ref_calculus_variations.bib

@book{attouchVariationalAnalysisSobolev2014,
  title = {Variational {{Analysis}} in {{Sobolev}} and {{BV Spaces}}: {{Applications}} to {{PDEs}} and {{Optimization}}, {{Second Edition}}},
  shorttitle = {Variational {{Analysis}} in {{Sobolev}} and {{BV Spaces}}},
  author = {Attouch, Hedy and Buttazzo, Giuseppe and Michaille, G{\'e}rard},
  year = {2014},
  month = oct,
  series = {{{MOS-SIAM Series}} on {{Optimization}}},
  edition = {Second Edition},
  publisher = {{SIAM-Society for Industrial and Applied Mathematics}},
  address = {Philadelphia},
  url = {https://epubs.siam.org/doi/book/10.1137/1.9781611973488},
  abstract = {This volume is an excellent guide for anyone interested in variational analysis, optimization, and PDEs. It offers a detailed presentation of the most important tools in variational analysis as well as applications to problems in geometry, mechanics, elasticity, and computer vision. Among the new elements in this second edition: the section of Chapter 5 on capacity theory and elements of potential theory now includes the concepts of quasi-open sets and quasi-continuity; Chapter 6 includes an increased number of examples in the areas of linearized elasticity system, obstacles problems, convection-diffusion, and semilinear equations; Chapter 11 has been expanded to include a section on mass transportation problems and the Kantorovich relaxed formulation of the Monge problem; a new subsection on stochastic homogenization in Chapter 12 establishes the mathematical tools coming from ergodic theory, and illustrates them in the scope of statistically homogeneous materials; Chapter 16 has been augmented by examples illustrating the shape optimization procedure; and Chapter 17 is an entirely new and comprehensive chapter devoted to gradient flows and the dynamical approach to equilibria.Audience: The book is intended for Ph.D. students, researchers, and practitioners who want to approach the field of variational analysis in a systematic way.Contents: Chapter 1: Introduction; Part I: Basic Variational Principles; Chapter 2: Weak Solution Methods in Variational Analysis; Chapter 3: Abstract Variational Principles; Chapter 4: Complements on Measure Theory; Chapter 5: Sobolev Spaces; Chapter 6: Variational Problems: Some Classical Examples; Chapter 7: The Finite Element Method; Chapter 8: Spectral Analysis of the Laplacian; Chapter 9: Convex Duality and Optimization; Part II: Advanced Variational Analysis; Chapter 10: Spaces BV and SBV; Chapter 11: Relaxation in Sobolev, BV, and Young Measures Spaces; Chapter 12: Convergence and Applications; Chapter 13: Integral Functionals of the Calculus of Variations; Chapter 14: Applications in Mechanics and Computer Vision; Chapter 15: Variational Problems with a Lack of Coercivity; Chapter 16: An Introduction to Shape Optimization Problems; Chapter 17: Gradient Flows},
  isbn = {978-1-61197-347-1},
  langid = {english}
}

@book{bruntCalculusVariations2003,
  title = {The {{Calculus}} of {{Variations}}},
  author = {van Brunt, Bruce},
  year = {2003},
  month = sep,
  series = {Universitext},
  publisher = {Springer},
  address = {New York},
  url = {https://doi.org/10.1007/b97436},
  abstract = {Suitable for advanced undergraduate and graduate students of mathematics, physics, or engineering, this introduction to the calculus of variations focuses on variational problems involving one independent variable. It also discusses more advanced topics such as the inverse problem, eigenvalue problems, and Noether's theorem. The text includes numerous examples along with problems to help students consolidate the material.},
  isbn = {978-0-387-40247-5},
  langid = {english}
}

@book{casselVariationalMethodsApplications2013,
  title = {Variational {{Methods}} with {{Applications}} in {{Science}} and {{Engineering}}},
  author = {Cassel, Kevin W.},
  year = {2013},
  month = jul,
  publisher = {Cambridge University Press},
  address = {Cambridge},
  url = {https://doi.org/10.1017/CBO9781139136860},
  abstract = {There is an ongoing resurgence of applications in which the calculus of variations has direct relevance. ~Variational Methods with Applications in Science and Engineering reflects the strong connection between calculus of variations and the applications for which variational methods form the fundamental foundation.~ The material is presented in a manner that promotes development of an intuition about the concepts and methods with an emphasis on applications, and the priority of the application chapters is to provide a brief introduction to a variety of physical phenomena and optimization principles from a unified variational point of view. ~The first part of the book provides a modern treatment of the calculus of variations suitable for advanced undergraduate students and graduate students in applied mathematics, physical sciences, and engineering. The second part gives an account of several physical applications from a variational point of view, such as classical mechanics, optics and electromagnetics, modern physics, and fluid mechanics.~ A unique feature of this part of the text is derivation of the ubiquitous Hamilton's principle directly from the first law of thermodynamics, which enforces conservation of total energy, and the subsequent derivation of the governing equations of many discrete and continuous phenomena from Hamilton's principle.~ In this way, the reader will see how the traditional variational treatments of statics and dynamics are unified with the physics of fluids, electromagnetic fields, relativistic mechanics, and quantum mechanics through Hamilton's principle. The third part covers applications of variational methods to optimization and control of discrete and continuous systems, including image and data processing as well as numerical grid generation.~ The application chapters in parts two and three are largely independent of each other so that the instructor or reader can choose a path through the topics that aligns with their interests.},
  isbn = {978-1-107-02258-4},
  langid = {english}
}

