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A regularity result for critical points of conformally invariant functionals

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Abstract

Let Ω be an open set inR 2 andI be a conformally invariant functional defined onH 1(Ω,R d). Letu be a critical point ofI. We show that, ifu is apriori assumed to be bounded, thenu is smooth in Ω, up to ∂Ω (ifu |δΩ is smooth). This is a partial (positive) answer to a conjecture of S. Hildebrandt [13]. As an application, we establish a regularity result for weak solutions to the equation of surfaces of prescribed mean curvature in a three-dimensional compact riemannian manifold.

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Choné, P. A regularity result for critical points of conformally invariant functionals. Potential Anal 4, 269–296 (1995). https://doi.org/10.1007/BF01071697

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Mathematics Subject Classifications (1991)

  • 35B65
  • 53A10
  • 58E12

Key words

  • Conformally invariant functional
  • prescribed mean curvature
  • mobile frame
  • compensation phenomena
  • jacobians
  • Lorentz spaces