Some relations between nonexpansive and order preserving mappings

Crandall, Michael G.; Tartar, Luc

doi:10.1090/S0002-9939-1980-0553381-X

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ISSN 1088-6826(online) ISSN 0002-9939(print)

Some relations between nonexpansive and order preserving mappings

Authors: Michael G. Crandall and Luc Tartar
Journal: Proc. Amer. Math. Soc. 78 (1980), 385-390
MSC: Primary 47H07
MathSciNet review: 553381
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Abstract: It is shown that nonlinear operators which preserve the integral are order preserving if and only if they are nonexpansive in ${L^1}$ and that those which commute with translation by a constant are order preserving if and only if they are nonexpansive in ${L^\infty }$ . Examples are presented involving partial differential equations, difference approximations and rearrangements.

[1] Haïm Brézis and Walter A. Strauss, Semi-linear second-order elliptic equations in 𝐿¹, J. Math. Soc. Japan 25 (1973), 565–590. MR 0336050 (49 #826)
[2] Michael G. Crandall and Andrew Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), no. 149, 1–21. MR 551288 (81b:65079), http://dx.doi.org/10.1090/S0025-5718-1980-0551288-3
[3] A. Douglis, Lectures on discontinuous solutions of first order nonlinear partial differential equations in several space variables, North British Symposium on Partial Differential Equations, 1972.
[4] Ignace I. Kolodner, On the Carleman’s model for the Boltzmann equation and its generalizations, Ann. Mat. Pura Appl. (4) 63 (1963), 11–32. MR 0168930 (29 #6186)
[5] S. N. Kružkov, Generalized solutions of nonlinear equations of the first order with several variables. I, Mat. Sb. (N.S.) 70 (112) (1966), 394–415 (Russian). MR 0199543 (33 #7687)
[6] Thomas G. Kurtz, Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics, Trans. Amer. Math. Soc. 186 (1973), 259–272 (1974). MR 0336482 (49 #1256), http://dx.doi.org/10.1090/S0002-9947-1973-0336482-1
[7] Michel Pierre, Un théorème général de génération de semi-groupes non linéaires, Israel J. Math. 23 (1976), no. 3-4, 189–199. MR 0428142 (55 #1171)
[8] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, N. J., 1951. MR 0043486 (13,270d)
[9] Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972 (46 #4102)
[10] Michael B. Tamburro, The evolution operator solution of the Cauchy problem for the Hamilton-Jacobi equation, Israel J. Math. 26 (1977), no. 3-4, 232–264. MR 0447777 (56 #6087)

Bibliography