equations, rearrangements, and functions of bounded lower oscillation, J. Math. Anal. Appl., 134, no. 1, 1988. | MR 958857 | Zbl 0684.35040

[222] Volpicelli, R., Comparison results for solutions of parabolic equations, Ricerche di Matematica, 42, 1993. | MR 1283813 | Zbl 0827.35018

[223] Weinberger, H. F., Upper and lower bounds for torsional rigidity, J. Math. and Phys., 32, 1953. | MR 55511 | Zbl 0051.41203

[224] Weinberger, H. F., An isoperimetric inequality for the N-dimensional free membrane problem, J. Rational Mech. Anal., 5, 1956. | MR 79286 | Zbl 0071.09902

[225] Weinberger, H. F., Symmetrization in uniformly elliptic problems, Studies in Mathematical Analysis and Related topics: Essays in honor of G. Szegö, Stanford University Press, Stanford, California, 1962. | MR 145191 | Zbl 0123.07202