Dissertationen zum Thema „Isoperimetric problems“

PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO
COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
PROGRAMA DE SUPORTE À PÓS-GRADUAÇÃO DE INSTS. DE ENSINO
O objetivo principal deste trabalho é resolver o problema isoperimétrico no plano de Minkowski, isto é, determinar dentre todas as curvas convexas, fechadas, simples e suaves de perímetro fixo de um plano munido com uma norma qualquer, qual é aquela que delimita a maior área. Mostraremos que a solução para este problema não é necessariamente o círculo como no caso euclideano e sim uma curva conhecida como isoperimetrix. Para isto, vamos demonstrar a desigualdade de Minkowski a partir do conceito de área mista. Em seguida, vamos determinar se há outros casos (além do caso euclideano) em que o círculo coincide com o isoperimetrix. Também iremos mostrar que o perímetro da bola nestes planos pode assumir qualquer valor real entre seis e oito, sendo seis quando a bola for um hexágono regular afim e oito quando for um paralelogramo.
The main objective of this work is to solve the isoperimetric problem in the Minkowski plane, i. e., determine among all smooth simple closed convex curves of a normed plane with fixed perimeter, what is that which defines the largest area. We will show that the solution to this problem is not necessarily the circle as in the Euclidean case, but a curve known as isoperimetrix. For this, we will demonstrate the Minkowski inequality from the concept of mixed area. Then, we determine if there are other cases (apart from the Euclidean case) in which the circle coincides with the isoperimetrix. We will also show that the ball perimeter in a normed plane can take any real value between six and eight. It is six when the ball is an affine regular hexagon and eight when it is a parallelogram.

