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This paper treats the problem of minimizing the norm of vectorfields in L1 with prescribed divergence. The ridge of . playsan important role in the analysis, and in the case where R2is a polygonal domain, the ridge is thoroughly analysed andsome examples are presented. In the case where Rn is a Lipschitzdomain and the divergence is a finite positive Borel measure,the infimum is calculated, and it is shown that if an extremalexists, then it is of the form 1 = Fd, where F is a nonnegativefunction and d(x) is the distance from x to the boundary .Finally, if R2 is a polygonal domain and the measure is representedby a nonnegative continuous function, then an explicit expressionfor the extremal is given, and it is proven that this extremalis unique. 相似文献
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