Gronwall’s inequality in $n$ independent variables

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.