The thesis is divided into two parts. The first part is mainly concerned with regularity issues for integro-differential (or nonlocal) elliptic and parabolic equations. In the same way that densities of particles with Brownian motion solve second order elliptic or parabolic equations, densities of particles with Lévy diffusion satisfy these more general nonlocal equations. In this context, fully nonlinear nonlocal equations arise in Stochastic control problems or differential games. The typical example of elliptic nonlocal operator is the fractional Laplacian, which is the only translation, rotation and scaling invariant nonlocal elliptic operator. There many classical regularity results for the fractional Laplacian ---whose ``inverse'' is the Riesz potential. For instance, the explicit Poisson kernel for a ball is an ``old'' result, as well as the linear solvability theory in L^p spaces. However, very little was known on boundary regularity for these problems. A main topic of this thesis is the study of this boundary regularity, which is qualitatively very different from that for second order equations. We stablish a new boundary regularity theory for fully nonlinear (and linear) elliptic integro-differential equations. Our proofs require a combination of original techniques and appropriate versions of classical ones for second order equations (such as Krylov's method). We also obtain new interior regularity results for fully nonlinear parabolic nonlocal equation with rough kernels. To do it, we develop a blow up and compactness method for viscosity solutions to fully nonlinear equations that allows us to prove regularity from Liouville type theorems.This method is a main contribution of the thesis. The new boundary regularity results mentioned above are crucially used in the proof of another main result of the thesis: the Pohozaev identity for the fractional Laplacian. This identity is has a flavor of integration by parts formula for the fractional Laplacian, with the important novely there appears a local boundary term (this was unusual with nolocal equations). In the second part of the thesis we give two instances of interaction between isoperimetry and Partial Differential Equations. In the first one we use the Alexandrov-Bakelman-Pucci method for elliptic PDE to obtain new sharp isoperimetric inequalities in cones with densities by generalizing a proof of the classical isoperimetric inequality due to Cabré. Our new results contain as particular cases the classical Wulff inequality and the isoperimetric inequality in cones of Lions and Pacella. In the second instance we use the isoperimetric inequality and the classical Pohozaev identity to establish a radial symmetry result for second order reaction-diffusion equations. The novelty here is to include discontinuous nonlinearities. For this, we extend a two-dimensional argument of P.-L. Lions from 1981 to obtain now results in higher dimensions
La tesi està dividida en dues parts. La primera part es centra principalment en questions de regularitat per equacions integro - iferencials (o no locals) el·líptiques i parbòliques. De la mateixa manera que les densitats de partícules amb un moviment Brownià resolen equacions el·líptiques o parbòliques de segon ordre, les densitats de partícules amb una difusió de tipus Lévy resolen aquestes equacions no locals més generals. En aquest context, les equacions completament no lineals sorgeixen de problemes de control estocàstic o "differential games''. L'exemple típic d'operador el·liptic no local és el laplacià fraccionari, el qual és l'únic d'aquests operadors que és invariant per translacions, rotacions, i reescalament. Hi ha molts resultats clàssics de regularitat per el laplacià fraccionari --- "l'invers'' del qual és el potencial de Riesz. Per exemple, el nucli de Poisson (explícit) per la bola és un resultat "vell'', així com la teoria de resolubilitat en espais L^p. No obstant això, se sabia ben poc sobre la regularitat a la vora per a aquests problemes. Un tema principal d'aquesta tesi és l'estudi d'aquesta regularitat a la vora, que és qualitativament molt diferent de la de les equacions de segon ordre . A la tesi s'estableix una nova teoria regularitat a la vora per completament no lineals ( i lineals ) equacions integro - diferencials el·líptiques . Les nostres demostracions requeixen una combinació de tècniques originals i versions apropiades de les clàssiques equacions de segon ordre ( com ara el mètode de Krylov ). També obtenim nous resultats de regularitat interior per equacions parabòliques no locals completament no lineals i amb "rough kernels''. A tal efecte, desenvolupem un mètode de blow-up i compacitat per a equacions completament no lineals que en permet provar regularitat a partir de teoremes de tipus Liouville. Aquest mètode és una contribució principal de la tesi. Els nous resultats de regularitat a la vora esmentats anteriorment són essencials en la prova d'un altre resultat principal de la tesi: la identitat Pohozaev per al Laplacià fraccionari. Aquesta identitat recorda a una fórmula d'integració per parts, però amb el Laplacià fraccionari. La novetat important és que apareix un terme de vora locals (això era inusual amb equacions no locals) . A la segona part de la tesi que donem dos exemples d'interacció entre isoperimetria i Equacions en Derivades Parcials. En el primer, s'utilitza el mètode d'Alexandrov- Bakelman-Pucci per a EDP el·líptiques a fi d'obtenir noves desigualtats isoperimètriques en cons convexos amb densitats, generalitzant una prova de la desigualtat isoperimètric clàssica de X. Cabré. Els nostres nous resultats contenen com a casos particularsla desigualtat clàssica de Wulff i la desigualtat isoperimètrica en cons de Lions-Pacella. En el segon exemple s'utilitza la desigualtat isoperimètrica i la identitat Pohozaev clàssica per establir un resultat de simetria radial per equacions de reacció-difusió de segon ordre. La novetat en aquest cas és que s'inclouen no-linealitats discontínues. Per a provar aquest resultat, estenem un argument en dues dimensions de P.-L. Lions de 1981 i podem obtenir ara resultass en dimensions superiors.

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We deal with extremum values problems. Our focus is the high school students. We present simple ideas and techniques on solving classical optimization problems. Among other problems we cite the classical isoperimetric ploblem and the Heron0s problem. We are based on the book Stories About Maxima and Minima by Tikhomirov which lead with these classical problems using only elementary mathematical subjects.
Estudamos problemas envolvendo valores extremos, com foco nos estudantes do Ensino Médio. Apresentamos de forma simples e resumida, algumas ideias e teorias para a solução de tais problemas. Dentre os quais citamos o Problema de Dido e o de Heron. O principal referencial teórico para confecção deste trabalho foi o livro de Tikhomirov intitulado Stories About Maxima and Minima. Baseados em tal livro, aplicamos métodos e teorias elementares para solucionarmos problemas clássicos de máximos e mínimos.

