On a Problem Posed by Brezis and Mironescu

TL;DR

This paper provides a positive resolution to an open problem from Brezis and Mironescu's work, establishing a key equality involving area-minimizing currents and smooth submanifolds with the same boundary.

Contribution

It proves that the least mass of area-minimizing currents equals the infimum of areas among smooth immersed submanifolds with the same boundary, confirming a conjecture in geometric measure theory.

Findings

Least mass of area-minimizing currents equals infimum among smooth submanifolds.

Validates a conjecture in the context of boundary regularity.

Connects minimal currents to smooth geometric solutions.

Abstract

The purpose of this note is to present a positive answer to an open problem proposed in the recent book \cite{Brezis-Mironescu} by H. Brezis and P. Mironescu. It has been stated in this book {\it Sobolev Maps to the Circle} as Proposition 4.3. We demonstrate, in particular, the value of the least mass of the area minimizing integral rectifiable currents with a given boundary equals to the infimum of areas among smoothly immersed submanifolds with the same boundary, under the assumption that the boundary is that of a smooth submanifold.