Some relations between nonexpansive and order preserving mappings

Crandall, Michael G.; Tartar, Luc

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

Some relations between nonexpansive and order preserving mappings
HTML articles powered by AMS MathViewer

by Michael G. Crandall and Luc Tartar PDF

Proc. Amer. Math. Soc. 78 (1980), 385-390 Request permission

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.

References

Haïm Brézis and Walter A. Strauss, Semi-linear second-order elliptic equations in $L^{1}$, J. Math. Soc. Japan 25 (1973), 565–590. MR 336050, DOI 10.2969/jmsj/02540565
Michael G. Crandall and Andrew Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), no. 149, 1–21. MR 551288, DOI 10.1090/S0025-5718-1980-0551288-3

Lectures on discontinuous solutions of first order nonlinear partial differential equations in several space variables

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 168930, DOI 10.1007/BF02412176
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
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 336482, DOI 10.1090/S0002-9947-1973-0336482-1
Michel Pierre, Un théorème général de génération de semi-groupes non linéaires, Israel J. Math. 23 (1976), no. 3-4, 189–199. MR 428142, DOI 10.1007/BF02761799
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
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
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 447777, DOI 10.1007/BF03007645