Let X be a space and let S ⊂ X with a measure of set size |S| and boundary size |∂S|. Fix a set C ⊂ X called the constraining set. The constrained isoperimetric problem asks when we can find a subset S of C that maximizes the Følner ratio FR(S) = |S|/|∂S|. We consider different measures for subsets of R^2,R^3,Z^2,Z^3 and describe the properties that must be satisfied for sets S that maximize the Folner ratio. We give explicit examples.

Nesta dissertação foram discutidas abordagens do problema isoperimétrico que podem ser aplicadas no ensino médio e para alunos de Licenciatura plena em Matemática. Foi realizada inicialmente uma abordagem histórica e posteriormente a discussão de casos particulares e gerais de desigualdade isoperimétrica tanto no plano como no espaço. A abordagem principal deste texto é no plano, no qual foram analisadas as áreas dos triângulos, quadriláteros e polígonos regulares dado um perímetro fixo.
In this dissertation isoperimetric problem approaches were discussed that can be applied in high school and full degree students in mathematics. It was initially performed a historical approach and then the discussion of individual and general cases of isoperimetric inequality both in the plane and in space . The main approach of this text is in the plan, in which the areas of the triangles were analyzed , quadrilaterals and regular polygons given a fixed perimeter.

In questo elaborato viene trattata la soluzione analitica del problema isoperimetrico nel piano proposta da Adolf Hurwitz. Utilizzando i risultati della teoria delle serie di Fourier e l’applicazione del Teorema di Gauss Green per il calcolo dell’area, dimostriamo che “fra tutte le curve semplici, chiuse, regolari e rettificabili di lunghezza fissata, la circonferenza è la sola a racchiudere la regione di maggior area”.

The solution for the edge-isoperimetric problem (EIP) of the square tessellation of plane is investigated and solved. Summaries of the stabilization theory and previous research dealing with the EIP are stated. These techniques enable us to solve the EIP of the cubical tessellation.

Graph-partitioning problems are a central topic of research in the study of algorithms and complexity theory. They are of interest to theoreticians with connections to error correcting codes, sampling algorithms, metric embeddings, among others, and to practitioners, as algorithms for graph partitioning can be used as fundamental building blocks in many applications. One of the central problems studied in this field is the sparsest cut problem, where we want to compute the cut which has the least ratio of number of edges cut to size of smaller side of the cut. This ratio is known as the expansion of the cut. In spite of over 3 decades of intensive research, the approximability of this parameter remains an open question. The study of this optimization problem has lead to powerful techniques for both upper bounds and lower bounds for various other problems, and interesting conjectures such as the SSE conjecture. Cheeger's Inequality, a central inequality in Spectral Graph Theory, establishes a bound on expansion via the spectrum of the graph. This inequality and its many (minor) variants have played a major role in the design of algorithms as well as in understanding the limits of computation. In this thesis we study three notions of expansion, namely edge expansion in graphs, vertex expansion in graphs and hypergraph expansion. We define suitable notions of spectra w.r.t. these notions of expansion. We show how the notion Cheeger's Inequality goes across these three problems. We study higher order variants of these notions of expansion (i.e. notions of expansion corresponding to partitioning the graph/hypergraph into more than two pieces, etc.) and relate them to higher eigenvalues of graphs/hypergraphs. We also study approximation algorithms for these problems. Unlike the case of graph eigenvalues, the eigenvalues corresponding to vertex expansion and hypergraph expansion are intractable. We give optimal approximation algorithms and computational lower bounds for computing them.

