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Surfaces Minimizing Boundary-Weighted Area

Module by: Leobardo Rosales. E-mail the author

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 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 cc-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,S, we define the cc-boundary-weighted area of SS to be

A c , ∂ ( S ) = A r e a ( S ) + c · ( l e n g t h ( ∂ S ) ) 2 . A c , ∂ ( S ) = A r e a ( S ) + c · ( l e n g t h ( ∂ S ) ) 2 .
(1)

For DRDR the disc of radius RR and ChCh the cylinder of radius RR and height hh we have:

A c , ∂ ( D R ) = ( π + 4 π 2 c ) R 2 , A c , ∂ ( C h ) = 2 π R h + 16 π 2 c R 2 . A c , ∂ ( D R ) = ( π + 4 π 2 c ) R 2 , A c , ∂ ( C h ) = 2 π R h + 16 π 2 c R 2 .
(2)

If c<112π,c<112π, then limh→0Ac,∂(Ch)<Ac,∂(DR).limh→0Ac,∂(Ch)<Ac,∂(DR).

Problem: Given a curve γ0γ0 in R3,R3, does there exist a surface SS with boundary

∂ S = γ 0 ∪ ⋃ i = 1 n γ i ∂ S = γ 0 ∪ ⋃ i = 1 n γ i
(3)

disjoint curves so that the cc-boundary-weighted area of SS is least amongst all surfaces having boundary at least γ0γ0?

Consider the upper-half of the truncated catenoid CRCR given by the graph of f(x,y)=cosh-1(r)f(x,y)=cosh-1(r) for 1<r<R,1<r<R, we have

A c , ∂ ( C R ) = π [ R R 2 - 1 + cosh - 1 R ] + 4 π 2 c ( R + 1 ) 2 . A c , ∂ ( C R ) = π [ R R 2 - 1 + cosh - 1 R ] + 4 π 2 c ( R + 1 ) 2 .
(4)

Boundary-Value Weighted Energy & Area

Let Ω=∪i=0n[a2i,a2i+1]Ω=∪i=0n[a2i,a2i+1] where a0=0,a2n+1=1a0=0,a2n+1=1 and aj<aj+1.aj<aj+1. We define the cc-boundary-value weighted Dirichlet energy of a function u:Ω→Ru:Ω→R to be:

E c , ∂ ( u ) = ∑ i = 0 n ∫ a 2 i a 2 i + 1 | u ' ( x ) | 2 d x + c · ∑ i = 0 2 n + 1 | u ( a i ) | . E c , ∂ ( u ) = ∑ i = 0 n ∫ a 2 i a 2 i + 1 | u ' ( x ) | 2 d x + c · ∑ i = 0 2 n + 1 | u ( a i ) | .
(5)

We seek to minimize Ec,∂(u)Ec,∂(u) over functions u:Ω→Ru:Ω→R subject to the constraints u(0)=0,u(1)=1.u(0)=0,u(1)=1.

Figure 1: A general function u:Ω→R,u:Ω→R, and the functions uϵ,Auϵ,A parameterized by ϵ,A.ϵ,A.
(a)
Figure 1(a) (graph1.png)
(b)
Figure 1(b) (graph2.png)

We can show that we only need to consider functions uϵ,Auϵ,A that vanish along the majority of the interval. We therefore need to minimize the function f(ϵ,A)f(ϵ,A) over (0,1)×(0,1),(0,1)×(0,1), given by

f ( ϵ , A ) = E ∂ , c ( u ϵ , A ) = ( 1 - A ) 2 ϵ + c ( A + 1 ) . f ( ϵ , A ) = E ∂ , c ( u ϵ , A ) = ( 1 - A ) 2 ϵ + c ( A + 1 ) .
(6)

For c≥2,c≥2, a minimizer is u1,0(x)=xu1,0(x)=x with Ω=[0,1]Ω=[0,1] which is unique for c>2.c>2. When c<2c<2 no minimizer exists, but can be approximated by the sequence uϵ,1-c2,uϵ,1-c2, letting ϵ→1.ϵ→1.

We define the cc-boundary-value weighted area of a function u:Ω→Ru:Ω→R to be:

A c , ∂ ( u ) = ∑ i = 0 n ∫ a 2 i a 2 i + 1 1 + | u ' ( x ) | 2 d x + c · ∑ i = 0 2 n + 1 | u ( a i ) | . A c , ∂ ( u ) = ∑ i = 0 n ∫ a 2 i a 2 i + 1 1 + | u ' ( x ) | 2 d x + c · ∑ i = 0 2 n + 1 | u ( a i ) | .
(7)

The minimizer for c≥2c≥2 is u1,0(x)=xu1,0(x)=x with Ω=[0,1],Ω=[0,1], unique for c>2.c>2. When c<2c<2, no minimizer exists.

Minimal Surface Equation

The MSE in two variables is the PDE:

M ( u ) = ∂ ∂ x ∂ u ∂ x 1 + | ∇ u | 2 + ∂ ∂ y ∂ u ∂ y 1 + | ∇ u | 2 . M ( u ) = ∂ ∂ x ∂ u ∂ x 1 + | ∇ u | 2 + ∂ ∂ y ∂ u ∂ y 1 + | ∇ u | 2 .
(8)

A function uu satisfies M(u)=0M(u)=0 over the unit disk DD if and only if the graph of uu 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 M(u)=0M(u)=0 in D,D, then uu attains its max/min only at the boundary, unless if uu is constant.

f(x,y)=cosh-1rf(x,y)=cosh-1r is a solution to the MSE for r>1.r>1.

Future Work

Our next task is to study the catenoid more closely. We wish to investigate the 2-D versions of the cc-boundary-value weighted Dirichlet energy and area. The Isoperimetric Inequality states that if SS is a region bounded by a curve γ,γ, then 4π·Area(S)≤length(γ).4π·Area(S)≤length(γ). 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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