A survey of the fundamental ideas which are the base of the socalled symmetrization method; a priori estimates in partial differential equations.
Referenze Bibliografiche
[1]
D. R. ADAMS,
A sharp inequality of J. Moser for higher order derivatives,
Ann. of Math.,
128,
1988. |
MR 960950 |
Zbl 0672.31008[2]
J. A. AGUILAR-
I. PERAL,
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]
A. ALBERICO-
V. FERONE,
Regularity properties of solutions of elliptic equations in $\mathbb{R}^{2}$ in limit cases,
Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (9)
6,
1995. |
MR 1382708 |
Zbl 0860.35015[4]
F. J. JR ALMGREN-
E. H. LIEB,
Symmetric decreasing rearrangement is sometimes continous,
J. Amer. Math. Soc., vol.
2, no. 4,
1989. |
MR 1002633 |
Zbl 0688.46014[6]
A. ALVINO,
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]
A. ALVINO,
Sulla diseguaglianza di Sobolev in spazi di Lorentz,
Boll. U.M.I. (5),
14-A,
1977. |
MR 438106 |
Zbl 0352.46020[8]
A. ALVINO,
Un caso limite della diseguaglianza di Sobolev in spazi di Lorentz,
Rend. Accad. Sci. Fis. Mat. Napoli,
44,
1977. |
Zbl 0412.46024[9] A. ALVINO-L. BOCCARDO-V. FERONE-L. ORSINA-G. TROMBETTI, in preparazione.
[10]
A. ALVINO-
J. I. DÍAZ-
P. L. LIONS-
G. TROMBETTI,
Équations elliptiques et symétrisation de Steiner,
C. R. Acad. Sci. Paris, Sér. I Math.,
314,
1992. |
MR 1168527 |
Zbl 0795.35022[11]
A. ALVINO-
J. I. DÍAZ-
P. L. LIONS-
G. TROMBETTI,
Elliptic equations and Steiner symmetrization,
Comm. Pure Appl. Math.,
49 (3),
1996. |
MR 1374171 |
Zbl 0856.35034[13]
A. ALVINO-
V. FERONE-
G. TROMBETTI,
Moser-type inequalities in Lorentz spaces,
Potential Analysis,
5,
1996. |
MR 1389498 |
Zbl 0856.46020[14]
A. ALVINO-
V. FERONE-
G. TROMBETTI,
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]
A. ALVINO-
V. FERONE-
G. TROMBETTI,
On the properties of some nonlinear eigenvalues,
SIAM J. Math. Anal.,
29,
1998. |
MR 1616519 |
Zbl 0908.35094[16]
A. ALVINO-
V. FERONE-
G. TROMBETTI,
Estimates for the gradient of nonlinear elliptic equations with $L^{1}$ data, in corso di stampa su
Ann. Mat. Pura Appl.. |
Zbl pre02216904[17]
A. ALVINO-
P. L. LIONS-
S. MATARASSO-
G. TROMBETTI,
Comparison results for solutions to elliptic problems via symmetrization,
Ann. Inst. H. Poincaré - Analyse Nonlinéaire,
16, 2,
1999. |
fulltext mini-dml |
MR 1674768 |
Zbl 0924.35038[18]
A. ALVINO-
P. L. LIONS-
G. TROMBETTI,
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]
A. ALVINO-
P. L. LIONS-
G. TROMBETTI,
On optimization problems with prescribed rearrangements,
Nonlinear Analysis T.M.A,
13,
1989. |
MR 979040 |
Zbl 0678.49003[20]
A. ALVINO-
P. L. LIONS-
G. TROMBETTI,
Comparison results for elliptic and parabolic equations via Schwartz symmetrization,
Ann. Inst. H. Poincaré - Analysis Nonlinéaire,
7,
1990. |
fulltext mini-dml |
MR 1051227 |
