Existence And Regularity Results For Some Shape Optimization Problems

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Existence and Regularity Results for Some Shape Optimization Problems

Bozhidar Velichkov
Existence and Regularity Results for Some Shape Optimization Problems Book PDF

Author : Bozhidar Velichkov
Publisher : Springer
Page : 349 pages
File Size : 48,8 Mb
Release : 2015-03-21
Category : Mathematics
ISBN : 9788876425271

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Existence and Regularity Results for Some Shape Optimization Problems by Bozhidar Velichkov Pdf

​We study the existence and regularity of optimal domains for functionals depending on the spectrum of the Dirichlet Laplacian or of more general Schrödinger operators. The domains are subject to perimeter and volume constraints; we also take into account the possible presence of geometric obstacles. We investigate the properties of the optimal sets and of the optimal state functions. In particular, we prove that the eigenfunctions are Lipschitz continuous up to the boundary and that the optimal sets subject to the perimeter constraint have regular free boundary. We also consider spectral optimization problems in non-Euclidean settings and optimization problems for potentials and measures, as well as multiphase and optimal partition problems.

New Trends in Shape Optimization

Aldo Pratelli,Günter Leugering
New Trends in Shape Optimization Book PDF

Author : Aldo Pratelli,Günter Leugering
Publisher : Birkhäuser
Page : 314 pages
File Size : 53,9 Mb
Release : 2015-12-01
Category : Mathematics
ISBN : 9783319175638

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New Trends in Shape Optimization by Aldo Pratelli,Günter Leugering Pdf

This volume reflects “New Trends in Shape Optimization” and is based on a workshop of the same name organized at the Friedrich-Alexander University Erlangen-Nürnberg in September 2013. During the workshop senior mathematicians and young scientists alike presented their latest findings. The format of the meeting allowed fruitful discussions on challenging open problems, and triggered a number of new and spontaneous collaborations. As such, the idea was born to produce this book, each chapter of which was written by a workshop participant, often with a collaborator. The content of the individual chapters ranges from survey papers to original articles; some focus on the topics discussed at the Workshop, while others involve arguments outside its scope but which are no less relevant for the field today. As such, the book offers readers a balanced introduction to the emerging field of shape optimization.

Regularity of the One-phase Free Boundaries

Bozhidar Velichkov
Regularity of the One phase Free Boundaries Book PDF

Author : Bozhidar Velichkov
Publisher : Springer Nature
Page : 249 pages
File Size : 46,8 Mb
Release : 2023-02-24
Category : Mathematics
ISBN : 9783031132384

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Regularity of the One-phase Free Boundaries by Bozhidar Velichkov Pdf

This open access book is an introduction to the regularity theory for free boundary problems. The focus is on the one-phase Bernoulli problem, which is of particular interest as it deeply influenced the development of the modern free boundary regularity theory and is still an object of intensive research. The exposition is organized around four main theorems, which are dedicated to the one-phase functional in its simplest form. Many of the methods and the techniques presented here are very recent and were developed in the context of different free boundary problems. We also give the detailed proofs of several classical results, which are based on some universal ideas and are recurrent in the free boundary, PDE and the geometric regularity theories. This book is aimed at graduate students and researches and is accessible to anyone with a moderate level of knowledge of elliptical PDEs.

Flows of Non-Smooth Vector Fields and Degenerate Elliptic Equations

Maria Colombo
Flows of Non Smooth Vector Fields and Degenerate Elliptic Equations Book PDF

Author : Maria Colombo
Publisher : Springer
Page : 250 pages
File Size : 48,6 Mb
Release : 2017-06-07
Category : Mathematics
ISBN : 9788876426070

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Flows of Non-Smooth Vector Fields and Degenerate Elliptic Equations by Maria Colombo Pdf

The first part of the book is devoted to the transport equation for a given vector field, exploiting the lagrangian structure of solutions. It also treats the regularity of solutions of some degenerate elliptic equations, which appear in the eulerian counterpart of some transport models with congestion. The second part of the book deals with the lagrangian structure of solutions of the Vlasov-Poisson system, which describes the evolution of a system of particles under the self-induced gravitational/electrostatic field, and the existence of solutions of the semigeostrophic system, used in meteorology to describe the motion of large-scale oceanic/atmospheric flows.​

Rigid Germs, the Valuative Tree, and Applications to Kato Varieties

Matteo Ruggiero
Rigid Germs the Valuative Tree and Applications to Kato Varieties Book PDF

Author : Matteo Ruggiero
Publisher : Springer
Page : 194 pages
File Size : 41,7 Mb
Release : 2016-04-28
Category : Mathematics
ISBN : 9788876425592

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Rigid Germs, the Valuative Tree, and Applications to Kato Varieties by Matteo Ruggiero Pdf

This thesis deals with specific features of the theory of holomorphic dynamics in dimension 2 and then sets out to study analogous questions in higher dimensions, e.g. dealing with normal forms for rigid germs, and examples of Kato 3-folds. The local dynamics of holomorphic maps around critical points is still not completely understood, in dimension 2 or higher, due to the richness of the geometry of the critical set for all iterates. In dimension 2, the study of the dynamics induced on a suitable functional space (the valuative tree) allows a classification of such maps up to birational conjugacy, reducing the problem to the special class of rigid germs, where the geometry of the critical set is simple. In some cases, from such dynamical data one can construct special compact complex surfaces, called Kato surfaces, related to some conjectures in complex geometry.

