On the Lagrange multiplier of a minimization problem related to the isoperimetric inequality Original Research Article

In this note, we study the sign of the Lagrange multiplier for a minimization problem related to the isoperimetric inequality for mappings. Though our minimization problem involves vector valued mappings defined on a domain in R2R2, our result is the same as the one for a minimization problem related to the Sobolev inequality for scalar valued functions in higher dimensions.