This thesis studies some problems derived from differential topology and differential geometry by techniques developed from geometric measure theory, variational calculus, and partial differential equations. It consists of two independent parts: Part I: An isoperimetric type inequality for chains on singular spaces. We find an isoperimetric type inequality for integral chains with support in a subset of \IR\spn, which satisfies some structural conditions but is not in the Lipschitz category. We also apply this inequality to derive some results in the subanalytic category for homologically mass minimizing currents. Part II: Energy minimizing sections of a fiber bundle. We show that a Dirichlet p-energy minimizing section of a fiber bundle is Holder continuous everywhere except possibly for a closed subset of Hausdorff dimension at most m−[p] −1, where m is the dimension of the base space of the fiber bundle and (p) is the greatest integer less than or equal to p.