Recent Advances in Partial Differential Equations and Applications

Vicenţiu D. Rădulescu,Adélia Sequeira,Vsevolod A. Solonnikov
Recent Advances in Partial Differential Equations and Applications Book PDF

Author : Vicenţiu D. Rădulescu,Adélia Sequeira,Vsevolod A. Solonnikov
Publisher : American Mathematical Soc.
Page : 404 pages
File Size : 55,9 Mb
Release : 2016-06-28
Category : Differential equations, Partial
ISBN : 9781470415211

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Recent Advances in Partial Differential Equations and Applications by Vicenţiu D. Rădulescu,Adélia Sequeira,Vsevolod A. Solonnikov Pdf

This volume contains the proceedings of the International Conference on Recent Advances in PDEs and Applications, in honor of Hugo Beirão da Veiga's 70th birthday, held from February 17–21, 2014, in Levico Terme, Italy. The conference brought together leading experts and researchers in nonlinear partial differential equations to promote research and to stimulate interactions among the participants. The workshop program testified to the wide-ranging influence of Hugo Beirão da Veiga on the field of partial differential equations, in particular those related to fluid dynamics. In his own work, da Veiga has been a seminal influence in many important areas: Navier-Stokes equations, Stokes systems, non-Newtonian fluids, Euler equations, regularity of solutions, perturbation theory, vorticity phenomena, and nonlinear potential theory, as well as various degenerate or singular models in mathematical physics. This same breadth is reflected in the mathematical papers included in this volume.

Variational Methods in Shape Optimization Problems

Dorin Bucur,Giuseppe Buttazzo
Variational Methods in Shape Optimization Problems Book PDF

Author : Dorin Bucur,Giuseppe Buttazzo
Publisher : Springer Science & Business Media
Page : 218 pages
File Size : 43,6 Mb
Release : 2006-09-13
Category : Mathematics
ISBN : 9780817644031

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Variational Methods in Shape Optimization Problems by Dorin Bucur,Giuseppe Buttazzo Pdf

Shape optimization problems are treated from the classical and modern perspectives Targets a broad audience of graduate students in pure and applied mathematics, as well as engineers requiring a solid mathematical basis for the solution of practical problems Requires only a standard knowledge in the calculus of variations, differential equations, and functional analysis Driven by several good examples and illustrations Poses some open questions.

Introduction to Shape Optimization

J. Haslinger,R. A. E. Makinen
Introduction to Shape Optimization Book PDF

Author : J. Haslinger,R. A. E. Makinen
Publisher : SIAM
Page : 291 pages
File Size : 50,9 Mb
Release : 2003-01-01
Category : Mathematics
ISBN : 0898718694

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Introduction to Shape Optimization by J. Haslinger,R. A. E. Makinen Pdf

The efficiency and reliability of manufactured products depend on, among other things, geometrical aspects; it is therefore not surprising that optimal shape design problems have attracted the interest of applied mathematicians and engineers. This self-contained, elementary introduction to the mathematical and computational aspects of sizing and shape optimization enables readers to gain a firm understanding of the theoretical and practical aspects so they may confidently enter this field. Introduction to Shape Optimization: Theory, Approximation, and Computation treats sizing and shape optimization comprehensively, covering everything from mathematical theory (existence analysis, discretizations, and convergence analysis for discretized problems) through computational aspects (sensitivity analysis, numerical minimization methods) to industrial applications. Applications include contact stress minimization for elasto-plastic bodies, multidisciplinary optimization of an airfoil, and shape optimization of a dividing tube. By presenting sizing and shape optimization in an abstract way, the authors are able to use a unified approach in the mathematical analysis for a large class of optimization problems in various fields of physics. Audience: the book is written primarily for students of applied mathematics, scientific computing, and mechanics. Most of the material is directed toward graduate students, although a portion of it is suitable for senior undergraduate students. Readers are assumed to have some knowledge of partial differential equations and their numerical solution, as well as modern programming language such as C++ Fortran 90.

