Unstable extremal surfaces of the “Shiffman functional” spanning rectifiable boundary curves

Abstract

In this paper we derive a sufficient condition for the existence of extremal surfaces of a parametric functional \(\mathcal{J}\) with a dominant area term, which do not furnish global minima of \(\mathcal{J}\) within the class \(\mathcal{C}^*(\Gamma )\) of H1,2-surfaces spanning an arbitrary closed rectifiable Jordan curve \(\Gamma\subset \mathbb{R}^3\) that merely has to satisfy a chord-arc condition. The proof is based on the “mountain pass result” of (Jakob in Calc Var 21:401–427, 2004) which yields an unstable \(\mathcal{J}\)-extremal surface bounded by an arbitrary simple closed polygon and Heinz’ ”approximation method” in (Arch Rat Mech Anal 38:257–267, 1970). Hence, we give a precise proof of a partial result of the mountain pass theorem claimed by Shiffman in (Ann Math 45:543–576, 1944) who only outlined a very sketchy and partially incorrect proof.