Standard facts about separating linear functionals will be used to determine how two cones $C$ and $D$ and their duals $C^{*}$ and $D^{*}$ may overlap. When $TV\to W$ is linear and $K\subset V$ and $D\subset W$ are cones, these results will be applied to $C=T\left(K\right)$ and $D$, giving a unified treatment of several theorems of the alternate which explain when $C$ contains an interior point of $D$. The case when $V=W$ is the space $H$ of $n\times n$ Hermitian matrices, $D$ is the $n\times n$ positive semidefinite matrices, and $T\left(X\right)=AX+X^{*}A$ yields new and known results about the existence of block diagonal...

Compactness in the space $L^p(0,T;B)$, $B$ being a separable Banach space, has been deeply investigated by J.P. Aubin (1963), J.L. Lions (1961, 1969), J. Simon (1987), and, more recently, by J.M. Rakotoson and R. Temam (2001), who have provided various criteria for relative compactness, which turn out to be crucial tools in the existence proof of solutions to several abstract time dependent problems related to evolutionary PDEs. In the present paper, the problem is examined in view of Young measure theory: exploiting...