A general theorem of existence of quasi absolutely minimal Lipschitz extensions

详细信息    查看全文

• 作者：Matthew J. Hirn (1)
  Erwan Y. Le Gruyer (2)
  
• 关键词：54C20 ; 58C25 ; 46T20 ; 49 ; XX ; 39B05
• 刊名：Mathematische Annalen
• 出版年：2014
• 出版时间：August 2014
• 年：2014
• 卷：359
• 期：3-4
• 页码：595-628
• 全文大小：359 KB
• 参考文献：1. Armstrong, S.N., Smart, C.K.: As easy proof of Jensen’s theorem on the uniqueness of infinity harmonic functions. Calc. Var. Partial Differ. Equ. 37(3-), 381-84 (2010) CrossRef
  2. Aronsson, G.: Minimization problems for the functional \(\sup _xf(x, f(x), f^{\prime }(x))\) . Arkiv f?r Matematik 6, 33-3 (1965) CrossRef
  3. Aronsson, G.: Minimization problems for the functional \(\sup _xf(x, f(x), f^{\prime }(x))\) II. Arkiv f?r Matematik 6, 409-31 (1966) CrossRef
  4. Aronsson, G.: Extension of functions satisfying Lipschitz conditions. Arkiv f?r Matematik 6, 551-61 (1967) CrossRef
  5. Aronsson, G., Crandall, M.G., Juutinen, P.: A tour of the theory of absolutely minimizing functions. Bull. Am. Math. Soc. 41, 439-05 (2004) CrossRef
  6. Barles, G., Busca, J.: Existence and comparison results for fully nonlinear degenerate elliptic equations. Commun. Partial Differ. Equ. 26(11-2), 2323-337 (2001) CrossRef
  7. Jensen, R.: Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient. Arch. Rational Mech. Anal. 123(1), 51-4 (1993) CrossRef
  8. Juutinen, P.: Absolutely minimizing Lipschitz extensions on a metric space. Annales Academiae Scientiarum Fennicae Mathematica 27, 57-7 (2002)
  9. Kelley, J.L.: Banach spaces with the extension property. Trans. Am. Math. Soc. 72(2), 323-26 (1952) CrossRef
  10. Kirszbraun, M.D.: über die zusammenziehende und Lipschitzsche Transformationen. Fundamenta Mathematicae 22, 77-08 (1934)
  11. Le Gruyer, E.: On absolutely minimizing Lipschitz extensions and PDE \(\Delta _{\infty }(u) = 0\) . NoDEA: Nonlinear Differ. Equ. Appl. 14(1-), 29-5 (2007)
  12. Le Gruyer, E.: Minimal Lipschitz extensions to differentiable functions defined on a Hilbert space. Geom. Funct. Anal. 19(4), 1101-118 (2009) CrossRef
  13. McShane, E.J.: Extension of range of functions. Bull. Am. Math. Soc. 40(12), 837-42 (1934) CrossRef
  14. Mickle, E.J.: On the extension of a transformation. Bull. Am. Math. Soc. 55(2), 160-64 (1949) CrossRef
  15. Milman, V.A.: Absolutely minimal extensions of functions on metric spaces. Matematicheskii Sbornik 190(6), 83-10 (1999) CrossRef
  16. Nachbin, L.: A theorem of the Hahn-Banach type for linear transformations. Trans. Am. Math. Soc. 68(1), 28-6 (1950) CrossRef
  17. Naor, A., Sheffield, S.: Absolutely minimal Lipschitz extension of tree-valued mappings. Mathematische Annalen 354(3), 1049-078 (2012) CrossRef
  18. Peres, Y., Schramm, O., Sheffield, S., Wilson, D.B.: Tug-of-war and the infinity Laplacian. J. Am. Math. Soc. 22(1), 167-10 (2009) CrossRef
  19. Schoenberg, I.: On a theorem of Kirszbraun and Valentine. Am. Math. Mon. 60, 620-22 (1953) CrossRef
  20. Sheffield, S., Smart, C.K.: Vector-valued optimal Lipschitz extensions. Commun. Pure. Appl. Math. 65(1), 128-54 (2012) CrossRef
  21. Valentine, F.A.: A Lipschitz condition preserving extension for a vector function. Am. J. Math. 67(1), 83-3 (1945) CrossRef
  22. Wells, J.C.: Differentiable functions on Banach spaces with Lipschitz derivatives. J. Differ. Geom. 8, 135-52 (1973)
  23. Wells, J.H., Williams, L.R.: Embeddings and Extensions in Analysis. Springer, Berlin (1975) CrossRef
  24. Whitney, H.: Analytic extensions of differentiable functions defined in closed sets. Trans. Am. Math. Soc. 36(1), 63-9 (1934) CrossRef
  
• 作者单位：Matthew J. Hirn (1)
  Erwan Y. Le Gruyer (2)
  
  1. Département d’Informatique, école normale supérieure, 45 rue d’Ulm, 75005?, Paris, France
  2. INSA de Rennes & IRMAR, 20, Avenue des Buttes de Co?smes CS 70839, 35708?, Rennes Cedex 7, France
  
• ISSN：1432-1807

文摘

In this paper we consider a wide class of generalized Lipschitz extension problems and the corresponding problem of finding absolutely minimal Lipschitz extensions. We prove that if a minimal Lipschitz extension exists, then under certain other mild conditions, a quasi absolutely minimal Lipschitz extension must exist as well. Here we use the qualifier “quasi-to indicate that the extending function in question nearly satisfies the conditions of being an absolutely minimal Lipschitz extension, up to several factors that can be made arbitrarily small.