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Construction of W 2 , n ( Ω ) function with gradient violating Lusin (N) condition
Author(s) -
Roskovec Tomáš
Publication year - 2016
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201500014
Subject(s) - measure (data warehouse) , mathematics , set (abstract data type) , construct (python library) , zero (linguistics) , function (biology) , data mining , computer science , linguistics , philosophy , evolutionary biology , biology , programming language
We construct f ∈ C 1 ( [ − 1 , 1 ] n ) , such that D f ∈ W 1 , n( ( − 1 , 1 ) n , R n )maps[ − 1 , 1 ] × { 0 } n − 1onto[ − 1 , 1 ] n . This shows that a mapping which maps a set of zero measure onto the set of positive measure can be a gradient mapping.