The research carried out in this thesis concerns two important class of stationary surfaces in Differential Geometry, namely isoperimetric surfaces and index one minimal surfaces. The former are solutions of the so called isoperimetric problem, which is to determine the regions of least perimeter among regions of same volume in a given manifold. The latter are critical points of the area functional with Morse index one, i.e., minimal surfaces which admits only one direction where the surface can be deformed so to decrease its area. These are usually constructed via mountain pass arguments. This work focus on the study of these objects when the ambient space is a 3-dimensional spherical space forms, i.e., space form with positive curvature. Our main results classify, at the level of topology, such stationary surfaces in the spherical space forms with large fundamental group. Our first result proves that the solutions of the isoperimetric problem in spherical space forms with large fundamental group are either spheres or tori. It was previously known that solutions with genus zero and one are respectively totally umbilical and flat. Combining our result and this geometric description, we derive that the solutions of the isoperimetric problem are either geodesic spheres or quotients of Clifford tori. Our second result proves that orientable minimal surfaces with index one in the aforementioned spherical space forms have genus at most two. This is a sharp estimate as one can use the continuous one-parameter min-max theory to construct in every 3-dimensional spherical space form an index one minimal surface with genus equal the Heegaard genus of such space which is known to be at most two. Our result confirms a conjecture of R. Schoen for an infinite class of 3-manifolds.

L'elaborato verte sull'evoluzione dei problemi di massimo e minimo in matematica a partire dai primi esempi sviluppati nell'ambito della geometria fino alla presentazione dei metodi sviluppati nell'ambito dell'analisi matematica per determinare in generale la soluzione di ogni problema di questo tipo. E' consuetudine in matematica collegare i concetti di massimo e di minimo a quello di derivata. L'obiettivo dell'elaborato è dimostrare quanto sia approssimativa questa visione dei massimi e dei minimi, ripercorrendo le origini e lo sviluppo dello studio dei matematici riguardo i problemi inerenti a essi. Nel primo capitolo dopo aver mostrato che i problemi in questione hanno avuto origini assai remote, sono presentati alcuni problemi di massimo e minimo inerenti alla geometria, affrontati sia da matematici dell'antica Grecia che da matematici di età moderna. Il blocco principale riguarda le tappe verso la risoluzione in geometria del problema isoperimetrico. Nel secondo capitolo vengono presi in esame gli studi sviluppati in Ottica fino al Seicento sulla legge di riflessione e sulla legge di rifrazione perché mostrano per la prima volta che la natura è governata da leggi descritte attraverso principi estremali. Nel terzo capitolo sono presentati due metodi dei massimi e minimi sviluppati nel corso del Seicento, che posero le basi, il primo, e determinarono, il secondo, la nascita di una nuova branca della matematica: l'analisi. Viene anche discusso quello che è stato il primo problema di minimo ad essere formulato che non si poteva risolvere con i metodi di massimi e minimi prima presentati. Nel quarto capitolo vengono analizzati l'origine, la nascita e lo sviluppo della branca dell'analisi matematica in cui vennero formulati dei metodi generali per la risoluzione di tutti quei problemi di geometria, meccanica e fisica in cui si richiedeva di trovare la curva che massimizzava (o minimizzava) certe quantità, definibili attraverso un integrale definito

