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.

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.

<b>Asymptotic cones.</b> A finitely generated group has a word metric, which one can scale and thereby view the group from increasingly distant vantage points. The group coalesces to an "asymptotic cone" in the limit (this is made precise using techniques of non-standard analysis). The reward is that in place of the discrete group one has a continuous object "that is amenable to attack by geometric (e.g. topological, infinitesimal) machinery" (to quote Gromov). We give coarse geometric conditions for a metric space X to have N-connected asymptotic cones. These conditions are expressed in terms of certain filling functions concerning filling N-spheres in an appropriately coarse sense. We interpret the criteria in the case where X is a finitely generated group &Gamma; with a word metric. This leads to upper bounds on filling functions for groups with simply connected cones -- in particular they have linearly bounded filling length functions. We prove that if all the asymptotic cones of &Gamma; are N-connected then &Gamma; is of type F_N+1 and we provide N-th order isoperimetric and isodiametric functions. Also we show that the asymptotic cones of a virtually polycyclic group &Gamma; are all contractible if and only if &Gamma; is virtually nilpotent. <b>Combable groups and almost-convex groups.</b> A combing of a finitely generated group &Gamma; is a normal form; that is a choice of word (a combing line) for each group element that satisfies a geometric constraint: nearby group elements have combing lines that fellow travel. An almost-convexity condition concerns the geometry of closed balls in the Cayley graph for &Gamma;. We show that even the most mild combability or almost-convexity restrictions on a finitely presented group already force surprisingly strong constraints on the geometry of its word problem. In both cases we obtain an n! isoperimetric function, and upper bounds of ~ n² on both the minimal isodiametric function and the filling length function.

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.

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.

Made available in DSpace on 2015-05-15T11:46:08Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-08-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / 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.

We show that the singular numbers of single layer potentials on smooth curves asymptotically behave like O(1/n). For the curves with singularities, as long as they contain a smooth sub-arc, the resulting single layer potentials are never trace-class. We provide upper bounds for the operator and the Hilbert-Schmidt norms of single layer potentials on smooth and chord-arc curves. Regarding the injectivity of single layer potentials on planar curves, we prove that among single layer potentials on dilations of a given curve, only one yields a non-injective single layer potential. A criterion for injectivity of single layer potentials on ellipses is given. We establish an isoperimetric inequality for Schatten p−norms of logarithmic potentials over quadrilaterals and its analogue for Newtonian potentials on parallelepipeds.

In this thesis we study qualitative as well as quantitative stability aspects of isometric and conformal maps from S^(n-1) to R^n, when n is greater or equal to 2 or 3 respectively. Starting from the classical theorem of Liouville, according to which the isometry group of S^(n-1) is the group of its rigid motions and the conformal group of S^(n-1) is the one of its Möbius transformations, we obtain stability results for these classes of mappings among maps from S^(n-1) to R^n in terms of appropriately defined deficits. Unlike classical geometric rigidity results for maps defined on domains of R^n and mapping into R^n, not only an isometric\ conformal deficit is necessary in this more flexible setting, but also a deficit measuring how much the maps in consideration distort S^(n-1) in a generalized sense. The introduction of the latter is motivated by the classical Euclidean isoperimetric inequality.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22975号 / 理博第4652号 / 新制||理||1668(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 山口 孝男, 教授 藤原 耕二, 教授 入谷 寛 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

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.