Energy estimates for area minimizing hypersurfaces with arbitrary boundaries (Q1971845)

The authors study a variant of the well-known concept of functions of least gradient integral \({\mathcal E}(u)= \int_\Omega|Du|dx\): instead of real-valued functions \(u\) they consider mappings \(u\) from the closed set \(\Omega\subset\mathbb{R}^n\) to the circle \(\mathbb{S}\), \(\mathbb{S}\) being represented as \(\mathbb{R}/\mathbb{Z}\) so that the gradient \(Du(x)\), \(x\in\Omega\), is again an element of \(\mathbb{R}^n\) in a natural way. Their investigation is motivated by possible applications to area minimizing hypersurfaces of \(\mathbb{R}^n\) which can be seen as level sets of functions minimizing the gradient integral. Given \(\Omega\) with bounded boundary \(\partial\Omega\), a Lipschitz function \(f:\Omega\to \mathbb{S}\) and a number \(M\) with Lipschitz constant \(\text{Lip}(f)\leq M\), the authors prove the existence of a minimizer of \({\mathcal E}(u)\) in the class of all mappings \(u:\Omega\to \mathbb{S}\) such that \(u|\partial\Omega= f|\partial\Omega\), \(u\) is homotopic to \(f\), and \(\text{Lip}(u)\leq M\). Moreover, using a suitable notion of the average of two homotopic mappings \(u,v:\Omega\to \mathbb{S}\), the following estimate is proved for a minimizer \(u\) and any competing mapping \(v\) in the above variational problem: \[ \int_\Omega (|Du||Dv|- Du\cdot Dv) dx\leq 2M({\mathcal E}(v)-{\mathcal E}(u)). \] The proofs use standard techniques of the calculus of variations.