Doctor of Philosophy
We consider the problem of minimising the eigenvalues of the Laplacian with Robin boundary conditions $\frac{\partial u}{\partial \nu} + \alpha u = 0$ and generalised Wentzell boundary conditions $\Delta u + \beta \frac{\partial u}{\partial \nu} + \gamma u = 0$ with respect to the domain $\Omega \subset \mathbb R^N$ on which the problem is defined. For the Robin problem, when $\alpha > 0$ we extend the Faber-Krahn inequality of Daners [Math. Ann. 335 (2006), 767--785], which states that the ball minimises the first eigenvalue, to prove that the minimiser is unique amongst domains of class $C^2$. The method of proof uses a functional of the level sets to estimate the first eigenvalue from below, together with a rearrangement of the ball's eigenfunction onto the domain $\Omega$ and the usual isoperimetric inequality. We then prove that the second eigenvalue attains its minimum only on the disjoint union of two equal balls, and set the proof up so it works for the Robin $p$-Laplacian. For the higher eigenvalues, we show that it is in general impossible for a minimiser to exist independently of $\alpha > 0$. When $\alpha < 0$, we prove that every eigenvalue behaves like $-\alpha^2$ as $\alpha \to -\infty$, provided only that $\Omega$ is bounded with $C^1$ boundary. This generalises a result of Lou and Zhu [Pacific J. Math. 214 (2004), 323--334] for the first eigenvalue. For the Wentzell problem, we (re-)prove general operator properties, including for the less-studied case $\beta < 0$, where the problem is ill-posed in some sense. In particular, we give a new proof of the compactness of the resolvent and the structure of the spectrum, at least if $\partial \Omega$ is smooth. We prove Faber-Krahn-type inequalities in the general case $\beta, \gamma \neq 0$, based on the Robin counterpart, and for the ``best'' case $\beta, \gamma > 0$ establish a type of equivalence property between the Wentzell and Robin minimisers for all eigenvalues. This yields a minimiser of the second Wentzell eigenvalue. We also prove a Cheeger-type inequality for the first eigenvalue in this case.

Nella tesi sono presentati numerosi problemi classici di minimo e massimo per illustrare i diversi metodi che si sono evoluti nel tempo, a partire dalle tecniche puramente geometriche utilizzate nell'antichità dai matematici greci, per passare ai matematici del XVII - XVIII secolo che hanno sfruttato i nascenti metodi infinitesimali; si conclude con un accenno ai più moderni metodi diretti.

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This work presents a research on problems of maxima and minima of the Euclidean geometry. Initially we present some preliminary results followed by statements that in essence use basic concepts of geometry. Below are some problems of maximizing area and minimizing perimeter of triangles and convex polygons, culminating in a proof of the isoperimetric inequality for polygons and review the general case. Solve some classical problems of geometry that are related to outliers and present other problems as proposed.
Este trabalho apresenta uma pesquisa sobre problemas de máximos e mínimos da Geometria Euclidiana. Inicialmente apresentamos alguns resultados preliminares seguidos de suas demonstrações que em sua essência usam conceitos básicos de geometria. Em seguida apresentamos alguns problemas de maximização de área e de minimização de perímetro em triângulos e polígonos convexos, culminando com uma prova da desigualdade isoperimétrica para polígonos e comentário do caso geral. Resolvemos alguns problemas clássicos de geometria que estão relacionados com valores extremos e apresentamos outros como problemas propostos.

Orientador: Sueli Irene Rodrigues Costa
Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: Neste trabalho são apresentadas abordagens do problema isoperimétrico que podem ser utilizadas no ensino médio ou ensino universitário. Estas incluem: i) aspectos históricos, ii) deduções formais do problema (dentre as curvas de perímetro fixo, a circunferência é a que engloba a maior área) utilizando apenas geometria euclidiana ou via cálculo diferencial, iii) contextualização em problemas de otimização a serem abordados também utilizando recursos computacionais e iv) descrição detalhada de material audiovisual produzido para o ensino médio, com a participação do autor, para um projeto com suporte MEC - UNICAMP
Abstract: This dissertation presents approaches to the isoperimetric problem that can be used in high school or university education. These include: i)historical aspects, ii) formal deductions of the problem (among the curves of fixed perimeter, the circle encompasses most area) using only Euclidean geometry or calculus iii) contextualization in optimization problems to be also addressed using computational resources iv) detailed description of audiovisual material produced for the high school, with the participation of the author
Mestrado
Matematica
Mestre em Matemática

Risolvere il problema isoperimetrico in R^2 significa determinare la figura piana avente area maggiore tra tutte le figure aventi ugual perimetro. In questo lavoro trattiamo la risoluzione del problema isoperimetrico in R^2 proposta da Hurwitz, il quale, basandosi esclusivamente sulle proprietà analitiche delle serie di Fourier, è riuscito a dimostrare che la circonferenza è l'unica curva piana, semplice, chiusa e rettificabile con l'area massima avendo fissato il perimetro.

