Higher Order Derivatives of Dirichlet’s Energy

Abstract

In this chapter we take the point of view of Jesse Douglas and consider minimal surfaces as critical points of Dirichlet’s integral within the class of harmonic surfaces X:B→ℝ³ that are continuous on the closure of the unit disk B and map ∂B=S ¹ homeomorphically onto a closed Jordan curve Γ of ℝ³. It will be assumed that Γ is smooth of class C ^∞ and nonplanar. Then any minimal surface bounded by Γ will be a nonplanar surface of class $C^{\infty}(\overline {B},\mathbb{R}^{3})$, and so we shall be allowed to take directional derivatives (i.e. “variations”) of any order of the Dirichlet integral along an arbitrary C ^∞-smooth path through the minimal surface.