Absolutely continuous functions of two variables in the sense of Caratheodory

Abstract

In this note, the notion of absolute continuity of functions of two variables is discussed. We recall that the set of functions of two variables absolutely continuous in the sense of Caratheodory coincides with the class of functions admitting a certain integral representation. We show that absolutely continuous functions in the sense of Caratheodory can be equivalently characterized in terms of their properties with respect to each of variables. These equivalent characterizations play an important role in the investigation of boundary value problems for partial differential equation of hyperbolic type with discontinuous right-hand side. We present several statements which are rather important when analyzing strong solutions of such problems by using the methods of real analysis but, unfortunately, are not formulated and proven precisely in the existing literature, which mostly deals with weak solutions or the case where the right-hand side of the equation is continuous.

Keywords

• Absolutely continuous function
• Caratheodory sense
• integral representation
• derivative of double integral