Etant donné une marche aléatoire sur un arbre, nous établissons pour les points de la frontière des critères de régularité analogues à des critères classiques relatifs au problème de Dirichlet pour le brownien en dimension n, dont celui de Wiener pour n égale 2. On montre l'équivalence, avec constante optimale, entre l'inégalité isopérimétrique forte pour les sous-ensembles d'un graphe dénombrable et l'inégalité pour les fonctions à variation finie nulle à l'infini. Plus généralement, on montre le résultat pour des inégalités isopérimétriques plus générales. On obtient l'inégalité isopérimétrique sur les réseaux des entiers comme application des résultats.

Nel mio lavoro di tesi prendo in esame alcuni problemi significativi di ottimizzazione unidimensionale, mostrando come, oltre al metodo del calcolo differenziale, esistano altri modi per trovarne la soluzione. I metodi utilizzati sfruttano la proprietà di alcune funzioni come parabole o funzioni goniometriche lineari, oppure si avvalgono di dimostrazioni di geometria sintetica, o ancora utilizzano le disuguaglianze. Questi metodi possono essere usati anche prima del quinto anno di scuola superiore, a differenza del metodo del calcolo differenziale che utilizza la derivata prima, ed eventualmente seconda, e che viene quindi padroneggiato a partire dalla metà del quinto anno scolastico. I problemi presi in esame sono problemi geometrici (ad esempio il problema isoperimetrico, il problema del triangolo di perimetro minimo inscritto in un triangolo acutangolo, il problema della rete stradale, ecc..), problemi trigonometrici, e problemi legati alla fisica (gittata massima, riflessione e rifrazione della luce). Il confronto tra i vari metodi di risoluzione mette in luce come, in alcuni casi, i metodi alternativi siano meno meccanici, più ragionati e più significativi rispetto all'uso delle derivate.

Cette thèse comporte deux parties. Dans la première partie, nous étudions une généralisation du problème variationnel de torsion élastique-plastique à des variétés. Nous montrons que dans le cas des variétés, le problème n'est pas équivalent à un problème d'obstacle, contrairement au cas euclidien, mais nous établissons l'équivalence lorsque le paramètre du problème tend vers l'infini. Nous montrons, comme dans le cas euclidien, que l'ensemble de non contact contient le cut locus de la variété, et converge vers ce dernier au sens de Hausforff. Nous montrons de plus que les miniseurs du problème sont uniformément semiconcaves. Nous en déduisons une approximation stable de cut locus, dans l'esprit du lambda axe médian de Chazal et Lieutier. Nous utilisons ensuite ce résultat pour calculer numériquement le cut locus de surfaces de géométries variées.Dans la seconde partie, nous étudions une extension d'un problème isopérimétrique non local. Précisément, on adjoint un potentiel de confinement au modèle de goutte liquide du noyau de Gamow. Nous étudions alors les minimiseurs de grand volume. Nous montrons que pour certains jeux de paramètres, les minimiseurs de grand volume convergent vers des boules, voire sont exactement des boules. Nous développons ensuite une méthode numérique pour ce problème variationnel. Cela permet de confirmer numériquement une conjecture de Choksi et Peletier en dimension 2 : dans ce cas les minimiseurs du modèle de Gamow semble être des boules si ils existent
This thesis is composed of two parts. In the first part, we study a generalization of the variational problem of elastic-plastic torsion problem to manifolds. We show that in the case of manifolds, the problem is not equivalent to an obstacle type problem, contrary to the euclidean case, but we establish the equivalence when the parameter of the problem goes to infinity. We show, as in the euclidean case, that the non contact set contains the cut locus of the manifold, and converges to the latter in the Hausdorff sense. What is more, we show that the minimizers of the problem are uniformly semiconcave. We deduce a stable approximation of the cut locus, in the spirit of the lambda medial axis of Chazal and Lieutier. We then use this result to compute numerically the cut locus of some surfaces of varied geometries.In the second part, we study an extension of a nonlocal isoperimetric problem. More precisely, we add a confinement potential to Gamow's liquid drop model for the nucleus. We then study large volume minimizers. We show that for certain sets of parameters, large volume minimizers converge to the ball, or may even exactly be the ball. Moreover, we develop a numerical method for this variational problem. Our results confirm numerically a conjecture of Choksi and Peletier, in dimension 2: it seems that minimizers of Gamow'sliquid drop model are balls as long as they exist

