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