About: Absolutely continuous functions of two varables in the sense of Carathéodory

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Description	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. 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. (en)
Title	Absolutely continuous functions of two varables in the sense of Carathéodory Absolutely continuous functions of two varables in the sense of Carathéodory (en)
skos:prefLabel	Absolutely continuous functions of two varables in the sense of Carathéodory Absolutely continuous functions of two varables in the sense of Carathéodory (en)
skos:notation	RIV/67985840:_____/10:00349672!RIV11-GA0-67985840
http://linked.open...avai/riv/aktivita	P Z
http://linked.open...avai/riv/aktivity	P(GA201/06/0254), Z(AV0Z10190503)
http://linked.open...iv/cisloPeriodika	154
http://linked.open...vai/riv/dodaniDat	2011
http://linked.open...aciTvurceVysledku	Šremr, Jiří
http://linked.open.../riv/druhVysledku	J - Článek v odborném periodiku
http://linked.open...iv/duvernostUdaju	S - Úplné a pravdivé údaje nepodléhající ochraně podle zvláštních právních předpisů
http://linked.open...titaPredkladatele	Matematický ústav AV ČR, v. v. i.
http://linked.open...dnocenehoVysledku	245133
http://linked.open...ai/riv/idVysledku	RIV/67985840:_____/10:00349672
http://linked.open...riv/jazykVysledku	eng - angličtina
http://linked.open.../riv/klicovaSlova	absolutely continuous function; Carathéodory sense; integral representation; derivative of double integral (en)
http://linked.open.../riv/klicoveSlovo	integral representation Carathéodory sense absolutely continuous function derivative of double integral
http://linked.open...odStatuVydavatele	US - Spojené státy americké
http://linked.open...ontrolniKodProRIV	[7D18C6E2EF13]
http://linked.open...i/riv/nazevZdroje	Electronic Journal of Differential Equations
http://linked.open...in/vavai/riv/obor	BA
http://linked.open...ichTvurcuVysledku	1 (xsd:int)
http://linked.open...cetTvurcuVysledku	1 (xsd:int)
http://linked.open...vavai/riv/projekt	Functional differential equations in Banach spaces
http://linked.open...UplatneniVysledku	2010
http://linked.open...v/svazekPeriodika	2010
http://linked.open...iv/tvurceVysledku	Šremr, Jiří
http://linked.open...n/vavai/riv/zamer	Rozvoj a prohloubení obecných matematických poznatků a jejich užití v dalších vědních oborech a v praxi
issn	1072-6691
number of pages	11 (xsd:int)
is http://linked.open...avai/riv/vysledek of	Absolutely continuous functions of two varables in the sense of Carathéodory Absolutely continuous functions of two varables in the sense of Carathéodory Absolutely continuous functions of two varables in the sense of Carathéodory

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