Zbl 0703.35007[21]
A. ALVINO-
P. L. LIONS-
G. TROMBETTI,
Comparison results for elliptic and parabolic equations via Symmetrization: a new approach,
Diff. Int. Eq.,
4,
1991. |
MR 1079609 |
Zbl 0735.35003[22]
A. ALVINO-
S. MATARASSO-
G. TROMBETTI,
Variational inequalities and rearrangements,
Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (9),
3,
1992. |
MR 1203167 |
Zbl 0803.35060[23]
A. ALVINO-
G. TROMBETTI,
Sulle migliori costanti di maggiorazione per una classe di equazioni ellittiche degeneri,
Ricerche di Matematica,
27,
1978. |
MR 527431 |
Zbl 0403.35027[24]
A. ALVINO-
G. TROMBETTI,
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]
A. ALVINO-
G. TROMBETTI,
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]
A. ALVINO-
G. TROMBETTI,
A lower bound for the first eigenvalue of an elliptic operator,
Math. Anal. Appl.,
94,
1983. |
MR 706368 |
Zbl 0525.35063[27]
A. ALVINO-
G. TROMBETTI,
Isoperimetric inequalities connected with torsion problem and capacity,
Boll. U.M.I. (6),
4-B,
1985. |
MR 831290[28]
V. ANDRIEVSKIĬ-
W. HANSEN-
N. NADIRASHVILI,
Isoperimetric inequalities for capacities in the plane,
Math. Ann.,
292, no. 2,
1992. |
MR 1149031 |
Zbl 0739.31001[29]
G. ARONSSON,
An integral inequality and plastic torsion,
Arch. Rational Mech. Anal.,
7,
1989. |
MR 540220 |
Zbl 0415.49005[31]
M. S. ASHBAUGH-
R. D. BENGURIA,
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]
M. S. ASHBAUGH-
R. D. BENGURIA,
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]
M. S. ASHBAUGH-
R. D. BENGURIA,
On Rayleigh's conjecture for the clamped plate and its generalization to three dimensions,
Duke Math. J.,
78, no. 1,
1995. |
fulltext mini-dml |
MR 1328749 |
Zbl 0833.35035[36]
T. AUBIN,
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]
A. II BAERNSTEIN,
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[39]
R. J. BAGBY,
Maximal functions and rearrangements: some new proofs,
Ind. Univ. Math. J., vol.
32, no. 6,
1983. |
MR 721570 |
Zbl 0534.42004[40]
C. BANDLE,
Inégalités isopérimétriques pour des membranes vibrantes,
C. R. Acad. Sci. Paris, Sér. A-B,
269,
1969. |
MR 252873 |
Zbl 0188.58101[42]
C. BANDLE,
Isoperimetric inequality for some eigenvalues of an inhomogeneous, free membrane,
SIAM J. Appl. Math.,
22,
1972. |
MR 313648 |
Zbl 0237.35069[43]
C. BANDLE,
A geometrical isoperimetric inequality and applications to problems of mathematical physics,
Comment. Math. Helv.,
49,
1974. |
MR 358100 |
Zbl 0292.52007[44]
C. BANDLE,
Isoperimetric inequalities for a class of nonlinear parabolic equations,
Z.A.M.P.,
27, no. 3,
1976. |
MR 415067 |
Zbl 0324.35051[45]
C. BANDLE,
On symmetrizations in parabolic equations,
J. Analyse Math.,
30,
1976. |
MR 442477 |
Zbl 0331.35036[46]
C. BANDLE,
Isoperimetric inequalities and applications,
Monographs and Studies in Mathematics,
7.
