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An inequality concerning rearrangements of functions and Hamilton-Jacobi equations. (English) Zbl 0787.35020
The authors prove an inequality concerning the decreasing rearrangement of functions. This inequality gives a comparison result between the viscosity solution of an initial boundary value problem for the Hamilton- Jacobi equation and the viscosity solution of the “symmetrized” problem.

MSC:
35F10 Initial value problems for linear first-order PDEs
35A15 Variational methods applied to PDEs
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
26D20 Other analytical inequalities
35D99 Generalized solutions to partial differential equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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[1] Alvino, A., Díaz, J. I., Lions, P. L. & Trombetti, G., Elliptic equations and Steiner symmetrization, C.R. Acad. Sci. Paris 314 (I) (1992), 1015-1020. · Zbl 0795.35022
[2] Alvino, A., Lions, P. L. & Trombetti, G., On optimization problems with prescribed rearrangements, Nonlinear Analysis T.M.A. 13 (1989), 185-220. · Zbl 0678.49003 · doi:10.1016/0362-546X(89)90043-6
[3] Alvino, A. & Trombetti, G., Sulle migliori costanti di maggiorazione per una classe di equazioni ellittiche degeneri, Ricerche Mat. 27 (1978), 413-428. · Zbl 0403.35027
[4] Bandle, C., On symmetrizations in parabolic equations, J. Anal. Math. 30 (1976), 98-112. · Zbl 0331.35036 · doi:10.1007/BF02786706
[5] Bandle, C., Isoperimetric Inequalities and Applications, Pitman, London, 1980. · Zbl 0436.35063
[6] 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.
[7] Crandall, M. G. & Lions, P. L., Viscosity solutions of Hamilton-Jacobi equations, Trans. Am. Math. Soc. 277 (1983), 1-42. · Zbl 0599.35024 · doi:10.1090/S0002-9947-1983-0690039-8
[8] De Giorgi, E., Su una teoria generale delia misura (r?1)-dimensionale in uno spazio ad r dimensioni, Ann. Mat. Pura e Appl. 36 (1954), 191-213. · Zbl 0055.28504 · doi:10.1007/BF02412838
[9] Federer, H., Geometric Measure Theory, Springer-Verlag, Heidelberg, 1969. · Zbl 0176.00801
[10] Fleming, W. & Rishel, R., An integral formula for total gradient variation, Arch. Math. 11 (1960), 218-222. · Zbl 0094.26301 · doi:10.1007/BF01236935
[11] Giarrusso, E., Estimates for generalized subsolutions of first order Hamilton-Jacobi equations and rearrangements, Nonlinear Analysis T.M.A. 18 (1992), 9-16. · Zbl 0758.35008 · doi:10.1016/0362-546X(92)90045-G
[12] Giarrusso, E. & Nunziante, D., Symmetrization in a class of first-order Hamilton-Jacobi equations, Nonlinear Analysis T.M.A. 8 (1984), 289-299. · Zbl 0543.35014 · doi:10.1016/0362-546X(84)90031-2
[13] Giarrusso, E. & Nunziante, D., Comparison theorems for a class of first order Hamilton-Jacobi equations, Ann. Fac. Sci. Toulouse 7 (1985), 57-73. · Zbl 0554.35007
[14] Hardy, G. H., Littlewood, J. E. & Pólya, G., Inequalities, Cambridge University Press, Cambridge, 1964.
[15] Lions, P. L., Generalized solutions of Hamilton-Jacobi equations, Pitman, London, 1982. · Zbl 0497.35001
[16] Mossino, J., Inégalités isopérimétriques et applications en physique, Collection Travaux en Cours, Hermann, Paris, 1984.
[17] Mossino, J. & Rakotoson, J. M., Isoperimetric inequalities in parabolic equations, Ann. Scuola Norm. Sup. Pisa 13 (1986), 51-73. · Zbl 0652.35053
[18] Talenti, G., Elliptic Equations and rearrangements, Ann. Scuola Norm. Sup. Pisa, (4) 3 (1976), 697-718. · Zbl 0341.35031
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