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Mathematische Annalen

, Volume 338, Issue 1, pp 119–146 | Cite as

Sharp upper bounds for a variational problem with singular perturbation

  • Sergio ContiEmail author
  • Camillo De Lellis
Article

Abstract

Let Ω be a C 2 bounded open set of \(\mathbb{R}^{2}\) and consider the functionals \(F_{\epsilon} (u) := \int\limits_{\Omega} \left\{\frac{(1-|{\nabla} u (x)|^{2})^{2}}{\epsilon} + {\epsilon} |D^{2} u (x)|^{2}\right\} {\rm d}x\) We prove that if \(u\in W^{1, \infty} (\Omega)\), |∇ u| = 1 a.e., and ∇ uBV, then \(\Gamma-\lim\limits_{\epsilon\downarrow0} F_\epsilon (u)=\frac13 \int\limits_{J_{\nabla u}} |[\nabla u]|^{3} {\rm d}\fancyscript{H}^{1}.\) The new result is the Γ- lim sup inequality.

Keywords

Variational Problem Singular Perturbation Quadratic Estimate Eikonal Equation Singular Perturbation Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Fachbereich MathematikUniversität Duisburg-EssenDuisburgGermany
  2. 2.Camillo De Lellis, Institut für MathematikUniversität ZürichZürichSwitzerland

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