Pitman (Advanced Publishing Program), Boston, Mass.-London,
1980. |
MR 572958 |
Zbl 0436.35063[47]
C. BANDLE-
E. GIARRUSSO,
Boundary blow up for semilinear elliptic equations with nonlinear gradient terms,
Adv. Differential Equations,
1, no. 1,
1996. |
MR 1357958 |
Zbl 0840.35034[48]
C. BANDLE-
J. MOSSINO,
Rearrangement in variational inequalities,
Ann. Mat. Pura Appl. (4),
138,
1984. |
MR 779535 |
Zbl 0563.49005[49]
C. BANDLE-
L. A. PELETIER,
Best Sobolev constants and Emden equations for the critical exponent in $S^{3}$,
Math. Ann.,
313, no. 1,
1999. |
MR 1666821 |
Zbl 0932.49023[50]
W. BECKNER,
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[52]
M. F. BETTA,
Estimates for solutions of nonlinear degenerate elliptic equations,
Atti Sem. Mat. Fis. Univ. Modena,
45,
1997. |
MR 1601934 |
Zbl 0894.35018[53]
M. F. BETTA-
F. BROCK-
A. MERCALDO-
M. R. POSTERARO,
A weighted isoperimetric inequality and applications to symmetrization, in corso di stampa su
J. Ineq. Appl.. |
Zbl 1029.26018[54]
M. F. BETTA-
V. FERONE-
A. MERCALDO,
Regularity for solutions of nonlinear elliptic equations,
Bull. Sci. Math.,
118,
1994. |
MR 1309088 |
Zbl 0842.35014[55]
M. F. BETTA-
A. MERCALDO,
Comparison and regularity results for a nonlinear elliptic equation,
Nonlinear Analalysis T.M.A.,
20,
1993. |
MR 1199064 |
Zbl 0786.35008[56]
M. F. BETTA-
A. MERCALDO,
Geometric inequalities related to Steiner symmetrization,
Diff. Int. Eq.,
10, no. 3,
1997. |
MR 1744857 |
Zbl 0897.49024[57]
T. BHATTACHARYA-
A. WEITSMAN,
Some estimates for the symmetrized first eigenfunction of the Laplacian,
Potential Analysis,
9,
1998. |
MR 1652553 |
Zbl 0930.35044[58]
G. A. BLISS,
An integral inequality,
J. London Math. Soc.,
5,
1930. |
Jbk 56.0434.02[59]
L. BOUKRIM,
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]
M. BRAMANTI,
Symmetrization in parabolic Neumann problems,
Appl. Anal., vol.
40, no. 1,
1991. |
MR 1121322 |
Zbl 0693.35019[61]
B. BRANDOLINI-
V. M. MONETTI-
L. RANDAZZO,
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]
H. J. BRASCAMP-
E. H. LIEB-
J. M. LUTTINGER,
A general rearrangement inequality for multiple integrals,
J. Funct. Anal.,
17,
1974. |
MR 346109 |
Zbl 0286.26005[63]
Y. BRENIER,
Polar factorization and monotone rearrangement of vector-valued functions,
Comm. Pure Appl. Math.,
54,
1991. |
MR 1100809 |
Zbl 0738.46011[64]
H. BREZIS,
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]
H. BREZIS-
S. WAINGER,
A note on limiting cases of Sobolev embeddings and convolution inequalities,
Communications in P.D.E.,
5,
1980. |
MR 579997 |
Zbl 0437.35071[67]
J. BROTHERS-
W. P. ZIEMER,
Minimal rearrangements of Sobolev functions,
J. Reine Angew. Math.,
384,
1988. |
MR 929981 |
Zbl 0633.46030[68]
P. BUONOCORE,
Isoperimetric inequalities in the torsion problem for multiply connected domains,
Z.A.M.P.,
36, no. 1,
1985. |
MR 785926 |
Zbl 0559.73010[69]
P. BUONOCORE,
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]
G. R. BURTON,
Rearrangements of functions, maximization of convex functionals, and vortex rings,
Math. Ann.,
276, no. 2,
1987. |
MR 870963 |
Zbl 0592.35049[71]
G. R. BURTON-
J. B. MCLEOD,
Maximization and minimization on classes of rearrangements,
Proc. Royal Soc. Edinburgh,
119-A, no. 3-4,
1991. |
MR 1135975 |
Zbl 0736.49006[73]
F. CHIACCHIO,
An existence result for a nonlinear problem in a limit case,
Le Matematiche (Catania),
53,
1998. |
MR 1710763 |
Zbl 1153.35321[74]
G. CHITI,
Rearrangements of functions and convergence in Orlicz spaces,
