Pub Date : 2001-12-12DOI: 10.1201/9781420035582.CH10
L. Carbone, R. D. Arcangelis
{"title":"Relaxation of Unbounded Functionals","authors":"L. Carbone, R. D. Arcangelis","doi":"10.1201/9781420035582.CH10","DOIUrl":"https://doi.org/10.1201/9781420035582.CH10","url":null,"abstract":"","PeriodicalId":345528,"journal":{"name":"Unbounded Functionals in the Calculus of Variations","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124353343","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2001-12-12DOI: 10.1201/9781420035582.CH9
L. Carbone, R. D. Arcangelis
{"title":"Integral Representation for Unbounded Functionals","authors":"L. Carbone, R. D. Arcangelis","doi":"10.1201/9781420035582.CH9","DOIUrl":"https://doi.org/10.1201/9781420035582.CH9","url":null,"abstract":"","PeriodicalId":345528,"journal":{"name":"Unbounded Functionals in the Calculus of Variations","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117193849","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2001-12-12DOI: 10.1201/9781420035582.ch1
L. Carbone, R. D. Arcangelis
Proposition 2.0 (i) Any set K ⊂ B is closed and convex iff IK is closed and convex. (ii) Any function f : B →]−∞,+∞] is lower semicontinuous and convex iff epi(f) is closed and convex. (iii) If {Ki}i∈I is a family of closed convex subsets of B, then ⋂ iKi is closed and convex. (iv) If {fi}i∈I is a family of lower semicontinuous convex functions B →] −∞,+∞], then their upper hull f(·) := supi fi(·) is lower semicontinuous and convex.
{"title":"Elements of Convex Analysis","authors":"L. Carbone, R. D. Arcangelis","doi":"10.1201/9781420035582.ch1","DOIUrl":"https://doi.org/10.1201/9781420035582.ch1","url":null,"abstract":"Proposition 2.0 (i) Any set K ⊂ B is closed and convex iff IK is closed and convex. (ii) Any function f : B →]−∞,+∞] is lower semicontinuous and convex iff epi(f) is closed and convex. (iii) If {Ki}i∈I is a family of closed convex subsets of B, then ⋂ iKi is closed and convex. (iv) If {fi}i∈I is a family of lower semicontinuous convex functions B →] −∞,+∞], then their upper hull f(·) := supi fi(·) is lower semicontinuous and convex.","PeriodicalId":345528,"journal":{"name":"Unbounded Functionals in the Calculus of Variations","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115175838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2001-12-12DOI: 10.1201/9781420035582.CH14
L. Carbone, R. D. Arcangelis
{"title":"Some Explicit Computations of Homogenized Energies in Mathematical Models Originating Unbounded Functionals","authors":"L. Carbone, R. D. Arcangelis","doi":"10.1201/9781420035582.CH14","DOIUrl":"https://doi.org/10.1201/9781420035582.CH14","url":null,"abstract":"","PeriodicalId":345528,"journal":{"name":"Unbounded Functionals in the Calculus of Variations","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121906475","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2001-12-12DOI: 10.1201/9781420035582.ch11
L. Carbone, R. D. Arcangelis
{"title":"Cut-off Functions and Partitions of Unity","authors":"L. Carbone, R. D. Arcangelis","doi":"10.1201/9781420035582.ch11","DOIUrl":"https://doi.org/10.1201/9781420035582.ch11","url":null,"abstract":"","PeriodicalId":345528,"journal":{"name":"Unbounded Functionals in the Calculus of Variations","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130839249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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