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Existence theorems for a class of nonconvex problems in the calculus of variations

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Abstract

We give some existence results of minima for a class of nonconvex functionals depending on the Laplacian. We minimize these functionals on the set of functionsu inW 2,p (Ω) ∩W 0 1,p (Ω) such that ∂u/∂n=0 on ∂Ω,p>1, with Ω either an annulus or the whole space ℝn. Our approach allows us to deal with integrands without any regularity conditions. The results are obtained first by showing that the corresponding convexified problem has at least one radially symmetric solution via a rotation; then, by using a Liapunov's theorem on the range of a vector-valued measure, we construct a function that is a solution to our problem.

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Additional information

The author wishes to thank Prof. A. Cellina for useful comments and Prof. G. Dal Maso for several helpful discussions during the preparation of this paper. He also is grateful to Prof. A. Salam and ICTP for the generous financial support that made this work possible.

Communicated by R. Conti

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Flores-Bazán, F. Existence theorems for a class of nonconvex problems in the calculus of variations. J Optim Theory Appl 78, 31–48 (1993). https://doi.org/10.1007/BF00940698

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Key Words

  • Calculus of variations
  • convex functionals
  • rotation groups
  • Fourier transforms
  • Liapunov's theorem
  • radially symmetric solutions