Appl. Anal.,
9,
1979. |
MR 536688 |
Zbl 0424.46023[75]
G. CHITI,
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]
K. M. CHONG-
N. M. RICE,
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]
A. CIANCHI,
Local minimizers and rearrangements,
Appl. Math. Optim.,
27, no. 3,
1993. |
MR 1201624 |
Zbl 0801.35005[79]
A. CIANCHI,
On the $L^{q}$ norm of functions having equidistributed gradients,
Nonlinear Anal.,
26, no. 12,
1996. |
MR 1386130 |
Zbl 0884.49011[80]
A. CIANCHI-
R. SCHIANCHI,
A priori sharp estimates for minimizers,
Boll. U.M.I. (7),
7-B, no. 4,
1993. |
MR 1255649 |
Zbl 0833.49001[82]
M. G. CRANDALL-
L. TARTAR,
Some relations between nonexpansive and order preserving mappings,
Proc. Amer. Math. Soc.,
78,
1980. |
MR 553381 |
Zbl 0449.47059[83]
P. S. CROOKE-
R. P. SPERB,
Isoperimetric inequalities in a class of nonlinear eigenvalue problems,
SIAM J. Math. Anal.,
9,
1978. |
MR 492835 |
Zbl 0385.35006[84]
J. A. CROWE-
J. A. ZWEIBEL,
Rearrangements of functions,
J. Funct. Anal.,
66,
1986. |
MR 839110 |
Zbl 0612.46027[85]
E. DE GIORGI,
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]
T. DEL VECCHIO,
Nonlinear elliptic equations with measure data,
Potential Analysis,
4, no. 2,
1995. |
MR 1323826 |
Zbl 0815.35023[87]
J. I. DÍAZ,
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]
J. I. DÍAZ-
J. MOSSINO,
Isoperimetric inequalities in the parabolic obstacle problems,
J. Math. Pures Appl. (9),
71, no. 3,
1992. |
MR 1172451 |
Zbl 0679.49011[89]
R. J. DOUGLAS,
Rearrangements of functions on unbounded domains,
Proc. Royal Soc. Edinburgh,
124-A,
1994. |
MR 1298584 |
Zbl 0818.49010[90]
G. F. DUFF,
Integral inequalities for equimeasurable rearrangements,
Canad. J. Math.,
22,
1970. |
MR 257304 |
Zbl 0194.36004[91]
H. EGNELL-
F. PACELLA-
M. TRICARICO,
Some remarks on Sobolev inequalities,
Nonlinear Anal.,
13, no. 6,
1989. |
MR 998512 |
Zbl 0687.46018[93]
J. B. EPPERSON,
A class of monotone decreasing rearrangements,
J. Math. Anal. Appl.,
150,
1990. |
MR 1059584 |
Zbl 0714.28002[94]
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. |
Jbk 49.0342.03[95]
A. FERONE-
V. FERONE-
R. VOLPICELLI,
Moser-type inequalities for solutions of linear elliptic equations with lower order terms,
Diff. Int. Eq.,
10,
1997. |
MR 1608021 |
Zbl 0940.35068[96]
A. FERONE-
R. VOLPICELLI,
Symmetrization in parabolic obstacle problems,
Bull. Sci. Math.,
120,
1996. |
MR 1420971 |
Zbl 0884.35080[97]
A. FERONE-
R. VOLPICELLI,
Some relations between pseudo-rearrangement and relative rearrangement, in corso di stampa su
Nonlinear Analysis T.M.A.. |
Zbl 1008.28003[98]
V. FERONE,
Symmetrization in a Neumann problem,
Le Matematiche (Catania),
41,
1986. |
MR 998687 |
Zbl 0687.35029[99]
V. FERONE,
Symmetrization results in electrostatic problems,
Ricerche di Matematica,
37,
1988. |
MR 1060493 |
Zbl 0711.31005[100]
V. FERONE-
A. MERCALDO,
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] V. FERONE-A. MERCALDO, in preparazione.
[102]
V. FERONE-
F. MURAT,
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]
V. FERONE-
F. MURAT,
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]
V. FERONE-
M. R. POSTERARO,
Maximization on classes of functions with fixed rearrangemet,
Diff. Int. Eq.,
4, no. 4,
1989. |
Zbl 0734.49002[105]
V. FERONE-
M. R. POSTERARO,
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]
V. FERONE-
M. R. POSTERARO-
J.-M. RAKOTOSON,
$L^{\infty}$-estimates for nonlinear elliptic problems with $p$-growth in the gradient,
