Nonnegativity of degenerated quadratic forms of the calculus of variations (Q1910808)

The question on the definiteness of the quadratic form \[ U(x)= \int^1_0(\langle A(t)\dot x(t),\dot x(t)\rangle+ \langle B(t)x(t),x(t)\rangle+ 2\langle C(t)\dot x(t),x(t)\rangle)dt+ \langle\Omega(x(0),x(1)),(x(0),x(1))\rangle, \] \[ N_0x(0)+ N_1x(1)=0,\quad x\in W^n_{2,1}[0,1] \] is investigated, when the form \(U\) is degenerated, i.e., the Legendre condition is satisfied and the strong Legendre condition is violated. The necessary as well as the sufficient conditions for the finiteness of the index are obtained. The sufficient conditions of positiveness of the form \(U\) are established. Moreover, for degenerated problems of the calculus of variations the necessary as well as the sufficient conditions for a local minimum are given.