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Motivated by some variational problems from a nonlocal model of mechanics,this work presents a set of sufficient conditions that guarantee a compactinclusion in the function space of $L^{p}$ vector fields defined on a domain$\Omega$ that is either a bounded domain in $\mathbb{R}^{d}$ or$\mathbb{R}^{d}$ itself. The criteria are nonlocal and are given with respectto nonlocal interaction kernels that may not be necessarily radially symmetric.Moreover, these criteria for vector fields are also different from those givenfor scalar fields in that the conditions are based on nonlocal interactionsinvolving only parts of the components of the vector fields. The $L^{p}$compactness criteria are utilized in demonstrating the convergence ofminimizers of parametrized nonlocal energy functionals.

$L^{p}$ compactness criteria with an application to variational convergence of some nonlocal energy functionals