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


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.


Variational Problem Singular Perturbation Quadratic Estimate Eikonal Equation Singular Perturbation Problem 
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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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