Minimizers of the vector-valued coarea formula
Hardt, Robert M.
Doctor of Philosophy thesis
The 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.
Geometric Measure Theory; Calculus of Variations; Partial Differential Equations