Absolutely continuous functions of two variables in the sense of Caratheodory

Electronic Journal of Differential Equations

关键词: Absolutely continuous function;    Caratheodory sense;    integral representation;    derivative of double integral;   

DOI  :  

来源: DOAJ

【 摘 要 】

In this note, the notion of absolute continuity of functions of twovariables is discussed. We recall that the set of functions of twovariables absolutely continuous in the sense of Caratheodorycoincides with the class of functions admitting a certain integralrepresentation. We show that absolutely continuous functions in thesense of Caratheodory can be equivalently characterized in termsof their properties with respect to each of variables. Theseequivalent characterizations play an important role in theinvestigation of boundary value problems for partial differentialequation of hyperbolic type with discontinuous right-hand side. Wepresent several statements which are rather important when analyzingstrong solutions of such problems by using the methods of realanalysis but, unfortunately, are not formulated and proven preciselyin the existing literature, which mostly deals with weak solutionsor the case where the right-hand side of the equation is continuous.

【 授权许可】

Unknown