On free boundary Plateau problem for general dimensional surfaces (Q2266437)

The author extends Reifenberg's results on the Plateau problem to a free boundary problem. The free boundary is defined homologically: The set where the free boundaries lie will be a fixed compact \(E\subset {\mathbb{R}}^ p\); for any compact subset X of \({\mathbb{R}}^ p\) we let FB(X) denote \(X\cap E\); the algebraic free boundary of X on E will be \(j_*(\ker i_*)\) where \(i:FB(X)\to X\) and \(j:FB(X)\to E\) are the inclusion maps. For a fixed subgroup \(\Gamma\) of \(H_{m-1}(E;G)\), where G is a fixed compact abelian group, we say X is a surface with free boundary including \(\Gamma\) if the algebraic free boundary of X contains \(\Gamma\). The author proves that there exists a surface with free boundary including \(\Gamma\) which is of least m-dimensional Hausdorff measure among all surfaces with free boundary including \(\Gamma\) and which is real analytic at \({\mathcal H}^ m\)-almost every point of \(X\setminus E\). The author also gives some examples to illustrate the definition of algebraic free boundary. There are some minor errors in the proof of Assertion 3, but these are easily remedied.