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Author: Francesco Maggi
Publisher:
ISBN: 9781139549738
Category : Geometric measure theory
Languages : en
Pages : 475
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Book Description
An engaging graduate-level introduction that bridges analysis and geometry. Suitable for self-study and a useful reference for researchers.
Author: Francesco Maggi
Publisher:
ISBN: 9781139549738
Category : Geometric measure theory
Languages : en
Pages : 475
Get Book Here
Book Description
An engaging graduate-level introduction that bridges analysis and geometry. Suitable for self-study and a useful reference for researchers.
Author: Francesco Maggi
Publisher: Cambridge University Press
ISBN: 1139560891
Category : Mathematics
Languages : en
Pages : 475
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Book Description
The marriage of analytic power to geometric intuition drives many of today's mathematical advances, yet books that build the connection from an elementary level remain scarce. This engaging introduction to geometric measure theory bridges analysis and geometry, taking readers from basic theory to some of the most celebrated results in modern analysis. The theory of sets of finite perimeter provides a simple and effective framework. Topics covered include existence, regularity, analysis of singularities, characterization and symmetry results for minimizers in geometric variational problems, starting from the basics about Hausdorff measures in Euclidean spaces and ending with complete proofs of the regularity of area-minimizing hypersurfaces up to singular sets of codimension 8. Explanatory pictures, detailed proofs, exercises and remarks providing heuristic motivation and summarizing difficult arguments make this graduate-level textbook suitable for self-study and also a useful reference for researchers. Readers require only undergraduate analysis and basic measure theory.
Author: Francesco Maggi
Publisher: Cambridge University Press
ISBN: 1107021030
Category : Mathematics
Languages : en
Pages : 475
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Book Description
An engaging graduate-level introduction that bridges analysis and geometry. Suitable for self-study and a useful reference for researchers.
Author: Ke Liang Xiao
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
"In this thesis, we explore how the theory of functions of bounded variation (BV) establishes an appropriate and versatile framework in the study of geometric variational problems. We begin with a presentation of some fundamental results on BV functions that will allow us to link them to Radon measures. In the special case of characteristic functions with bounded variation, we present structural results on sets of finite perimeter, including a generalization of the Gauss-Green Theorem. This machinery will allow us to assign a notion of perimeter to any set of finite Lebesgue measure, hence allowing non- smooth competitors to be considered in minimization problems involving the surface area. We will then address Plateau's problem and the first variation of the area functional. Finally, we will present the ideas of Steiner symmetrization to provide a proof of the Isoperimetric inequality"--
Author: Giusti
Publisher: Springer Science & Business Media
ISBN: 1468494864
Category : Mathematics
Languages : en
Pages : 250
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Book Description
The problem of finding minimal surfaces, i. e. of finding the surface of least area among those bounded by a given curve, was one of the first considered after the foundation of the calculus of variations, and is one which received a satis factory solution only in recent years. Called the problem of Plateau, after the blind physicist who did beautiful experiments with soap films and bubbles, it has resisted the efforts of many mathematicians for more than a century. It was only in the thirties that a solution was given to the problem of Plateau in 3-dimensional Euclidean space, with the papers of Douglas [DJ] and Rado [R T1, 2]. The methods of Douglas and Rado were developed and extended in 3-dimensions by several authors, but none of the results was shown to hold even for minimal hypersurfaces in higher dimension, let alone surfaces of higher dimension and codimension. It was not until thirty years later that the problem of Plateau was successfully attacked in its full generality, by several authors using measure-theoretic methods; in particular see De Giorgi [DG1, 2, 4, 5], Reifenberg [RE], Federer and Fleming [FF] and Almgren [AF1, 2]. Federer and Fleming defined a k-dimensional surface in IR" as a k-current, i. e. a continuous linear functional on k-forms. Their method is treated in full detail in the splendid book of Federer [FH 1].