@book{courantMethodsMathematicalPhysics1989,
  title = {Methods of {{Mathematical Physics}}, {{Vol}}. 1},
  author = {Courant, Richard and Hilbert, David},
  year = {1989},
  month = jan,
  publisher = {Wiley-VCH},
  address = {Weinheim},
  abstract = {Since the first volume of this work came out in Germany in 1924, this book, together with its second volume, has remained standard in the field. Courant and Hilbert's treatment restores the historically deep connections between physical intuition and mathematical development, providing the reader with a unified approach to mathematical physics. The present volume represents Richard Courant's second and final revision of 1953.},
  isbn = {978-0-471-50447-4},
  langid = {english}
}

@book{gelfandCalculusVariations2000,
  title = {Calculus of {{Variations}}},
  author = {Gelfand, I. M. and Fomin, S. V.},
  year = {2020},
  edition = {Reprint of the 1963 edition},
  publisher = {Dover Publications},
  address = {Mineola, N.Y},
  isbn = {978-0-486-41448-5},
  langid = {english}
}

@book{kotFirstCourseCalculus2014,
  title = {A {{First Course}} in the {{Calculus}} of {{Variations}}},
  author = {Kot, Mark},
  year = {2014},
  month = oct,
  series = {Student {{Mathematical Library}}},
  publisher = {American Mathematical Society},
  address = {Providence, Rhode Island},
  url = {https://bookstore.ams.org/stml-72/},
  abstract = {This book is intended for a first course in the calculus of variations, at the senior or beginning graduate level. The reader will learn methods for finding functions that maximize or minimize integrals. The text lays out important necessary and sufficient conditions for extrema in historical order, and it illustrates these conditions with numerous worked-out examples from mechanics, optics, geometry, and other fields. The exposition starts with simple integrals containing a single independent variable, a single dependent variable, and a single derivative, subject to weak variations, but steadily moves on to more advanced topics, including multivariate problems, constrained extrema, homogeneous problems, problems with variable endpoints, broken extremals, strong variations, and sufficiency conditions. Numerous line drawings clarify the mathematics. Each chapter ends with recommended readings that introduce the student to the relevant scientific literature and with exercises that consolidate understanding.},
  isbn = {978-1-4704-1495-5},
  langid = {english}
}

@book{LagrangeMultiplierApproach,
  title = {Lagrange {{Multiplier Approach}} to {{Variational Problems}} and {{Applications}}},
  author = {Ito, Kazufumi and Kunisch Karl},
  year = {2008},
  series = {Advances in {{Design}} and {{Control}}},
  publisher = {{Society for Industrial and Applied Mathematics}},
  address = {Philadelphia, PA},
  url = {https://epubs.siam.org/doi/book/10.1137/1.9780898718614},
  urldate = {2022-04-30},
  abstract = {Lagrange multiplier theory provides a tool for the analysis of a general class of nonlinear variational problems and is the basis for developing efficient and powerful iterative methods for solving these problems. This comprehensive monograph analyzes Lagrange multiplier theory and shows its impact on the development of numerical algorithms for problems posed in a function space setting. The book is motivated by the idea that a full treatment of a variational problem in function spaces would not be complete without a discussion of infinite-dimensional analysis, proper discretization, and the relationship between the two. The authors develop and analyze efficient algorithms for constrained optimization and convex optimization problems based on the augumented Lagrangian concept and cover such topics as sensitivity analysis, convex optimization, second order methods, and shape sensitivity calculus. General theory is applied to challenging problems in optimal control of partial differential equations, image analysis, mechanical contact and friction problems, and American options for the Black---Scholes model.},
  langid = {english}
}

@book{langtangenIntroductionNumericalMethods2019a,
  title = {Introduction to {{Numerical Methods}} for {{Variational Problems}}},
  author = {Langtangen, Hans Petter and Mardal, Kent-Andre},
  year = {2019},
  month = sep,
  series = {Texts in {{Computational Science}} and {{Engineering}}},
  publisher = {Springer},
  address = {Cham},
  url = {https://doi.org/10.1007/978-3-030-23788-2},
  abstract = {This textbook teaches finite element methods from a computational point of view. It focuses on how to develop flexible computer programs with Python, a programming language in which a combination of symbolic and numerical tools is used to achieve an explicit and practical derivation of finite element algorithms. The finite element library FEniCS is used throughout the book, but the content is provided in sufficient detail to ensure that students with less mathematical background or mixed programming-language experience will equally benefit. All program examples are available on the Internet.},
  isbn = {978-3-030-23789-9},
  langid = {english}
}

@book{SIAMBookstore,
  title = {Semismooth {{Newton Methods}} for {{Variational Inequalities}} and {{Constrained Optimization Problems}} in {{Function Spaces}}},
  author = {Ulbrich, Michael},
  year = {2011},
  series = {{{MOS-SIAM Series}} on {{Optimization}}},
  publisher = {{Society for Industrial and Applied Mathematics}},
  url = {https://my.siam.org/Store/Product/viewproduct/?ProductId=106657},
  urldate = {2022-04-30},
  abstract = {Semismooth Newton methods are a modern class of remarkably powerful and versatile algorithms for solving constrained optimization problems with partial differential equations (PDEs), variational inequalities, and related problems. This book provides a comprehensive presentation of these methods in function spaces, striking a balance between thoroughly developed theory and numerical applications.  Although largely self-contained, the book also covers recent developments in the field, such as state-constrained problems, and offers new material on topics such as improved mesh independence results. The theory and methods are applied to a range of practically important problems, including      optimal control of nonlinear elliptic differential equations,      obstacle problems, and     flow control of instationary Navier--Stokes fluids.  In addition, the author covers adjoint-based derivative computation and the efficient solution of Newton systems by multigrid and preconditioned iterative methods.},
  isbn = {978-1-61197-068-5}
}