Shape Optimization Problems

Hideyuki Azegami
Shape Optimization Problems Book PDF

Author : Hideyuki Azegami
Publisher : Springer Nature
Page : 646 pages
File Size : 46,6 Mb
Release : 2020-09-30
Category : Mathematics
ISBN : 9789811576188

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Shape Optimization Problems by Hideyuki Azegami Pdf

This book provides theories on non-parametric shape optimization problems, systematically keeping in mind readers with an engineering background. Non-parametric shape optimization problems are defined as problems of finding the shapes of domains in which boundary value problems of partial differential equations are defined. In these problems, optimum shapes are obtained from an arbitrary form without any geometrical parameters previously assigned. In particular, problems in which the optimum shape is sought by making a hole in domain are called topology optimization problems. Moreover, a problem in which the optimum shape is obtained based on domain variation is referred to as a shape optimization problem of domain variation type, or a shape optimization problem in a limited sense. Software has been developed to solve these problems, and it is being used to seek practical optimum shapes. However, there are no books explaining such theories beginning with their foundations. The structure of the book is shown in the Preface. The theorems are built up using mathematical results. Therefore, a mathematical style is introduced, consisting of definitions and theorems to summarize the key points. This method of expression is advanced as provable facts are clearly shown. If something to be investigated is contained in the framework of mathematics, setting up a theory using theorems prepared by great mathematicians is thought to be an extremely effective approach. However, mathematics attempts to heighten the level of abstraction in order to understand many things in a unified fashion. This characteristic may baffle readers with an engineering background. Hence in this book, an attempt has been made to provide explanations in engineering terms, with examples from mechanics, after accurately denoting the provable facts using definitions and theorems.

Shape Optimization And Optimal Design

John Cagnol,Michael P. Polis,Jean-Paul Zolesio
Shape Optimization And Optimal Design Book PDF

Author : John Cagnol,Michael P. Polis,Jean-Paul Zolesio
Publisher : CRC Press
Page : 451 pages
File Size : 44,8 Mb
Release : 2017-08-02
Category : Mathematics
ISBN : 9780203904169

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Shape Optimization And Optimal Design by John Cagnol,Michael P. Polis,Jean-Paul Zolesio Pdf

This volume presents developments and advances in modelling passive and active control systems governed by partial differential equations. It emphasizes shape analysis, optimal shape design, controllability, nonlinear boundary control, and stabilization. The authors include essential data on exact boundary controllability of thermoelastic plates with variable transmission coefficients.

Non-Smooth and Complementarity-Based Distributed Parameter Systems

Michael Hintermüller,Roland Herzog,Christian Kanzow,Michael Ulbrich,Stefan Ulbrich
Non Smooth and Complementarity Based Distributed Parameter Systems Book PDF

Author : Michael Hintermüller,Roland Herzog,Christian Kanzow,Michael Ulbrich,Stefan Ulbrich
Publisher : Springer Nature
Page : 518 pages
File Size : 54,8 Mb
Release : 2022-02-18
Category : Mathematics
ISBN : 9783030793937

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Non-Smooth and Complementarity-Based Distributed Parameter Systems by Michael Hintermüller,Roland Herzog,Christian Kanzow,Michael Ulbrich,Stefan Ulbrich Pdf

Many of the most challenging problems in the applied sciences involve non-differentiable structures as well as partial differential operators, thus leading to non-smooth distributed parameter systems. This edited volume aims to establish a theoretical and numerical foundation and develop new algorithmic paradigms for the treatment of non-smooth phenomena and associated parameter influences. Other goals include the realization and further advancement of these concepts in the context of robust and hierarchical optimization, partial differential games, and nonlinear partial differential complementarity problems, as well as their validation in the context of complex applications. Areas for which applications are considered include optimal control of multiphase fluids and of superconductors, image processing, thermoforming, and the formation of rivers and networks. Chapters are written by leading researchers and present results obtained in the first funding phase of the DFG Special Priority Program on Nonsmooth and Complementarity Based Distributed Parameter Systems: Simulation and Hierarchical Optimization that ran from 2016 to 2019.

Trends in PDE Constrained Optimization

Günter Leugering,Peter Benner,Sebastian Engell,Andreas Griewank,Helmut Harbrecht,Michael Hinze,Rolf Rannacher,Stefan Ulbrich
Trends in PDE Constrained Optimization Book PDF

Author : Günter Leugering,Peter Benner,Sebastian Engell,Andreas Griewank,Helmut Harbrecht,Michael Hinze,Rolf Rannacher,Stefan Ulbrich
Publisher : Springer
Page : 539 pages
File Size : 51,6 Mb
Release : 2014-12-22
Category : Mathematics
ISBN : 9783319050836

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Trends in PDE Constrained Optimization by Günter Leugering,Peter Benner,Sebastian Engell,Andreas Griewank,Helmut Harbrecht,Michael Hinze,Rolf Rannacher,Stefan Ulbrich Pdf

