Our research has been devoted to minimizing the
Summary: This report summarizes work done as part of the Calculus of Variations PFUG under Rice University's VIGRE program. VIGRE is a program of Vertically Integrated Grants for Research and Education in the Mathematical Sciences under the direction of the National Science Foundation. A PFUG is a group of Postdocs, Faculty, Undergraduates and Graduate students formed around the study of a common problem. This module investigates the boundaryweighted area of surfaces, and asks the Plateau problem for this functional. An analogous functional, the boundaryvalue weighted area and energy, is discussed in the onedimensional setting. This work was studied in the Rice University VIGRE class MATH499 in the Fall of 2010.
Our research has been devoted to minimizing the
Given a surface
For
If
Problem:
Given a curve
disjoint curves so that the
Consider the upperhalf of the truncated catenoid
Let
We seek to minimize

We can show that we only need to consider functions
For
We define the
The minimizer for
The MSE in two variables is the PDE:
A function
Our next task is to study the catenoid more closely. We wish to investigate the 2D versions of the
We thank the guidance offered by our PFUG leader Dr. Leobardo Rosales. We also thank our faculty sponsors in the Department of Mathematics, Dr. Robert Hardt and Dr. Michael Wolf. We also thank the undergraduate group members Sylvia Casas de Leon, James Hart, Marissa Lawson, Conor Loftis, Aneesh Mehta, and Trey Villafane. This work was supported by NSF grant No. DMS0739420.