Cette thèse traite de quelques problèmes de transmission à deux exposants provenant de la physique mathématique. Plus précisément, on étudie le problème à frontière libre de Muskat lié à l'extraction du pétrole et les problèmes de renforcement en elastcit2 et en élasticité. Dans la première partie, on établit d'abord des inégalités isopérimétriques optimales pour un problème de Muskat à flux fixe et a géométrie quelconque par comparaison avec un problème analogue à symétrie sphérique. On estime en particulier le temps critique à partir duquel le problème d'évolution dégénère. Ensuite, on considère le problème de transmission qui correspond a la partie stationnaire du problème de Muskat a deux exposants et on prédit le comportement asymptotique de ce problème de transmission, en particulier lorsque les coefficients deviennent très petits ou tres grands. Dans la deuxième partie, on étudie le renforcement d'un domaine donne par une fine couche d'un matériau dur et non homogène. Les problèmes non linéaires considérés comportement deux exposants éventuellement différents. Premièrement, on examine un modèle simplifie ou le matériau renforçant est isotrope. Deuxièmement, on analyse le problème de la rigidité à la torsion avec un renforcement non isotrope. Sous des hypothèses affaiblies, on montre que le problème limite est encore explicite, lorsque l'épaisseur de la couche est décrite par une fonction de jauge liée à l'anisotropie du matériau renforçant.

The goal of this diploma thesis is to gather and describe ways the optimisation problems had been solved before the calculus was invented. Some of the methods identified through history are generalised and these which could be utilised at secondary or primary school education are described more in depth. Thesis descri- bes the method, where educator follows historical development of the problem in classroom and what benefits it might bring, as well. There is a section dedicated to evaluation of experiment which goal was to investigate understanding of pupils to certain concepts in the field of optimisation, mainly concerning isoperimetric problem. It was also focused on the pupils fe- edback to different approaches of optimisation problem solving. 1

We consider some aspects of the global geometry of cellular complexes. Motivated by techniques in graph theory, we develop combinatorial versions of isoperimetric and Poincare inequalities, and use them to derive various geometric and topological estimates. This has a progression of three major topics: 1. We define isoperimetric inequalities for normed chain complexes. In the graph case, these quantities boil down to various notions of graph expansion. We also develop some randomized algorithms which provide (in expectation) solutions to these isoperimetric problems. 2. We use these isoperimetric inequalities to derive topological and geometric estimates for certain models of random simplicial complexes. These models are generalizations of the well-known models of random graphs. 3. Using these random complexes as examples, we show that there are simplicial complexes which cannot be embedded into Euclidean space while faithfully preserving the areas of minimal surfaces.

