Christoffel's Problem and the Generalized Green's Function for a Shifted Laplacian on the Hypersphere Open Access

Albritton, Dallas Lai-Ahn (2014)

Permanent URL: https://etd.library.emory.edu/concern/etds/qj72p756k?locale=en
Published

Abstract

Christoffel's problem gives rise naturally to an elliptic partial differential equation Delta h + nh = Phi on the n-dimensional unit sphere S^n, where under certain conditions h may represent the support function of a non-degenerate convex body in R^(n+1) and Phi/n the mean radius of curvature prescribed on S^n. We construct a closed form of the generalized Green's function for the differential operator Delta + n on the hypersphere by reducing the original equation to an ordinary differential equation and choosing the undetermined constants in the correct way. We compare the results with existing expressions for the generalized Green's function in the literature and investigate an incorrect claim about the choice of constants.

Table of Contents

Contents

1 Introduction 1 1.1 Christoffel's Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Content of this Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Preliminaries 3 2.1 Convex Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.2 Properties of the Support Function . . . . . . . . . . . . . . . . . . . 4 2.1.3 Relation to Christoffel's Problem . . . . . . . . . . . . . . . . . . . . 5 2.2 The Laplacian on the Unit Sphere . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.1 Eigenvalues and Eigenfunctions . . . . . . . . . . . . . . . . . . . . . 6 2.2.2 The Shifted Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.3 Invertibility on a Subspace . . . . . . . . . . . . . . . . . . . . . . . . 7 3 The Generalized Green's Function for a Shifted Laplacian 8 3.1 Basic Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1.2 Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.1.3 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.1.4 Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Reduction to an Ordinary Differential Equation . . . . . . . . . . . . . . . . 11 3.3 The Case of n Even . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3.1 Closed Form for Qn_1 . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3.2 Closed Form for Sn_l . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3.3 Closed Form for In_1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3.4 Choice of C1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3.5 Closed Form for g_n . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3.6 Choice of C2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3.7 The Generalized Green's Function for n = 2 . . . . . . . . . . . . . . 17 3.4 The Case of n Odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.4.1 Closed Form for Qn_1 . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.4.2 Closed Form for Sn_1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.4.3 Closed Form for In_1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.4.4 Cancellation of log Terms . . . . . . . . . . . . . . . . . . . . . . . . 19 3.4.5 Choice of C1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.4.6 Closed Form for g_n . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.4.7 The Generalized Green's Function for n = 3 . . . . . . . . . . . . . . 20 4 Comparison of Closed Form Expressions 22 4.1 Reduction From Spectral Expansion [Szm07] . . . . . . . . . . . . . . . . . . 22 4.1.1 Closed Form for n Even . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.1.2 Closed Form for n Odd . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Construction in [Oli11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2.1 A Note About Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . 24 4.2.2 Choice of the Constant a . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2.3 The Case of n Even . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2.4 The Case of n Odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5 Conclusion 27 5.0.5 Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.0.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6 Appendix 29 6.1 Formulas From [Szm06] and [Szm07] . . . . . . . . . . . . . . . . . . . . . . 29 6.1.1 The Case n = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.1.2 The Case n = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.1.3 The General Case n Even . . . . . . . . . . . . . . . . . . . . . . . . 30 6.1.4 The General Case n Odd . . . . . . . . . . . . . . . . . . . . . . . . . 31 6.2 Proof of Minor Claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6.2.1 Equation (20) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6.2.2 Equation (21) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6.2.3 Equation (22) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6.2.4 Equation (24) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.2.5 Equation (25) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.2.6 Equation (26) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.2.7 Equation (27) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6.2.8 Equation (29) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6.2.9 Equation (36) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 6.2.10 Equation (38) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 6.2.11 Equation (39) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 6.2.12 Equation (40) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6.2.13 Equation (43) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6.2.14 Equation (56) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6.2.15 Equation (60) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

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