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Nuovi risultati sulla semicontinuità inferiore di certi funzionali integrali

Luigi Ambrosio (1985)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Given an open subset Ω of n and a Borel function f : Ω × × n [ 0 , + [ , conditions on f are given which assure the lower semicontinuity of the functional Ω f ( x , u , D u ) d x with respect to different topologies.

On a problem of Mazur from "The Scottish Book" concerning second partial derivatives

Volodymyr Mykhaylyuk, Anatolij Plichko (2015)

Colloquium Mathematicae

We comment on a problem of Mazur from “The Scottish Book" concerning second partial derivatives. We prove that if a function f(x,y) of real variables defined on a rectangle has continuous derivative with respect to y and for almost all y the function F y ( x ) : = f y ' ( x , y ) has finite variation, then almost everywhere on the rectangle the partial derivative f y x ' ' exists. We construct a separately twice differentiable function whose partial derivative f x ' is discontinuous with respect to the second variable on a set of positive...

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