Hardt, Robert M.2012-09-062012-09-062012-09-062012-09-062012-052012-09-05May 2012Carroll, Colin. "Minimizers of the vector-valued coarea formula." (2012) Diss., Rice University. <a href="https://hdl.handle.net/1911/64635">https://hdl.handle.net/1911/64635</a>.https://hdl.handle.net/1911/64635The vector-valued coarea formula provides a relationship between the integral of the Jacobian of a map from high dimensions down to low dimensions with the integral over the measure of the fibers of this map. We explore minimizers of this functional, proving existence using both a variational approach and an approach with currents. Additionally, we consider what properties these minimizers will have and provide examples. Finally, this problem is considered in metric spaces, where a third existence proof is given.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.Geometric measure theoryCalculus of variationsPartial differential equationsThesis2012-09-06123456789/ETD-2012-05-85