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

Authors: Michael G. Crandall and Luc Tartar
Journal: Proc. Amer. Math. Soc. 78 (1980), 385-390
MSC: Primary 47H07
DOI: https://doi.org/10.1090/S0002-9939-1980-0553381-X
MathSciNet review: 553381
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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] H. Brezis and W. Strauss, Semilinear elliptic equations in ${L^1}$ , J. Math. Soc. Japan 25 (1973), 15-26. MR 0336050 (49:826)
[2] M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. (to appear). MR 551288 (81b:65079)
[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. 63 (1963), 11-32. MR 0168930 (29:6186)
[5] S. N. Kružkov, Generalized solutions of first order nonlinear equations in several independent variables, Mat. Sb. 70 (1966), 394-415. MR 0199543 (33:7687)
[6] T. Kurtz, Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics, Trans. Amer. Math. Soc. 186 (1973), 259-272. MR 0336482 (49:1256)
[7] M. Pierre, Un théorème de generation de semi-groupes nonlinéaires, Israel J. Math. 23 (1976), 189-199. MR 0428142 (55:1171)
[8] G. Pólya and G. Szegö, Isoperimetric inequalities in mathmematical physics, Ann. of Math. Studies, no. 27, Princeton Univ. Press, Princeton, N.J., 1951. MR 0043486 (13:270d)
[9] E. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, N.J., 1971. MR 0304972 (46:4102)
[10] M. B. Tamburro, The evolution operator approach to the Hamiltion-Jacobi equation, Israel J. Math. 26 (1977), 232-264. MR 0447777 (56:6087)