TITLE

Unstable extremal surfaces of the �Shiffman functional� spanning rectifiable boundary curves

AUTHOR(S)
Jakob, Ruben
PUB. DATE
March 2007
SOURCE
Calculus of Variations & Partial Differential Equations;Mar2007, Vol. 28 Issue 3, p383
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
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 H 1,2-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.
ACCESSION #
23468775

 

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