Boundary regularity for minimizing currents with prescribed mean curvature
The authors prove complete boundary regularity for energy minimizing integer multiplicity rectifiable \(n\) currents \(T\) in \(\mathbb{R}^{n+1}\) of prescribed mean curvature \(H\) with boundary \(B= \partial T\) represented by an oriented smooth submanifold of dimension \(n-1\) in \(\mathbb{R}^{n+1}\). Also, combining their boundary regularity theorem with the interior regularity theory, they formulate and prove a theorem regarding the Plateau problem for surfaces with prescribed mean curvature. The notion of energy minimality here refers to the energy functional \(E_ H\) associated with a prescribed mean curvature function \(H: \mathbb{R}^{n+1}\to \mathbb{R}\), i.e., \(E_ H(T)= M(T)- V_ H(T)\), where \(M(T)\) is the mass (= total \(n\) area) of \(T\) and \(V_ H(T)\) is the \(H\) weighted volume enclosed by \(T\) and a fixed integer multiplicity rectifiable reference \(n\) current \(T_ 0\) with \(\partial T_ 0= B\). The references contain 35 papers, but the indispensable work for the preceding results is \textit{R. Hardt} and \textit{L. Simon} [Ann. Math., II. Ser. 110, 439-486 (1979; Zbl 0457.49029)].
scientific article; zbMATH DE number 484525
scientific article