Let $A$ be a $2m$ order differential operator in a hermitian vector bundle $E$ over a compact riemannian manifold $\bar{\Omega }$ with boundary $\Gamma$ ; and denote by $A_B$ the realization defined by a normal differential boundary condition $B\rho u=0$ ($u\in H^{2m}\left(E\right)$, $\rho u=$ Cauchy data). We characterize, by an explicit condition on $A$ and $B$ near $\Gamma$, the realizations $A_B$ for which there exists an integro-differential sesquilinear form $a_B(u,\nu )$ on $H^m\left(E\right)$ such that $(Au,\nu )=a_B(u,\nu )$ on $D\left(A_B\right)$; moreover we show that these are exactly the realizations satisfying a weak semiboundedness estimate:...