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Jacobi condition for elliptic forms in Hilbert spaces


In this paper, we give the definitions of conjugate and semi-conjugate points for a quadratic elliptic form in a Hilbert space, and we state the corresponding necessary and sufficient conditions (the Jacobi conditions) for the form to be positive (nonnegative). We apply the abstract results to a one-dimensional problem in the calculus of variations where both endpoints are allowed to vary. The conditions that we obtain complete and generalize previously known results.

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This research was partly supported by MURST Research Grant, Teoria del Controllo dei Sistemi Dinamici.

Communicated by R. Conti

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Zezza, P. Jacobi condition for elliptic forms in Hilbert spaces. J Optim Theory Appl 76, 357–380 (1993).

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Key Words

  • Conjugate points
  • Jacobi condition
  • necessary and sufficient optimality conditions