J. Ineq. Appl., vol.
3,
1999. |
MR 1733106 |
Zbl 0928.35060[107]
V. FERONE-
M. R. POSTERARO-
R. VOLPICELLI,
An inequality concerning rearrangements of functions and Hamilton-Jacobi equations,
Arch. Rational Mech. Anal.,
125,
1993. |
MR 1245072 |
Zbl 0787.35020[108]
D. G. DE FIGUEIREDO-
P. L. LIONS,
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]
W. H. FLEMING-
R. RISHEL,
An integral formula for total gradient variation,
Arch. Math.,
11,
1960. |
MR 114892 |
Zbl 0094.26301[110]
A. FRIEDMAN-
J. B. MCLEOD,
Strict inequalities for integrals of decreasingly rearranged functions,
Proc. Royal Soc. Edinburgh,
102-A, no. 3-4,
1986. |
MR 852361 |
Zbl 0601.49012[112]
E. GIARRUSSO,
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]
E. GIARRUSSO-
D. NUNZIANTE,
Su un problema di autovalori per una classe di equazioni ellittiche degeneri,
Le Matematiche (Catania),
36,
1981. |
MR 762915[114]
E. GIARRUSSO-
D. NUNZIANTE,
Symmetrization in a class of first-order Hamilton-Jacobi equations,
Nonlinear Analysis T.M.A.,
8,
1984. |
MR 739660 |
Zbl 0543.35014[115]
E. GIARRUSSO-
D. NUNZIANTE,
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]
E. GIARRUSSO-
G. TROMBETTI,
Estimates for solutions of elliptic equations in a limit case,
Bull. Austral. Math. Soc., vol.
36,
1987. |
MR 923824 |
Zbl 0641.35019[117]
N. GRENON-ISSELKOU-
J. MOSSINO,
Existence de solutions bornées pour certaines équations elliptiques quasilinéaires,
C. R. Acad. Sci. Paris,
321,
1995. |
MR 1340081 |
Zbl 0837.35045[118]
W. HANSEN-
N. NADIRASHVILI,
Isoperimetric inequalities for capacities,
Harmonic analysis and discrete potential theory (Frascati, 1991),
Plenum, New York,
1992. |
MR 1222458[119] G. H. HARDY-J. E. LITTELWOOD-G. PÓLYA G., Inequalities, Cambridge University Press, 1964.
[121]
J. HERSCH,
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]
J. HERSCH,
Isoperimetric monotonicity: some properties and conjectures (connections between isoperimetric inequalities),
SIAM Rev.,
30, no. 4,
1988. |
MR 967959 |
Zbl 0669.35089[123]
K. HILDÉN,
Symmetrization of functions in Sobolev spaces and the isoperimetric inequality,
Manuscripta Math.,
18,
1976. |
MR 409773 |
Zbl 0365.46031[124]
B. KAWOHL,
Rearrangements and convexity of level sets in PDE,
Lecture Notes in Mathematics,
1150.
Springer-Verlag, Berlin-New York,
1985. |
MR 810619 |
Zbl 0593.35002[126]
J. B. KELLER,
The shape of the strongest column,
Arch. Rational Mech. Anal.,
5,
1960. |
MR 128160[128]
S. KESAVAN,
On a comparison theorem via symmetrization,
Proc. Royal Soc.Edinburgh,
119-A, no. 1-2,
1991. |
MR 1130603 |
Zbl 0762.35005[129]
M. T. KOHLER-JOBIN,
Démonstration de l'inégalité isopérimétrique $P \lambda^2 \geq \pi j_0^4 / 2$, conjecturée par Polya et Szegö,
C. R. Acad. Sc. Paris, t.
281,
1975. |
MR 385691 |
Zbl 0303.52003[130]
M. T. KOHLER-JOBIN,
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]
M. T. KOHLER-JOBIN,
Une méthode de comparaison isopérimétrique de fonctionelles de domaines de la physique mathématique I. Première part: une démonstration de la conjecture isopérimétrique $P\lambda^2 \geq \pi j^{4}_{0}/2$ de Polya et Szegö,
J. Math. Phys. Appl.,
29,
1978. |
Zbl 0427.73056[132]
E. KRAHN,
Über eine von Rayleigh formulierte minimaleigenschaft des kreises,
Math. Ann.,
94,
1924. |
Jbk 51.0356.05[133]
E. B. LIEB,
Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation,
Studies in Appl. Math.,
57, no. 2,
1977. |
Zbl 0369.35022[134]
E. H. LIEB,
Sharp constants in the Hardy-Littelwood-Sobolev and related inequalities,
Ann. of Math.,
118,
1983. |
MR 717827 |
Zbl 0527.42011[135]
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. |
MR 631400 |
Zbl 0467.35008[136]
P.-L. LIONS-
F. PACELLA-
M. TRICARICO,
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]
V. A. LISKEVICH,
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]
C. MADERNA-
C. D. PAGANI-
S. SALSA,
Quasilinear elliptic equations with quadratic growth in the gradient,
J. Differential Equations,
97,
1992. |
MR 1161311 |
Zbl 0785.35039[139]
C. MADERNA-
S. SALSA,
Symmetrization in Neumann problems,
Appl. Anal., vol.
