A Liouville property for gradient graphs and a Bernstein problem for Hamiltonian stationary equations

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• 作者：Micah W. Warren
• 刊名：manuscripta mathematica
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：150
• 期：1-2
• 页码：151-157
• 全文大小：369 KB
• 参考文献：1.Gilbarg, D., Trudinger, N.S: Elliptic partial differential equations of second order. In: Classics in Mathematics. Springer, Berlin, Reprint of the 1998 edition. MR 1814364 (2001k:35004) (2001)
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  5.Schoen, R., Wolfson, J.: The Volume Functional for Lagrangian Submanifolds. Lectures on Partial Differential Equations, New Stud. Adv. Math., vol. 2, Int. Press, Somerville, MA, pp. 181–191. MR 2055848 (2005f:53141) (2003)
  6.Yuan, Y.: A Bernstein problem for special Lagrangian equations. Invent. Math. 150(1), 117–125. MR 1930884 (2003k:53060) (2002)
  7.Yuan, Y.: Global solutions to special Lagrangian equations. Proc. Am. Math. Soc. 134(5), 1355–1358 (electronic). MR 2199179 (2006k:35111) (2006)
  
• 作者单位：Micah W. Warren (1)
  
  1. Department of Mathematics, University of Oregon, Eugene, OR, 97403, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Algebraic Geometry
  Topological Groups and Lie Groups
  Geometry
  Number Theory
  Calculus of Variations and Optimal Control
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1785

文摘

Using a rotation of Yuan, we observe that the gradient graph of any semi-convex function is a Liouville manifold, that is, does not admit non-constant bounded harmonic functions. As a corollary, we find that any solution of the fourth order Hamiltonian stationary equation satisfying $$\theta \geq \left( n - 2\right) \frac{\pi}{2} + \delta$$for some \({\delta > 0}\) must be a quadratic. Mathematics Subject Classification Primary 35J30 Secondary 31B05 53D12 The author is partially supported by NSF Grant DMS-1438359.