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

Jakob, Ruben
March 2007
Calculus of Variations & Partial Differential Equations;Mar2007, Vol. 28 Issue 3, p383
Academic Journal
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.


Related Articles

  • Direct Block Backward Differentiation Formulas for Solving Second Order Ordinary Differential Equations. Ibrahim, Zarina Bibi; Suleiman, Mohamed; Othman, Khairil Iskandar // Proceedings of World Academy of Science: Engineering & Technolog;Jun2008, Vol. 42, p57 

    In this paper, a direct method based on variable step size Block Backward Differentiation Formula which is referred as BBDF2 for solving second order Ordinary Differential Equations (ODEs) is developed. The advantages of the BBDF2 method over the corresponding sequential variable step variable...

  • On the positive solutions of certain semi-linear elliptic equations. Agarwal, Ravi P.; Mustafa, Octavian G.; Popescu, Liviu // Bulletin of the Belgian Mathematical Society - Simon Stevin;Feb2009, Vol. 16 Issue 1, p49 

    We establish that the elliptic equation Δu + f (x, u) + g(|x|)x · ∇u = 0, where x ∈ ℝn, n ≥ 3, and |x| > A > 0, has a positive solution which decays to 0 as |x| → +∞ undermild restrictions on the functions f , g. The main theorem improves substantially...

  • APPLICATION OF SUMUDU AND DOUBLE SUMUDU TRANSFORMS TO CAPUTO-FRACTIONAL DIFFERENTIAL EQUATIONS. Jarad, F.; Tas, K. // Journal of Computational Analysis & Applications;Apr2012, Vol. 14 Issue 3, p475 

    The definition, properties and applications of the Sumudu transform to ordinary differential equations are described in [1-3]. In this manuscript we derive the formulae for the Sumudu and double Sumudu transforms of ordinary and partial fractional derivatives and apply them in solving...

  • On Some New Impulsive Integral Inequalities. Jianli Li // Journal of Inequalities & Applications;2008, Vol. 2008, p1 

    The article discusses a study which establish new impulsive integral inequalities connected to certain integral inequalities coming from the theory of differential equalities. The study aims to create new impulsive integral inequalities like the Bihari's inequalities. It states that differential...

  • HAHN-BANACH THEOREM IN GENERALIZED 2-NORMED SPACES. Lewandowska, Zofia; Moslehian, Mohammad Sal; Moghaddam, Assieh Saadatpour // Communications in Mathematical Analysis;2006, Vol. 1 Issue 2, p109 

    In this paper we prove an extension Hahn-Banach theorem in the context of generalized 2-normed spaces.

  • BOUNDARY VALUE PROBLEMS FOR DIFFERENTIAL EQUATIONS WITH FRACTIONAL ORDER. Benchohra, Mouffak; Hamani, Samira; Ntouyas, Sotiris K. // Surveys in Mathematics & its Applications;2008, Vol. 3, p1 

    In this paper, we shall establish sufficient conditions for the existence of solutions for a first order boundary value problem for fractional differential equations

  • L p-Poincaré inequality for general symmetric forms. Yong Hua Mao // Acta Mathematica Sinica;Dec2009, Vol. 25 Issue 12, p2055 

    L p Poincaré inequalities for general symmetric forms are established by new Cheeger’s isoperimetric constants. L p super-Poincaré inequalities are introduced to describe the equivalent conditions for the L p compact embedding, and the criteria via the new Cheeger’s constants...

  • The partial averaging of fuzzy differential inclusions. Skripnik, Natalia V. // Journal of Advanced Research in Differential Equations;2011, Vol. 3 Issue 1, p52 

    In this paper the substantiation of the method of partial averaging for fuzzy differential inclusions is considered. Such variant of the averaging method happens to be useful when the average of some mappings does not exist or their presence in the inclusion does not complicate its research....

  • A note on the uniform limit of transitive dynamical systems. Fedeli, Alessandro; Le Donne, Attilio // Bulletin of the Belgian Mathematical Society - Simon Stevin;Feb2009, Vol. 16 Issue 1, p59 

    In this note we study the dynamical behaviour of the uniform limit of a sequence of continuous self-maps on a compact metric space satisfying (topological) transitivity or other related properties. Moreover, some conditions for the transitivity of a limit are given.


Read the Article


Sign out of this library

Other Topics