9,
1979. |
MR 553957 |
Zbl 0422.35027[140]
C. MADERNA-
S. SALSA,
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]
E. MAKAI,
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]
E. MAKAI,
A proof of Saint-Venant's theorem on torsional rigidity,
Acta Math. Acad. Sci. Hungar.,
17,
1966. |
MR 202053 |
Zbl 0151.36305[143]
V. G. MAZ'JA,
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]
A. MERCALDO,
Boundedness of minimizers of degenerate functionals,
Diff. Int. Eq.,
9,
1996. |
MR 1371706 |
Zbl 0858.49006[145]
A. MERCALDO,
A remark on comparison results for first-order Hamilton-Jacobi equations,
Nonlinear Analysis T.M.A.,
9,
1997. |
Zbl 0885.35014[146]
J. MOSER,
A sharp form of an inequality by N. Trudinger,
Ind. Univ. Math. J., vol.
20,
1971. |
MR 301504 |
Zbl 0203.43701[147]
J. MOSSINO,
Inégalitée isopérimétriques et applications en physique,
Travaux en Cours.
Hermann, Paris,
1984. |
MR 733257 |
Zbl 0537.35002[149]
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, no. 3,
1981. |
fulltext mini-dml |
MR 630581 |
Zbl 0476.35031[150]
F. MUGELLI-
G. TALENTI,
Sobolev inequalities in 2-D hyperbolic space: a borderline case,
J. Ineq. Appl., vol.
2, no. 3,
1998. |
MR 1671679 |
Zbl 0935.26005[151]
D. MUGNAI-
G. TALENTI,
Estimating the resolvent of elliptic second-order partial differential operators,
Le Matematiche (Catania),
51 (
1996), no. 2,
1997. |
MR 1488072 |
Zbl 1010.35034[152]
N. NADIRASHVILI,
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]
E. T. OKLANDER,
Interpolación, espacios de Lorentz y teorema de Marchinkiewicz,
Cursos y Seminarios de Matemática, Fasc. 20 Universidad de Buenos Aires, Buenos Aires,
1965. |
MR 217591 |
Zbl 0143.35702[155]
F. PACELLA-
M. TRICARICO,
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]
L. E. PAYNE,
Inequalities for eigenvalues of membranes and plates,
J. Rational Mech. Anal.,
4,
1955. |
MR 70834 |
Zbl 0064.34802[157]
L. E. PAYNE,
News isoperimetric inequalities for eigenvalues and other physical quantities,
Comm. Pure Appl. Math.,
9,
1956. |
MR 81433 |
Zbl 0074.31405[158]
L. E. PAYNE,
Inequalities for eigenvalues of supported and free plates,
Quart. Appl. Math.,
16,
1958. |
MR 96440 |
Zbl 0084.20705[159]
L. E. PAYNE,
Isoperimetric inequalities for eigenvalues and their applications,
Autovalori e autosoluzioni, Centro Internazionale Matematico Estivo 20 ciclo, Chieti,
1962. |
Zbl 0201.57403[160]
L. E. PAYNE,
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]
L. E. PAYNE,
Isoperimetric inequalities and their applications,
SIAM Rev.,
9,
1967. |
MR 218975 |
Zbl 0154.12602[162]
L. E. PAYNE-
G. PÓLYA-
H. F. WEINBERGER,
On the ratio of consecutive eigenvalues,
J. Math. and Phys.,
35,
1956. |
MR 84696 |
Zbl 0073.08203[163]
L. E. PAYNE-
M. E. RAYNER,
An isoperimetric inequality for the first eigenfunction in the fixed membrane problem,
Z.A.M.P.,
23,
1972. |
MR 313649 |
Zbl 0241.73080[164]
L. E. PAYNE-
M. E. RAYNER,
Some isoperimetric norm bounbs for solutions of the Helmholtz equation,
Z.A.M.P.,
24,
1973. |
MR 324202 |
Zbl 0256.35023[165]
L. E. PAYNE-
H. F. WEINBERGER,
Some isoperimetric inequalities for membrane frequencies and torsional rigidity,
J. Math. Anal. Appl.,
2,
1961. |
MR 149735 |
Zbl 0098.39201[167]
M. PĂSIĆ,
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]
H. POINCARÉ,
Figures d'equilibre d'une masse fluide,
Naud, Paris,
1903. |
Jbk 34.0757.05[169]
G. PÓLYA,
Torsional rigidity, principal frequency, eletrostatic capacity and symmetrization,
Quart. Appl. Math.,
6,
1948. |
MR 26817 |
Zbl 0037.25301[170]
G. PÓLYA,
On the eigenvalues of vibrating membranes,
Proc. London Math. Soc.,
11,
1961. |
MR 129219 |
Zbl 0107.41805[171]
G. PÓLYA-
G. SZEGÖ,
Isoperimetric inequalities in mathematical physics,
Ann. of Math. Studies, n.
