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On A Minimization Problem for Vector Fields in L 1
Author(s) -
Janfalk Ulf
Publication year - 1996
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/28.2.165
Subject(s) - mathematics , infimum and supremum , lipschitz continuity , domain (mathematical analysis) , measure (data warehouse) , divergence (linguistics) , boundary (topology) , mathematical analysis , function (biology) , norm (philosophy) , ridge , measurable function , combinatorics , vector valued function , borel measure , pure mathematics , probability measure , bounded function , paleontology , linguistics , philosophy , database , law , biology , evolutionary biology , computer science , political science
This paper treats the problem of minimizing the norm of vector fields in L 1 with prescribed divergence. The ridge of Ω. plays an important role in the analysis, and in the case where Ω ⊂ R 2 is a polygonal domain, the ridge is thoroughly analysed and some examples are presented. In the case where Ω ⊂ R n is a Lipschitz domain and the divergence is a finite positive Borel measure, the infimum is calculated, and it is shown that if an extremal exists, then it is of the form υ 1 = − F ▿ d , where F is a nonnegative function and d ( x ) is the distance from x ∈ Ω to the boundary ∂Ω. Finally, if Ω ⊂ R 2 is a polygonal domain and the measure is represented by a nonnegative continuous function, then an explicit expression for the extremal is given, and it is proven that this extremal is unique.