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Absolutely continuous functions of two variables in the sense of Carathéodory. (English) Zbl 1200.26016
Summary: The notion of absolute continuity of functions of two variables is discussed. We recall that the set of functions of two variables that is absolutely continuous in the sense of Carathéodory coincides with the class of functions admitting a certain integral representation. We show that absolutely continuous functions in the sense of Carathéodory can be equivalently characterized in terms of their properties with respect to each of the 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, they 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.

26B30 Absolutely continuous real functions of several variables, functions of bounded variation
26B05 Continuity and differentiation questions
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