27,
Princeton University Press,
1951. |
MR 43486 |
Zbl 0044.38301[172]
G. PÓLYA-
A. WEINSTEIN,
On the torsional rigidity of multiply connected crosssections,
Ann. of Math.,
52,
1950. |
MR 40159 |
Zbl 0038.37502[173]
S. POORNIMA,
An embedding theorem for the Sobolev space $W^{1,1}$,
Bull. Sci. Math. (2),
107,
1983. |
MR 719267 |
Zbl 0529.46025[175]
M. R. POSTERARO,
On the solutions of the equation $\Delta u = e^u$ blowing up on the boundary,
C. R. Acad. Sci. Paris, Ser. I Math,
332,
1996. |
MR 1381781 |
Zbl 0848.35037[176]
M. H. PROTTER,
Lower bounds for the first eigenvalue of elliptic equations,
Ann. of Math.,
71,
1960. |
MR 111923 |
Zbl 0095.20004[177]
J.-M. RAKOTOSON,
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]
J.-M. RAKOTOSON,
Some properties of the relative rearrangement,
J. Math. Anal. Appl.,
135, no. 2,
1988. |
MR 967224 |
Zbl 0686.28003[179]
J.-M. RAKOTOSON,
A differentiability result for the relative rearrangement,
Diff. Int. Eq.,
2, no. 3,
1989. |
MR 983687 |
Zbl 0772.35018[180]
M. RAKOTOSON,
Relative rearrangement for highly nonlinear equations,
Nonlinear Anal.,
24, no. 4,
1995. |
MR 1315691 |
Zbl 0830.35036[181]
J.-M. RAKOTOSON-
B. SIMON,
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]
J.-M. RAKOTOSON-
R. TEMAM,
Relative rearrangement in quasilinear elliptic variational inequalities,
Ind. Univ. Math. J.,
36, no. 4,
1987. |
MR 916743[183]
J.-M. RAKOTOSON-
R. TEMAM,
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. |
Jbk 27.0701.05[185]
J. V. RYFF,
Measure preserving transformations and rearrangements,
J. Math. Anal. Appl.,
31,
1970. |
MR 419734 |
Zbl 0214.13701[186] B. DE ST. VENANT, Memoire sur la torsion des prismes, Memor. pres. divers. savants Acad. Sci., 14, 1856.
