Stefan Hildebrandt Knihy

This volume is the first in a three-part treatise on minimal surfaces, focusing on boundary value problems. It serves as a revised and expanded version of earlier monographs. The book opens with fundamental concepts of surface theory in three-dimensional Euclidean space, introducing minimal surfaces as stationary points of area or surfaces with zero mean curvature. A minimal surface is defined as a nonconstant harmonic mapping that is conformally parametrized and may have branch points. The classical theory of minimal surfaces is explored, featuring numerous examples, Björling’s initial value problem, reflection principles, and important theorems by Bernstein, Heinz, Osserman, and Fujimoto. The second part addresses Plateau’s problem and its modifications, presenting a new elementary proof that the area and Dirichlet integral share the same infimum for admissible surfaces spanning a prescribed contour. This leads to a simplified solution for minimizing both area and Dirichlet integral, along with new proofs of Riemann and Korn-Lichtenstein's mapping theorems, and a solution to the simultaneous Douglas problem for contours with multiple components. The volume also covers stable minimal surfaces, deriving curvature estimates and presenting uniqueness and finiteness results. Additionally, it develops a theory of unstable solutions to Plateau’s problems based on Courant’s mountain pass lemma and solves Dirichlet’s problem for non

This well-organized and coherent collection of papers leads the reader to the frontiers of present research in the theory of nonlinear partial differential equations and the calculus of variations and offers insight into some exciting developments. In addition, most articles also provide an excellent introduction to their background, describing extensively as they do the history of those problems presented, as well as the state of the art and offer a well-chosen guide to the literature. Part I contains the contributions of geometric nature: From spectral theory on regular and singular spaces to regularity theory of solutions of variational problems. Part II consists of articles on partial differential equations which originate from problems in physics, biology and stochastics. They cover elliptic, hyperbolic and parabolic cases.

Focusing on minimal surfaces with free boundaries, the book explores their boundary behavior and presents key results, including asymptotic expansions and Gauss-Bonnet formulas. It tackles the challenges of deriving regularity proofs for non-minimizers through indirect reasoning and monotonicity formulas. Geometric properties, enclosure theorems, and isoperimetric inequalities are examined, alongside discussions of obstacle problems and Plateau’s problem in Riemannian manifolds. The final chapter introduces a novel approach to the absence of interior branch points in area-minimizing solutions.

The exploration of minimal surfaces is deepened through a focus on existence, regularity, and uniqueness theorems for surfaces with partially free boundaries, highlighting the concept of "edge-crawling." A priori estimates for higher-dimensional minimal surfaces and singular integral minimizers are also discussed, leading to significant Bernstein theorems. Additionally, the book presents a comprehensive global theory, addressing the Douglas problem with Teichmüller theory, deriving index theorems, and introducing a topological perspective through Fredholm vector fields, all reflecting Smale's vision.

The book explores classical variational calculus, appealing to analysts, geometers, and physicists. Volume 1 focuses on the formal framework and nonparametric field theory, while Volume 2 covers parametric variational problems, Hamilton-Jacobi theory, and first-order partial differential equations. It emphasizes inner variations, revealing insights like monotonicity formulas and conservation laws, particularly through Emmy Noether's principles. Additionally, it examines Legendre-Jacobi theory, one-dimensional field theory, and the role of convexity in variational calculus, highlighting the concept of null Lagrangians.

Focusing on the classical variational calculus, this book serves as a comprehensive resource for analysts, geometers, and physicists. Volume 1 introduces the foundational concepts and nonparametric field theory, while Volume 2 explores parametric variational problems, Hamilton-Jacobi theory, and first-order partial differential equations. Unique emphasis is placed on inner variations, leading to insights on conservation laws and symmetries, particularly through Emmy Noether's principles. Additionally, it covers Legendre-Jacobi theory, one-dimensional field theory, and the application of convexity in variational calculus.

Seifenblasen und andere Minimalflächen - genau besehen von 2 Mathematikern

Der zweite Band dieses Lehrbuchs der Analysis umfaßt den Stoff des zweiten Semesters eines mathematischen Grundstudiums für Studierende der Mathematik, Physik und Informatik. Der klare und übersichtliche Aufbau berücksichtigt, daß schon frühzeitig die mathematischen Hilfsmittel erörtert werden, die zum Verständnis der physikalischen Grundvorlesungen unerläßlich sind. In Verbindung mit Band 1 ist so ein Leitfaden für das Studium der Analysis entstanden, der das in den ersten beiden Studiensemestern zu erwerbende mathematische Grundwissen umfaßt. Ausführliche Beweise und Erläuterungen sowie zahlreiche Beispiele und interessante Übungsaufgaben eignen es sehr gut für das Selbststudium. Ein klarer und übersichtlicher Aufbau und eine geschickte Gliederung des Stoffes ermöglichen, das erste Studium auf Kernbereiche zu beschränken. Geometrische Intuition und historische Motivation in Verbindung mit einer maßvollen Abstraktion kennzeichnen diese moderne Einführung in die Analysis.

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.