Gronwall’s inequality in 𝑛 independent variables

Young, Eutiquio C.

doi:10.1090/S0002-9939-1973-0320493-1

Gronwall’s inequality in $n$ independent variables
HTML articles powered by AMS MathViewer

by Eutiquio C. Young PDF

Proc. Amer. Math. Soc. 41 (1973), 241-244 Request permission

Abstract:

The paper presents an extension of Gronwall’s inequality to $n$ independent variables. The inequality is established by solving a characteristic initial value problem by the Riemann method. Thus a Riemann function associated with a hyperbolic partial differential equation appears in the inequality. By using a representation of the Riemann function, the result is shown to coincide with an earlier result obtained by Walter using an entirely different approach.

References

T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. of Math. (2) 20 (1919), no. 4, 292–296. MR 1502565, DOI 10.2307/1967124
Richard Bellman, The stability of solutions of linear differential equations, Duke Math. J. 10 (1943), 643–647. MR 9408
J. Conlan and J. B. Diaz, Existence of solutions of an $n$-th order hyperbolic partial differential equation, Contributions to Differential Equations 2 (1963), 277–289 (1963). MR 155106
Wolfgang Walter, Differential- und Integral-Ungleichungen und ihre Anwendung bei Abschätzungs- und Eindeutigkeits-problemen, Springer Tracts in Natural Philosophy, Vol. 2, Springer-Verlag, Berlin-New York, 1964 (German). MR 0172076
Donald R. Snow, A two independent variable Gronwall-type inequality, Inequalities, III (Proc. Third Sympos., Univ. California, Los Angeles, Calif., 1969; dedicated to the memory of Theodore S. Motzkin), Academic Press, New York, 1972, pp. 333–340. MR 0338537
Donald R. Snow, Gronwall’s inequality for systems of partial differential equations in two independent variables, Proc. Amer. Math. Soc. 33 (1972), 46–54. MR 298188, DOI 10.1090/S0002-9939-1972-0298188-1

The solution of the characteristic and the Cauchy boundary value problem for the Bianchi partial differential equation in $n$ independent variables by a generalization of Riemann’s method