On shape optimizing the ratio of the first two eigenvalues of the Laplacian

76 Pages·1991·1.4 MB·English

by  Haeberly, Jean-Pierre A

Computer Department
Science
TECHNICAL REPORT
On Shape Optimizing the Ratio of the First Two
Eigenvalues of the Laplacian
Jean-Pierre A. Haeberly
Technical Report 586
October 1991
NEW
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On Shape Optimizing the Ratio of the First Two
Eigenvalues of the Laplacian
Jean-Pierre A. Haeberly
Technical Report 586
October 1991

On Shape Optimizing the Ratio of the First
Two
Eigenvalues of the Laplacian
Jean-Pierre A. Haeberly
October 1991
A dissertation in the Department of Computer Science submitted to the
faculty of the Graduate School of Arts and Science in partial fulfillment of
the requirements forthe degreeofMastersofScience at New York University.
Approved
(Signed)
Michael Overton
Thesis Advisor

Abstract
We investigate numerically a 1956 conjecture ofPayne, Polya, and Wein-
berger. The conjecture asserts that the ratio of the first two eigenvalues of
the Laplacian on a bounded domain il ofthe plane with Dirichlet boundary
conditions reaches its minimum value precisely when fi is a disk. A crucial
feature of this problem is the loss of smoothness of the objective function
at the solution. The following results form the core of our numerical treat-
ment. First, we construct finite dimensional families of deformations of a
disk equipped with a uniform triangulation. This permits the formulation
ofa discrete model ofthe problem via finite element techniques. Second, we
build on the work of M. Overton to derive optimality conditions in terms
of Clarke's generalized gradients for nonsmooth functions. These ideas are
then combined into an algorithm and implemented in Fortran.

Contents
1 Introduction 2
2 Preliminaries 4
3 Max characterization of ^^=1 A, for symmetric matrices 8
—
4 The generalized gradient of ^*=i A. : S^"" 3? 18
—
S^
5 The differential of ^v x Q" » 20
:
6 The generalized gradient of ^*_j A,(x) 26
7 The ratio ofthe first two eigenvalues ofa symmetric definite
pencil 30
8 The gap between the first two eigenvalues of a symmetric
definite pencil 35
Optimality Conditions 36
10 Perturbations ofa disk 40
11 Description of the Algorithm 45
12 Numerical Results 48
A Appendix 50