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Let $F: [0, \infty) \to [0, \infty)$ be a strictly increasing $C^2$ functionwith $F(0)=0$. We unify the concepts of $F$-harmonic maps, minimalhypersurfaces, maximal spacelike hypersurfaces, and Yang-Mills Fields, andintroduce $F$-Yang-Mills fields, $F$-degree, $F$-lower degree, and generalizedYang-Mills-Born-Infeld fields (with the plus sign or with the minus sign) onmanifolds. When $F(t)=t, \frac 1p(2t)^{\frac p2}, \sqrt{1+2t} -1,$ and$1-\sqrt{1-2t},$ the $F$-Yang-Mills field becomes an ordinary Yang-Mills field,$p$-Yang-Mills field, a generalized Yang-Mills-Born-Infeld field with the plussign, and a generalized Yang-Mills-Born-Infeld field with the minus sign on amanifold respectively. We also introduce the $E_{F,g}-$energy functional (resp.$F$-Yang-Mills functional) and derive the first variational formula of the$E_{F,g}-$energy functional (resp. $F$-Yang-Mills functional) withapplications. In a more general frame, we use a unified method to study thestress-energy tensors that arise from calculating the rate of change of variousfunctionals when the metric of the domain or base manifold is changed. Thesestress-energy tensors, linked to $F$-conservation laws yield monotonicityformulae. A "macroscopic" version of these monotonicity inequalities enables usto derive some Liouville type results and vanishing theorems for $p-$forms withvalues in vector bundles, and to investigate constant Dirichlet boundary valueproblems for 1-forms. In particular, we obtain Liouville theorems for$F-$harmonic maps (e.g. $p$-harmonic maps), and $F-$Yang-Mills fields (e.g.generalized Yang-Mills-Born-Infeld fields on manifolds). We also obtaingeneralized Chern type results for constant mean curvature type equations for$p-$forms on $\Bbb{R}^m$ and on manifolds $M$ with the global doubling propertyby a different approach. The case $p=0$ and $M=\mathbb{R}^m$ is due to Chern.

On Vanishing Theorems For Vector Bundle Valued p-Forms And Their Applications