No description available.

No description available.

Thesis (Ph. D.)--University of Missouri-Columbia, 1999. / Typescript. Vita. Includes bibliographical references (leaves 79-83). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1999. / Typescript. Vita. Includes bibliographical references (leaves 79-83). Also available on the Internet.

No description available.

We study two related inequalities that arise in Harmonic Analysis: restriction type inequalities and isoperimetric inequalities. The (Lp, Lq) Restriction type inequalities have been the subject of much interest since they were first conceived in the 1960s. The classical restriction type inequality involving surfaces of non-vanishing curvature is only fully resolved in two dimensions and there have been a lot of recent developments to establish the conjectured (p,q) range in higher dimensions. However, it also interesting to consider what can be said for curves where the curvature does vanish. In particular we build upon a restriction result for homogeneous polynomial surfaces, using what is considered the natural weight - the one induced by the affine curvature of the surface. This is known to hold with a non-universal constant which depends in some way on the coefficients of the polynomial. In this dissertation we shall quantify that relationship. Restriction estimates (for curves or surfaces) using the affine curvature weight can be shown to lead to an affine isoperimetric inequality for such curves or surfaces. We first prove, directly, this inequality for polynomial curves, where the constant depends only on the degree of the underlying polynomials. We then adapt this method, to prove an isoperimetric inequality for a wide class of curves, which includes curves for which a restriction estimate is not yet known. Next we state and prove an analogous result of the relative affine isoperimetric inequality, which applies to unbounded convex sets. Lastly we demonstrate that this relative affine isoperimetric inequality for unbounded sets is in fact equivalent to the classical affine isoperimetric inequality.

Lin, Jessey. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 79-80). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.

No description available.

This thesis considers two isoperimetric inequalities for the eigenvalues of the Laplacian on a family of spherically symmetric Riemannian manifolds. The Payne-Pólya-Weinberger Conjecture (PPW) states that for a bounded domain Ω in Euclidean space Rn, the ratio λ1(Ω)/λ0(Ω) of the first two eigenvalues of the Dirichlet Laplacian is bounded by the corresponding eigenvalue ratio for the Dirichlet Laplacian on the ball BΩof equal volume. The Szegö-Weinberger inequality states that for a bounded domain Ω in Euclidean space Rn, the first nonzero eigenvalue of the Neumann Laplacian μ1(Ω) is maximized on the ball BΩ of the same volume. In the first three chapters we will look at the known work for the manifolds Rn and Hn. Then we will take a family a spherically symmetric manifolds given by Rn with a spherically symmetric metric determined by a radially symmetric function f. We will then give a PPW-type upper bound for the eigenvalue gap, λ1(Ω) − λ0(Ω), and the ratio λ1(Ω)/λ0(Ω) on a family of symmetric bounded domains in this space. Finally, we prove the Szegö-Weinberger inequality for this same class of domains.

The first part of this thesis is devoted to symmetrizations. Symmetrizations are tranformations of functions that preserve many properties of functions and enhance their symmetry. In the calculus of variation they are a simple and powerful tool to prove that minimizers of functionals are symmetric functions. In this work, the approximation of symmetrizations by simpler symmetrizations is investigated: The existence of a universal approximating sequence is proved, sufficient conditions for deterministic and random sequences to be approximating are given. These approximation methods are then used to prove some symmetry properties of critical points obtained by minimax methods: For example if there is a solution obtained by the mountain pass theorem, then there is a symmetric solution with the same energy. This part ends with a study of the properties of anisotropic symmetrizations i.e. symmetrizations performed with respect to noneuclidean norms. The second part is devoted to L^1 estimates.  In general, the second derivative of the solution of the Poisson equation with L^1 data fails to be in L^1. Recently it was proved that if the data is a L^1 divergence-free vector-field, then even if in general it is false that the second derivative of the solution is in L^1, all the consequences thereof by Sobolev embeddings hold. Elementary proofs of such results, as well as a generalization with a second order operator replacing the divergence, are given. / La première partie de cette thèse est consacrée aux symétrisations. Les symétrisations sont des transformations de fonctions qui préservent de nombreuses propriétés des fonctions et qui améliorent leur symétrie. Elles sont un outil simple et puissant pour montrer dans le calcul des variations que les minimiseurs de certaines fonctionnelles sont des fonctions symétriques. Dans ce travail, nous étudions l'approximation des symétrisations par des symétrisations plus simples. Nous prouvons l'existence d'une suite approximante universelle et nous donnons des conditions suffisantes pour que des suites déterministes et aléatoires soient approximantes. Nous utilisons ensuite ces méthodes d'approximation pour prouver des propriétés de symétrie de points critiques obtenus par des méthodes de minimax. Par exemple, s'il y a une solution obtenue par le théorème du col, alors il y a une solution symétrique de même énergie. Nous achevons cette partie par une étude des symétrisations anisotropes (symétrisations par rapport à des normes non euclidiennes). La seconde partie est consacrée aux estimations L^1. En général, les dérivées secondes de la solution de l'équation de Poisson avec des données L^1 ne sont pas dans L^1. Recemment, on a prouvé que si les données sont un champ de vecteurs L^1 à divergence nulle, même si en général les dérivées secondes ne sont toujours pas dans L^1, toutes les conséquences qui en suivraient par les injections de Sobolev sont vraies. Nous donnons des preuves élémentaires de ces résultats, avec une extension où la divergence est remplacée par un opérateur différentiel du second ordre.