In this thesis we study the isoperimetric problem on trees and graphs with bounded treewidth. Let G = (V,E) be a finite, simple and undirected graph. For let δ(S,G)= {(u,v) ε E : u ε S and v ε V – S }be the edge boundary of S. Given an integer i, 1 ≤ i ≤ | V| , let the edge isoperimetric value of G at I be defined as be(i,G)= mins v;|s|= i | δ(S,G)|. For S V, let φ(S,G) = {u ε V – S : ,such that be the vertex boundary of S. Given an integer i, 1 ≤ i ≤ | V| , let the vertex isoperimetric value of G at I be defined as bv(i,G)= The edge isoperimetric peak of G is defined as be(G) =. Similarly the vertex isoperimetric peak of G is defined as bv(G)= .The problem of determining a lower bound for the vertex isoperimetric peak in complete k-ary trees of depth d,Tdkwas recently considered in[32]. In the first part of this thesis we provide lower bounds for the edge and vertex isoperimetric peaks in complete k-ary trees which improve those in[32]. Our results are then generalized to arbitrary (rooted)trees. Let i be an integer where . For each i define the connected edge isoperimetric value and the connected vertex isoperimetric value of G at i as follows: is connected and is connected A meta-Fibonacci sequence is given by the reccurence a(n)= a(x1(n)+ a1′(n-1))+ a(x2(n)+ a2′(n -2)), where xi: Z+ → Z+ , i =1,2, is a linear function of n and ai′(j)= a(j) or ai′(j)= -a(j), for i=1,2. Sequences belonging to this class have been well studied but in general their properties remain intriguing. In the second part of the thesis we show an interesting connection between the problem of determining and certain meta-Fibonacci sequences. In the third part of the thesis we study the problem of determining be and bv algorithmically for certain special classes of graphs. Definition 0.1. A tree decomposition of a graph G = (V,E) is a pair where I is an index set, is a collection of subsets of V and T is a tree whose node set is I such that the following conditions are satisfied: (For mathematical equations pl see the pdf file)

碩士
臺灣大學
數學研究所
98
In the thesis we follow the demonstration of Prof. M. Ritoré to solve the isoperimetric problem on roatationally and equatorially symmetric spheres with monotone Gauss curvature from the poles. We first classify all the curves with constant geodesic curvature on a sphere with the above properties. Then we apply Sturm''s comparison theorem successively to get the final only possible curve enclosing an isoperimetric domain. On regions with constant Gauss curvature we also invoke the Bol-Fiala inequality to conclude that inside such regions a geodesic circle has the minimal length encircling a domain with a given area.

This treatise deals with the isoperimetric problem in finite projective planes. We prove that certain sets, called (0,n,n+1)-sets, are solutions to this problem. This class of sets includes all the previously known solutions to the isoperimetric problem, as well as two new types of solutions which exist in every finite projective plane. We prove a characterization theorem for (0,n,n+1)-sets with many points. We solve the isoperimetric problem for large set size, and for q + 3 points if q is even. We find all the (0,n,n+1)-sets in planes of order at most 8 and develop techniques for proving that some (0,n,n+1)-sets in larger order planes do not exist. We solve the isoperimetric problem in the planes of order at most 7 (the solution was known only for planes of order at most 4), proving that nested solutions exist in these planes. We prove that no nested solutions exist in PG(2,8). We give examples of (0,2,3)-sets in planes of order 7, 8 and 16 which are new solutions to the isoperimetric problem not included in the infinite classes mentioned above, and we investigate Latin squares and Steiner triple systems associated with these examples.

In the present work we study isoperimetric problem and its description by isoperimetric inequality. The legend of Dido, which inspired formulation of the isoperimetric problem, is described in the first chapter. The following chapters are devoted to elementary proofs of isoperimetric inequality for polygons as well as for curves. The last chapter focuses on related problem than isoperimetric that is isodiametric problem. This is described Reuleaux polygon that constitutes a means for proof of isodiametric inequality.

We investigate stability for certain geometric and functional inequalities and address the regularity of the free boundary for a problem arising in optimal transport theory. More specifically, stability estimates are obtained for the relative isoperimetric inequality inside convex cones and the Gaussian log-Sobolev inequality for a two parameter family of functions. Thereafter, away from a ``small" singular set, local C^{1,\alpha} regularity of the free boundary is achieved in the optimal partial transport problem. Furthermore, a technique is developed and implemented for estimating the Hausdorff dimension of the singular set. We conclude with a corresponding regularity theory on Riemannian manifolds.
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