Relative minimizers of energy are relative minimizers of area

Let Γ be a closed, regular Jordan curve in ${{mathbb R}^3}$ which is of class C ^1,μ , 0 <  μ <  1, and denote by ${{mathcal C}(Gamma)}$ the class of the disk-type surfaces ${X : B to {mathbb R}^3}$ with continuous, monotonic boundary values, mapping ${partial B}$ onto Γ. One easily sees that any minimal surface ${X in {mathcal C}(Gamma)}$ is a relative minimizer of energy, i.e. of Dirichlet’s integral D, if it is a relative minimizer of the area functional A. Here we prove conversely: If an immersed ${X in {mathcal C}(Gamma)}$ is a C ¹-relative minimizer of D in ${{mathcal C}(Gamma)}$ , then it also is a C ^1,μ -relative minimizer of A in ${{mathcal C}(Gamma)}$ .