# 4.4 Surfaces minimizing boundary-weighted area

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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 boundary-weighted area of surfaces, and asks the Plateau problem for this functional. An analogous functional, the boundary-value weighted area and energy, is discussed in the one-dimensional setting. This work was studied in the Rice University VIGRE class MATH499 in the Fall of 2010.

## Introduction

Our research has been devoted to minimizing the $c$ -boundary-weighted area of surfaces. We have also studied the boundary-value weighted Dirichlet energy over the interval [0,1]. These two problems have required us to study the minimal surface equation over the plane, using techniques from geometric calculus of variations.

## Problem

Given a surface $S,$ we define the $c$ -boundary-weighted area of $S$ to be

${A}_{c,\partial }\left(S\right)=Area\left(S\right)+c·{\left(length\left(\partial S\right)\right)}^{2}.$

For ${\mathcal{D}}_{R}$ the disc of radius $R$ and ${C}_{h}$ the cylinder of radius $R$ and height $h$ we have:

${A}_{c,\partial }\left({\mathcal{D}}_{R}\right)=\left(\pi +4{\pi }^{2}c\right){R}^{2},\phantom{\rule{4pt}{0ex}}{A}_{c,\partial }\left({C}_{h}\right)=2\pi Rh+16{\pi }^{2}c{R}^{2}.$

If $c<\frac{1}{12\pi },$ then ${lim}_{h\to 0}{A}_{c,\partial }\left({C}_{h}\right)<{A}_{c,\partial }\left({\mathcal{D}}_{R}\right).$

Problem: Given a curve ${\gamma }_{0}$ in ${\mathbb{R}}^{3},$ does there exist a surface $S$ with boundary

$\partial S={\gamma }_{0}\cup \bigcup _{i=1}^{n}{\gamma }_{i}$

disjoint curves so that the $c$ -boundary-weighted area of $S$ is least amongst all surfaces having boundary at least ${\gamma }_{0}$ ?

Consider the upper-half of the truncated catenoid ${\mathfrak{C}}_{R}$ given by the graph of $f\left(x,y\right)={cosh}^{-1}\left(r\right)$ for $1 we have

${A}_{c,\partial }\left({\mathfrak{C}}_{R}\right)=\pi \left[R\sqrt{{R}^{2}-1}+{cosh}^{-1}R\right]+4{\pi }^{2}c{\left(R+1\right)}^{2}.$

## Boundary-value weighted energy&Area

Let $\Omega ={\cup }_{i=0}^{n}\left[{a}_{2i},{a}_{2i+1}\right]$ where ${a}_{0}=0,{a}_{2n+1}=1$ and ${a}_{j}<{a}_{j+1}.$ We define the $c$ -boundary-value weighted Dirichlet energy of a function $u:\Omega \to \mathbb{R}$ to be:

${E}_{c,\partial }\left(u\right)=\sum _{i=0}^{n}{\int }_{{a}_{2i}}^{{a}_{2i+1}}|{u}^{\text{'}}{\left(x\right)|}^{2}\phantom{\rule{4pt}{0ex}}dx+c·\sum _{i=0}^{2n+1}|u\left({a}_{i}\right)|.$

We seek to minimize ${E}_{c,\partial }\left(u\right)$ over functions $u:\Omega \to \mathbb{R}$ subject to the constraints $u\left(0\right)=0,u\left(1\right)=1.$

We can show that we only need to consider functions ${u}_{ϵ,A}$ that vanish along the majority of the interval. We therefore need to minimize the function $f\left(ϵ,A\right)$ over $\left(0,1\right)×\left(0,1\right),$ given by

$f\left(ϵ,A\right)={E}_{\partial ,c}\left({u}_{ϵ,A}\right)=\frac{{\left(1-A\right)}^{2}}{ϵ}+c\left(A+1\right).$

For $c\ge 2,$ a minimizer is ${u}_{1,0}\left(x\right)=x$ with $\Omega =\left[0,1\right]$ which is unique for $c>2.$ When $c<2$ no minimizer exists, but can be approximated by the sequence ${u}_{ϵ,1-\frac{c}{2}},$ letting $ϵ\to 1.$

We define the $c$ -boundary-value weighted area of a function $u:\Omega \to \mathbb{R}$ to be:

${A}_{c,\partial }\left(u\right)=\sum _{i=0}^{n}{\int }_{{a}_{2i}}^{{a}_{2i+1}}\sqrt{1+|{u}^{\text{'}}{\left(x\right)|}^{2}}\phantom{\rule{4pt}{0ex}}dx+c·\sum _{i=0}^{2n+1}|u\left({a}_{i}\right)|.$

The minimizer for $c\ge \sqrt{2}$ is ${u}_{1,0}\left(x\right)=x$ with $\Omega =\left[0,1\right],$ unique for $c>\sqrt{2}.$ When $c<\sqrt{2}$ , no minimizer exists.

## Minimal surface equation

The MSE in two variables is the PDE:

$\mathcal{M}\left(u\right)=\frac{\partial }{\partial x}\left(\frac{\frac{\partial u}{\partial x}}{\sqrt{1+{|\nabla u|}^{2}}}\right)+\frac{\partial }{\partial y}\left(\frac{\frac{\partial u}{\partial y}}{\sqrt{1+{|\nabla u|}^{2}}}\right).$

A function $u$ satisfies $\mathcal{M}\left(u\right)=0$ over the unit disk $\mathcal{D}$ if and only if the graph of $u$ has the least surface area amongst all other graphs of functions with the same boundary values. Solutions to the MSE satisfy the Maximum Principle: if $\mathcal{M}\left(u\right)=0$ in $\mathcal{D},$ then $u$ attains its max/min only at the boundary, unless if $u$ is constant.

$f\left(x,y\right)={cosh}^{-1}r$ is a solution to the MSE for $r>1.$

## Future work

Our next task is to study the catenoid more closely. We wish to investigate the 2-D versions of the $c$ -boundary-value weighted Dirichlet energy and area. The Isoperimetric Inequality states that if $S$ is a region bounded by a curve $\gamma ,$ then $4\pi ·Area\left(S\right)\le length\left(\gamma \right).$ We need to see how this theory applies.

## Acknowledgements

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. DMS-0739420.

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