Author: LawrenceCraig Evans
Publisher: Routledge
ISBN: 1351432826
Category : Mathematics
Languages : en
Pages : 286
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Book Description
This book provides a detailed examination of the central assertions of measure theory in n-dimensional Euclidean space and emphasizes the roles of Hausdorff measure and the capacity in characterizing the fine properties of sets and functions. Topics covered include a quick review of abstract measure theory, theorems and differentiation in Mn, lower Hausdorff measures, area and coarea formulas for Lipschitz mappings and related change-of-variable formulas, and Sobolev functions and functions of bounded variation. The text provides complete proofs of many key results omitted from other books, including Besicovitch's Covering Theorem, Rademacher's Theorem (on the differentiability a.e. of Lipschitz functions), the Area and Coarea Formulas, the precise structure of Sobolev and BV functions, the precise structure of sets of finite perimeter, and Alexandro's Theorem (on the twice differentiability a.e. of convex functions). Topics are carefully selected and the proofs succinct, but complete, which makes this book ideal reading for applied mathematicians and graduate students in applied mathematics.
Author: Steven G. Krantz
Publisher: Springer Science & Business Media
ISBN: 0817646795
Category : Mathematics
Languages : en
Pages : 344
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Book Description
This textbook introduces geometric measure theory through the notion of currents. Currents, continuous linear functionals on spaces of differential forms, are a natural language in which to formulate types of extremal problems arising in geometry, and can be used to study generalized versions of the Plateau problem and related questions in geometric analysis. Motivating key ideas with examples and figures, this book is a comprehensive introduction ideal for both self-study and for use in the classroom. The exposition demands minimal background, is self-contained and accessible, and thus is ideal for both graduate students and researchers.
Author: Marc Pegon
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
This thesis is dedicated to the study of two separate geometric variational problems involving nonlocal energies: firstly, the geometry and singularities of fractional harmonic maps,and secondly, an iso perimetric problem with a repulsive integrable potential inspired by Gamow's liquid drop model for the atomic nucleus. On the first topic, we improve already-known results for minimizing 1/2-harmonic maps when the target manifold is a sphere by reducing the upperbound on the Haudorff dimension of the singular set, i.e., the set of points of discontinuity. Wealso characterize so-called minimizing 1/2-harmonic tangent maps from the plane into the unit circle S1, shedding light on the behavior of minimizing 1/2-harmonic maps from R2into S1 near singularities. Finally, when s ∈ (0, 1), we prove partial regularity results for s-harmonic maps into spheres in the stationary and minimizing case, obtaining sharp estimates on the Hausdorffd imension of the set of singularities, depending on the value of s. As for the second topic of the thesis, we study a minimization problem on sets of finite perimeter under a volume constraint, where the functional is the sum of a cohesive perimeter term and a repulsive term given by a general integrable symmetric kernel on Rn. We show that under reasonable assumptions on the behavior near the origin and on some of the moments of this kernel - which include physically relevant Bessel potentials - the problem admits large mass (or volume) minimizers. In addition,after normalization, those minimizers converge to the unit ball as the mass goes to infinity. By studying the stability of the ball, we show that without these assumptions, symmetry breaking can occur, that is, there are cases when the problem admits minimizers which cannot be the ball.
Author: Leon Simon
Publisher:
ISBN: 9780867844290
Category : Geometric measure theory
Languages : en
Pages : 286
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Book Description
Author: Andrea Braides
Publisher: Springer Nature
ISBN: 303069917X
Category : Mathematics
Languages : en
Pages : 134
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Book Description
This book introduces the reader to important concepts in modern applied analysis, such as homogenization, gradient flows on metric spaces, geometric evolution, Gamma-convergence tools, applications of geometric measure theory, properties of interfacial energies, etc. This is done by tackling a prototypical problem of interfacial evolution in heterogeneous media, where these concepts are introduced and elaborated in a natural and constructive way. At the same time, the analysis introduces open issues of a general and fundamental nature, at the core of important applications. The focus on two-dimensional lattices as a prototype of heterogeneous media allows visual descriptions of concepts and methods through a large amount of illustrations.
