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On the duality between p -modulus and probability measures

Luigi Ambrosio, Simone Di Marino, Giuseppe Savaré (2015)

Journal of the European Mathematical Society

Motivated by recent developments on calculus in metric measure spaces ( X , d , m ) , we prove a general duality principle between Fuglede’s notion [15] of p -modulus for families of finite Borel measures in ( X , d ) and probability measures with barycenter in L q ( X , m ) , with q dual exponent of p ( 1 , ) . We apply this general duality principle to study null sets for families of parametric and non-parametric curves in X . In the final part of the paper we provide a new proof, independent of optimal transportation, of the equivalence...

On the lower semicontinuity of certain integral functionals

Ennio De Giorgi, Giuseppe Buttazzo, Gianni Dal Maso (1983)

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

Si dimostra che il funzionale Ω f ( u , D u ) d x è semicontinuo inferiormente su W l o c 1 , 1 ( Ω ) , rispetto alla topologia indotta da L l o c 1 ( Ω ) , qualora l’integrando f ( s , p ) sia una funzione non-negativa, misurabile in s , convessa in p , limitata nell’intorno dei punti del tipo ( s , 0 ) , e tale che la funzione s f ( s , 0 ) sia semicontinua inferiormente su 𝐑 .

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