Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 154, pp. 1-11.
Title: Absolutely continuous functions of two variables in the sense of
Caratheodory
Author: Jiri Sremr (Academy of Sciences, Czech epublic)
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.
Submitted March 19, 2009. Published October 28, 2010.
Math Subject Classifications: 26B30, 26B05.
Key Words: Absolutely continuous function; Caratheodory sense;
integral representation; derivative of double integral.