Mariano Giaquinta Knihy

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.

Non-scalar variational problems arise in various fields, including geometry, where they relate to harmonic maps between Riemannian manifolds and minimal immersions. These issues also appear in physics, particularly in classical a-models. In continuum mechanics, non-linear elasticity serves as another example, while the Oseen-Frank theory of liquid crystals and Ginzburg-Landau theory of superconductivity necessitate the treatment of variational problems to model complex phenomena. The primary focus is often on identifying energy-minimizing representatives within homology or homotopy classes of maps, as well as minimizers with specific topological singularities, topological charges, and stable deformations. Over the past few decades, there has been an increasing interest and understanding of the general theory surrounding these geometric variational problems. However, the absence of a regularity theory in the non-scalar case, in contrast to the scalar case, complicates matters. This is due to the presence of singularities in vector-valued minimizers, which are often linked to concentration phenomena in energy density. Consequently, discerning a weak formulation or fully grasping the implications of various weak formulations becomes a complex task.

This monograph (in two volumes) deals with non scalar variational problems arising in geometry, as harmonic mappings between Riemannian manifolds and minimal graphs, and in physics, as stable equilibrium configuations in nonlinear elasticity or for liquid crystals. The presentation is selfcontained and accessible to non specialists. Topics are treated as far as possible in an elementary way, illustrating results with simple examples; in principle, chapters and even sections are readable independently of the general context, so that parts can be easily used for graduate courses. Open questions are often mentioned and the final section of each chapter discusses references to the literature and sometimes supplementary results. Finally, a detailed Table of Contents and an extensive Index are of help to consult this monograph

This book explores classical variational calculus, appealing to analysts, geometers, and physicists. Volume 1 focuses on the formal apparatus of variational calculus and nonparametric field theory, while Volume 2 addresses parametric variational problems, Hamilton-Jacobi theory, and classical first-order partial differential equations. Future discussions will cover developments from Hilbert's 19th and 20th problems, emphasizing direct methods and regularity theory. The text highlights the often-overlooked theory of inner variations—variations of independent variables—which provides valuable insights such as monotonicity formulas, conformality relations, and conservation laws. The combined variation of dependent and independent variables leads to Emmy Noether's general conservation laws, a critical tool for exploiting symmetries. Additional sections delve into Legendre-Jacobi theory and field theories, offering a thorough examination of one-dimensional field theory for both nonparametric and parametric integrals, and its connections to Hamilton-Jacobi theory, geometrical optics, and point mechanics. The book also investigates various approaches to utilizing convexity in the calculus of variations, with field theory presenting a particularly nuanced method. Furthermore, it emphasizes the significance of the concept of a null Lagrangian, which plays a crucial role in several contexts.

Non-scalar variational problems appear in different fields. In geometry, for in stance, we encounter the basic problems of harmonic maps between Riemannian manifolds and of minimal immersions; related questions appear in physics, for example in the classical theory of a-models. Non linear elasticity is another example in continuum mechanics, while Oseen-Frank theory of liquid crystals and Ginzburg-Landau theory of superconductivity require to treat variational problems in order to model quite complicated phenomena. Typically one is interested in finding energy minimizing representatives in homology or homotopy classes of maps, minimizers with prescribed topological singularities, topological charges, stable deformations i. e. minimizers in classes of diffeomorphisms or extremal fields. In the last two or three decades there has been growing interest, knowledge, and understanding of the general theory for this kind of problems, often referred to as geometric variational problems. Due to the lack of a regularity theory in the non scalar case, in contrast to the scalar one - or in other words to the occurrence of singularities in vector valued minimizers, often related with concentration phenomena for the energy density - and because of the particular relevance of those singularities for the problem being considered the question of singling out a weak formulation, or completely understanding the significance of various weak formulations becames non trivial. Inhaltsverzeichnis 1. Regular Variational Integrals.- 2. Finite Elasticity and Weak Diffeomorphisms.- 3. The Dirichlet Integral in Sobolev Spaces.- 4. The Dirichlet Energy for Maps into S2.- 5. Some Regular and Non Regular Variational Problems.- 6. The Non Parametric Area Functional.- Symbols.

The book addresses the crucial role of mathematics in intellectual development, highlighting its decline in educational settings. It critiques the current teaching methods that focus on rote problem-solving rather than fostering genuine understanding and independence. The authors emphasize the need for a more engaging approach that connects mathematics with real-life applications and other disciplines, while avoiding excessive technicality. They advocate for a presentation of mathematics that inspires curiosity and comprehension, rather than mere routine learning.

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.

Focusing on the study of approximation and discrete processes, this volume is structured into two parts. The first part covers numerical systems including reals, integers, and complex numbers, introducing sequences and rethinking limits and continuity. It also explores combinatorial calculus and the fundamental theorem of algebra. The second part delves into discrete processes, discussing infinite summation and power series, culminating in an introduction to discrete dynamical systems, encouraging further exploration in the field.

Examines linear structures, the topology of metric spaces, and continuity in infinite dimensions, with detailed coverage at the graduate level Includes applications to geometry and differential equations, numerous beautiful illustrations, examples, exercises, historical notes, and comprehensive index May be used in graduate seminars and courses or as a reference text by mathematicians, physicists, and engineers