Optimization problems subject to constraints governed by partial differential equations (PDEs) are among the most challenging problems in the context of industrial, economical and medical applications. Almost the entire range of problems in this field of research was studied and further explored as part of the Deutsche Forschungsgemeinschaft (DFG) priority program 1253 on “Optimization with Partial Differential Equations” from 2006 to 2013. The investigations were motivated by the fascinating potential applications and challenging mathematical problems that arise in the field of PDE constrained optimization. New analytic and algorithmic paradigms have been developed, implemented and validated in the context of real-world applications. In this special volume, contributions from more than fifteen German universities combine the results of this interdisciplinary program with a focus on applied mathematics. The book is divided into five sections on “Constrained Optimization, Identification and Control”, “Shape and Topology Optimization”, “Adaptivity and Model Reduction”, “Discretization: Concepts and Analysis” and “Applications”. Peer-reviewed research articles present the most recent results in the field of PDE constrained optimization and control problems. Informative survey articles give an overview of topics that set sustainable trends for future research. This makes this special volume interesting not only for mathematicians, but also for engineers and for natural and medical scientists working on processes that can be modeled by PDEs.

Partial Differential Equations On Multistructures

Felix Mehmeti,Joachim Von Below,Serge Nicaise
Partial Differential Equations On Multistructures Book PDF

Author : Felix Mehmeti,Joachim Von Below,Serge Nicaise
Publisher : CRC Press
Page : 288 pages
File Size : 49,6 Mb
Release : 2001-04-10
Category : Mathematics
ISBN : 9780824745042

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Partial Differential Equations On Multistructures by Felix Mehmeti,Joachim Von Below,Serge Nicaise Pdf

This text is based on lectures presented at the International Conference on Partial Differential Equations (PDEs) on Multistructures, held in Luminy, France. It contains advances in the field, compiling research on the analyses and applications of multistructures - including treatments of classical theories, specific characterizations and modellings of multistructures, and discussions on uses in physics, electronics, and biology.

Shape Optimization and Spectral Theory

Antoine Henrot
Shape Optimization and Spectral Theory Book PDF

Author : Antoine Henrot
Publisher : De Gruyter Open
Page : 474 pages
File Size : 51,7 Mb
Release : 2017-05-08
Category : Electronic
ISBN : 3110550857

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Shape Optimization and Spectral Theory by Antoine Henrot Pdf

"Shape optimization and spectral theory" is a survey book aiming to give an overview of recent results in spectral geometry and its links with shape optimization. It covers most of the issues which are important for people working in PDE and differential geometry interested in sharp inequalities and qualitative behaviour for eigenvalues of the Laplacian with different kind of boundary conditions (Dirichlet, Robin and Steklov). This includes: existence of optimal shapes, their regularity, the case of special domains like triangles, isospectrality, quantitative form of the isoperimetric inequalities, optimal partitions, universal inequalities and numerical results. Much progress has been made in these extremum problems during the last ten years and this edited volume presents a valuable update to a wide community interested in these topics. List of contributors Antunes Pedro R.S., Ashbaugh Mark, Bonnaillie-Noel Virginie, Brasco Lorenzo, Bucur Dorin, Buttazzo Giuseppe, De Philippis Guido, Freitas Pedro, Girouard Alexandre, Helffer Bernard, Kennedy James, Lamboley Jimmy, Laugesen Richard S., Oudet Edouard, Pierre Michel, Polterovich Iosif, Siudeja Bartlomiej A., Velichkov Bozhidar

A Course in the Calculus of Variations

Filippo Santambrogio
A Course in the Calculus of Variations Book PDF

Author : Filippo Santambrogio
Publisher : Springer Nature
Page : 354 pages
File Size : 53,6 Mb
Release : 2024-01-18
Category : Mathematics
ISBN : 9783031450365

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A Course in the Calculus of Variations by Filippo Santambrogio Pdf

This book provides an introduction to the broad topic of the calculus of variations. It addresses the most natural questions on variational problems and the mathematical complexities they present. Beginning with the scientific modeling that motivates the subject, the book then tackles mathematical questions such as the existence and uniqueness of solutions, their characterization in terms of partial differential equations, and their regularity. It includes both classical and recent results on one-dimensional variational problems, as well as the adaptation to the multi-dimensional case. Here, convexity plays an important role in establishing semi-continuity results and connections with techniques from optimization, and convex duality is even used to produce regularity results. This is then followed by the more classical Hölder regularity theory for elliptic PDEs and some geometric variational problems on sets, including the isoperimetric inequality and the Steiner tree problem. The book concludes with a chapter on the limits of sequences of variational problems, expressed in terms of Γ-convergence. While primarily designed for master's-level and advanced courses, this textbook, based on its author's instructional experience, also offers original insights that may be of interest to PhD students and researchers. A foundational understanding of measure theory and functional analysis is required, but all the essential concepts are reiterated throughout the book using special memo-boxes.