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Title: | Payne-Pólya-Weinberger Type Inequalities |

Authors: | Arbon, Ryan |

Advisors: | Sarnak, Peter |

Department: | Mathematics |

Class Year: | 2021 |

Abstract: | We develop the spectral theory of the Laplacian on domains Ω ⊂ R n, including the classical result showing completeness of the Laplacian spectrum and the variational characterization of eigenvalues via the Rayleigh-Ritz formulation. These results are meant as an introduction to the Payne–P´olya– Weinberger Inequality, which states that the fundamental ratio, λ2/λ1, as a function of a domain Ω ∈ R n is maximized by the n-ball. We consider the Polygonal Payne–P´olya–Weinberger Conjecture, which states that among planar k-polygons, the regular k-polygon maximizes λ2/λ1. We show that among the class of triangles, λ2/λ1 is uniquely maximized by the equilateral triangle, and that among quadrilaterals λ2/λ1 is uniquely maximized by the square. To do so, we use perturbative techniques to establish local maximality together with a continuity estimate to reduce the problem to finitely many computations accomplished by a computer. This provides an alternative proof of the k = 3 case, a proof for the convex k = 4 case, and the sketch of a proof for the non-convex k = 4 case. |

URI: | http://arks.princeton.edu/ark:/88435/dsp01k35697423 |

Type of Material: | Princeton University Senior Theses |

Language: | en |

Appears in Collections: | Mathematics, 1934-2021 |

Files in This Item:

File | Description | Size | Format | |
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ARBON-RYAN-THESIS.pdf | 690.16 kB | Adobe PDF | Request a copy |

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