One-dimensional variational problems are often neglected in favor of problems which use multiple integrals and partial differential equations, which are typically more difficult to handle. However, these problems and their associated ordinary differential equations do exhibit many of the same challenges and complexity of higher-dimensional problems, while being accessible to more students. This book for graduate students provides the first modern introduction to this subject. It emphasizes direct methods and provides an exceptionally clear view of the underlying theory. Except for standard material on measures, integration and convex functions, the book develops all of the necessary mathematical tools, including basic results for one-dimensional Sobolev spaces, absolutely continuous functions, and functions of bounded variation.

One-dimensional Variational Problems: An Introduction (Oxford Lecture Series in Mathematics and Its Applications)

Science & Math

Giaquinta, Mariano

Hildebrandt, Stefan

Buttazzo, Giuseppe

Mathematics

Clarendon Press

The Parsimonious Universe: Shape and Form in the Natural World

Why does nature prefer some shapes and not others? The variety of sizes, shapes, and irregularities in nature is endless. Skillfully integrating striking full-color illustrations, the authors describe the efforts by scientists and mathematicians since the Renaissance to identify and describe the principles underlying the shape of natural forms. But can one set of laws account for both the symmetry and irregularity as well as the infinite variety of nature's designs? A complete answer to this question is likely never to be discovered. Yet, it is fascinating to see how the search for some simple universal laws down through the ages has increased our understanding of nature. The Parsimonious Universe looks at examples from the world around us at a non-mathematical, non-technical level to show that nature achieves efficiency by being stingy with the energy it expends.

Politics & Social Sciences

Tromba, Anthony

Philosophy

Copernicus

Looks at the history of the calculus of variations, describes the connection between soap films and mathematical structure, and discusses the design of buildings and living things

Mathematics and Optimal Form (Scientific American Library)

W.H.Freeman & Co.

Reference

Das vorliegende Lehrbuch ist als Leitfaden für eine zwei- oder dreisemestrige Analysis-Vorlesung gedacht und richtete sich an Studierende der Mathematik und Physik sowie an mathematisch interessierte Studierende der Informatik und der exakten Wissenschaften. Ausführliche Beweise und Erläuterungen sowie zahlreiche Beispiele und interessante Übungsaufgaben eignen es sehr gut für das mathematische Selbststudium. Ein klarer und übersichtlicher Aufbau und eine geschickte Gliederung des Stoffes ermöglichen, das erste Studium auf Kernbereiche zu beschränken. Dem Dozenten werden vielfältige Möglichkeiten geboten, je nach Art der Vorlesung verschiedene Schwerpunkte zu setzen und geeignete Wege zur Darstellung des Stoffes zu wählen. Geometrische Intuition und historische Motivation in Verbindung mit einer maßvollen Abstraktion kennzeichnen diese moderne Einführung in die Analysis.

Analysis 1 (Springer-Lehrbuch) (German Edition)

Foreign Language Study & Reference

Springer

This two-volume treatise is a standard reference in the field. It pays special attention to the historical aspects and the origins partly in applied problems―such as those of geometric optics―of parts of the theory. It contains an introduction to each chapter, section, and subsection and an overview of the relevant literature in the footnotes and bibliography. It also includes an index of the examples used throughout the book.

Calculus of Variations I (Grundlehren der mathematischen Wissenschaften, 310)

This book by two of the foremost researchers and writers in the field is the first part of a treatise that covers the subject in breadth and depth, paying special attention to the historical origins of the theory. Both individually and collectively these volumes have already become standard references.

Calculus of Variations II (Grundlehren der mathematischen Wissenschaften, 311)

Calculus of Variations I (Grundlehren der mathematischen Wissenschaften)

This book by two of the foremost researchers and writers in the field is the first part of a treatise that covers the subject in breadth and depth, paying special attention to the historical origins of the theory. Both individually and collectively these volumes have already become standard references.

Calculus of Variations II (Grundlehren der mathematischen Wissenschaften)

Minimal Surfaces is the first volume of a three volume treatise on minimal surfaces (Grundlehren Nr. 339-341). Each volume can be read and studied independently of the others. The central theme is boundary value problems for minimal surfaces. The treatise is a substantially revised and extended version of the monograph Minimal Surfaces I, II (Grundlehren Nr. 295 &amp; 296). The first volume begins with an exposition of basic ideas of the theory of surfaces in three-dimensional Euclidean space, followed by an introduction of minimal surfaces as stationary points of area, or equivalently, as surfaces of zero mean curvature. The final definition of a minimal surface is that of a nonconstant harmonic mapping X: \Omega\to\R^3 which is conformally parametrized on \Omega\subset\R^2 and may have branch points. Thereafter the classical theory of minimal surfaces is surveyed, comprising many examples, a treatment of Björling´s initial value problem, reflection principles, a formula of the second variation of area, the theorems of Bernstein, Heinz, Osserman, and Fujimoto. The second part of this volume begins with a survey of Plateau´s problem and of some of its modifications. One of the main features is a new, completely elementary proof of the fact that area A and Dirichlet integral D have the same infimum in the class C(G) of admissible surfaces spanning a prescribed contour G. This leads to a new, simplified solution of the simultaneous problem of minimizing A and D in C(G), as well as to new proofs of the mapping theorems of Riemann and Korn-Lichtenstein, and to a new solution of the simultaneous Douglas problem for A and D where G consists of several closed components. Then basic facts of stable minimal surfaces are derived; this is done in the context of stable H-surfaces (i.e. of stable surfaces of prescribed mean curvature H), especially of cmc-surfaces (H = const), and leads to curvature estimates for stable, immersed cmc-surfaces and to Nitsche´s uniqueness theorem and Tomi´s finiteness result. In addition, a theory of unstable solutions of Plateau´s problems is developed which is based on Courant´s mountain pass lemma. Furthermore, Dirichlet´s problem for nonparametric H-surfaces is solved, using the solution of Plateau´s problem for H-surfaces and the pertinent estimates.

Minimal Surfaces (Grundlehren Der Mathematischen Wissenschaften)

Dierkes, Ulrich

Sauvigny, Friedrich

See the best price to sell, buy, or rent books by Hildebrandt, Stefan. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

Books by Hildebrandt, Stefan

See the best price to sell, buy, or rent books by Hildebrandt, Stefan. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

One-dimensional Variational Problems: An Introduction (Oxford Lecture Series in Mathematics and Its Applications)

The Parsimonious Universe: Shape and Form in the Natural World

Mathematics and Optimal Form (Scientific American Library)

The Parsimonious Universe: Shape and Form in the Natural World

Analysis 1 (Springer-Lehrbuch) (German Edition)

Calculus of Variations I (Grundlehren der mathematischen Wissenschaften, 310)

Calculus of Variations II (Grundlehren der mathematischen Wissenschaften, 311)

Calculus of Variations I (Grundlehren der mathematischen Wissenschaften)

Calculus of Variations II (Grundlehren der mathematischen Wissenschaften)

Minimal Surfaces (Grundlehren Der Mathematischen Wissenschaften)