[187]
F. SCHULZ-
V. VERA DE SERIO,
Symmetrization with respect to a measure,
Trans. Amer. Math. Soc.,
337,
1993. |
MR 1088477 |
Zbl 0844.49025[188]
W. SCHUMANN,
On isoperimetric inequalities in plasticity,
Quart. Appl. Math.,
16,
1958. |
MR 103658 |
Zbl 0086.39104[189]
B. SCHWARZ,
Bounds for the principal frequency of the nonhomogeneous membrane and for the generalized Dirichlet integral,
Pacific J. Math.,
7,
1957. |
fulltext mini-dml |
MR 92918 |
Zbl 0081.31703[190]
E. SPERNER,
Zur symmetrisierung für funktionen auf sphären,
Math. Z.,
134,
1973. |
MR 340558 |
Zbl 0283.26015[191]
E. SPERNER,
Symmetrisierung für funktionen mehrerer reeller variabel,
Manuscripta Math.,
11,
1974. |
MR 328000 |
Zbl 0268.26011[192]
E. SPERNER,
Spherical symmetrization and eigenvalue estimates,
Math. Z.,
176,
1976. |
MR 606173 |
Zbl 0464.49009[193]
W. SPIEGEL,
Über die symmetrisierung stetiger funktionen im euklidishen raum,
Archiv. der Math.,
24,
1973. |
MR 412365 |
Zbl 0274.52011[194]
J. STEINER,
Einfache Beweise der isoperimetrischen Hauptsätze,
Ges. Werke II, Berlin,
1882. |
Zbl 018.0600cj[195]
G. SZEGÖ,
Über einige neue Extremalaufgaben der Potential-theorie,
Math. Z.,
31,
1930. |
Jbk 56.1069.03[196]
G. SZEGÖ,
Inequalities for certain eigenvalues of a membrane of given area,
J. Rational Mech. Anal.,
3,
1954. |
MR 61749 |
Zbl 0055.08802[198]
G. TALENTI,
Best constant in Sobolev inequality,
Ann. Mat. Pura Appl., (4)
110,
1976. |
MR 463908 |
Zbl 0353.46018[200]
G. TALENTI,
Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces,
Ann. Mat. Pura Appl. (4),
120,
1979. |
MR 551065 |
Zbl 0419.35041[202]
G. TALENTI,
On the first eigenvalue of the clamped plate,
Ann. Mat. Pura Appl. (4),
129,
1981. |
MR 648335 |
Zbl 0475.73050[203]
G. TALENTI,
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]
G. TALENTI,
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]
G. TALENTI,
Assembling a rearrangement,
Arch. Rational Mech. Anal.,
98, no. 4,
1987. |
MR 872748 |
Zbl 0619.35113[206]
G. TALENTI,
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[208]
G. TALENTI,
Rearrangements and PDE,
Inequalities (Birmingham, 1987),
Lecture Notes in Pure and Appl. Math.,
129,
Dekker, New York,
1991. |
MR 1112579 |
Zbl 0733.35013[209]
G. TALENTI,
On isoperimetric theorems of mathematical physics,
Handbook of convex geometry, vol. A, B,
North-Holland, Amsterdam,
1993. |
MR 1243005 |
Zbl 0804.35005[210]
G. TALENTI,
The standard isoperimetric theorem,
Handbook of convex geometry, vol. A, B,
North-Holland, Amsterdam,
1993. |
MR 1242977 |
Zbl 0799.51015[213]
M. TRANSIRICO,
Symmetrization in a nonlinear degenerate parabolic problem,
Ann. Mat. Pura Appl. (4),
149,
1987. |
MR 932792 |
Zbl 0655.35048[214]
C. TROMBETTI,
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]
G. TROMBETTI-
G. L. VAZQUEZ,
Symmetrization results for elliptic equations with lower order terms,
Ann. Fac. Sci. Toulouse,
7,
1985. |
fulltext mini-dml |
Zbl 0617.35036[216]
N. S. TRUDINGER,
On embeddings into Orlicz spaces and some applications,
J. Math. Mech., vol.
17,
1967. |
MR 216286 |
Zbl 0163.36402[217]
N. S. TRUDINGER,
On new isoperimetric inequalities and symmetrization,
J. Reine Angew. Math.,
488,
1997. |
MR 1465371 |
Zbl 0883.52006[219]
J. L. VAZQUEZ,
Symétrization pour $u_t=\Delta\phi(u)$ 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]
C. VOAS-
D. YANIRO,
Symmetrization and optimal control for elliptic equations,
Proc. Amer. Math. Soc.,
99, no. 3,
1987. |
MR 875390 |
Zbl 0609.49016[221]
C. VOAS-
D. YANIRO,
Elliptic equations, rearrangements, and functions of bounded lower oscillation,
J. Math. Anal. Appl.,
134, no. 1,
1988. |
MR 958857 |
Zbl 0684.35040[222]
R. VOLPICELLI,
Comparison results for solutions of parabolic equations,
Ricerche di Matematica,
42,
1993. |
MR 1283813 |
Zbl 0827.35018[223]
H. F. WEINBERGER,
Upper and lower bounds for torsional rigidity,
J. Math. and Phys.,
32,
1953. |
MR 55511 |
Zbl 0051.41203[224]
H. F. WEINBERGER,
An isoperimetric inequality for the $N$-dimensional free membrane problem,
J. Rational Mech. Anal.,
5,
1956. |
MR 79286 |
Zbl 0071.09902[225]
H. F. WEINBERGER,
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