On A Minimization Problem for Vector Fields in L 1

Abstract

This paper treats the problem of minimizing the norm of vector fields in L1 with prescribed divergence. The ridge of Ω. plays an important role in the analysis, and in the case where Ω ⊂ R2 is a polygonal domain, the ridge is thoroughly analysed and some examples are presented. In the case where Ω ⊂ Rn 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 υ1 = −F▿d, where F is a nonnegative function and d(x) is the distance from x ∈ Ω to the boundary ∂Ω. Finally, if Ω ⊂ R2 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.