Locally optimal configurations for the two-phase torsion problem in the ball

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2 Citations (Scopus)


We consider the unit ball Ω⊂RN (N≥2) filled with two materials with different conductivities. We perform shape derivatives up to the second order to find out precise information about locally optimal configurations with respect to the torsional rigidity functional. In particular we analyse the role played by the configuration obtained by putting a smaller concentric ball inside Ω. In this case the stress function admits an explicit form which is radially symmetric: this allows us to compute the sign of the second order shape derivative of the torsional rigidity functional with the aid of spherical harmonics. Depending on the ratio of the conductivities a symmetry breaking phenomenon occurs.

Original languageEnglish
Pages (from-to)33-48
Number of pages16
JournalNonlinear Analysis, Theory, Methods and Applications
Publication statusPublished - 2017 Oct


  • Elliptic PDE
  • Optimization problem
  • Shape derivative
  • Torsion problem


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