The Calculus of Variations
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Cloaking Via Change of Variables for Second Order Quasi-Linear Elliptic Differential Equations Maher Belgacem, Abderrahman Boukricha
Cloaking via change of variables for second order quasi-linear elliptic differential equations Maher Belgacem, Abderrahman Boukricha To cite this version: Maher Belgacem, Abderrahman Boukricha. Cloaking via change of variables for second order quasi- linear elliptic differential equations. 2017. hal-01444772 HAL Id: hal-01444772 https://hal.archives-ouvertes.fr/hal-01444772 Preprint submitted on 24 Jan 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Cloaking via change of variables for second order quasi-linear elliptic differential equations Maher Belgacem, Abderrahman Boukricha University of Tunis El Manar, Faculty of Sciences of Tunis 2092 Tunis, Tunisia. Abstract The present paper introduces and treats the cloaking via change of variables in the framework of quasi-linear elliptic partial differential operators belonging to the class of Leray-Lions (cf. [14]). We show how a regular near-cloak can be obtained using an admissible (nonsingular) change of variables and we prove that the singular change- of variable-based scheme achieve perfect cloaking in any dimension d ≥ 2. We thus generalize previous results (cf. [7], [11]) obtained in the context of electric impedance tomography formulated by a linear differential operator in divergence form. -
Geometric Integration Theory Contents
Steven G. Krantz Harold R. Parks Geometric Integration Theory Contents Preface v 1 Basics 1 1.1 Smooth Functions . 1 1.2Measures.............................. 6 1.2.1 Lebesgue Measure . 11 1.3Integration............................. 14 1.3.1 Measurable Functions . 14 1.3.2 The Integral . 17 1.3.3 Lebesgue Spaces . 23 1.3.4 Product Measures and the Fubini–Tonelli Theorem . 25 1.4 The Exterior Algebra . 27 1.5 The Hausdorff Distance and Steiner Symmetrization . 30 1.6 Borel and Suslin Sets . 41 2 Carath´eodory’s Construction and Lower-Dimensional Mea- sures 53 2.1 The Basic Definition . 53 2.1.1 Hausdorff Measure and Spherical Measure . 55 2.1.2 A Measure Based on Parallelepipeds . 57 2.1.3 Projections and Convexity . 57 2.1.4 Other Geometric Measures . 59 2.1.5 Summary . 61 2.2 The Densities of a Measure . 64 2.3 A One-Dimensional Example . 66 2.4 Carath´eodory’s Construction and Mappings . 67 2.5 The Concept of Hausdorff Dimension . 70 2.6 Some Cantor Set Examples . 73 i ii CONTENTS 2.6.1 Basic Examples . 73 2.6.2 Some Generalized Cantor Sets . 76 2.6.3 Cantor Sets in Higher Dimensions . 78 3 Invariant Measures and the Construction of Haar Measure 81 3.1 The Fundamental Theorem . 82 3.2 Haar Measure for the Orthogonal Group and the Grassmanian 90 3.2.1 Remarks on the Manifold Structure of G(N,M).... 94 4 Covering Theorems and the Differentiation of Integrals 97 4.1 Wiener’s Covering Lemma and its Variants . -
33 .1 Implicit Differentiation 33.1.1Implicit Function
Module 11 : Partial derivatives, Chain rules, Implicit differentiation, Gradient, Directional derivatives Lecture 33 : Implicit differentiation [Section 33.1] Objectives In this section you will learn the following : The concept of implicit differentiation of functions. 33 .1 Implicit differentiation \ As we had observed in section , many a times a function of an independent variable is not given explicitly, but implicitly by a relation. In section we had also mentioned about the implicit function theorem. We state it precisely now, without proof. 33.1.1Implicit Function Theorem (IFT): Let and be such that the following holds: (i) Both the partial derivatives and exist and are continuous in . (ii) and . Then there exists some and a function such that is differentiable, its derivative is continuous with and 33.1.2Remark: We have a corresponding version of the IFT for solving in terms of . Here, the hypothesis would be . 33.1.3Example: Let we want to know, when does the implicit expression defines explicitly as a function of . We note that and are both continuous. Since for the points and the implicit function theorem is not applicable. For , and , the equation defines the explicit function and for , the equation defines the explicit function Figure 1. y is a function of x. A result similar to that of theorem holds for function of three variables, as stated next. Theorem : 33.1.4 Let and be such that (i) exist and are continuous at . (ii) and . Then the equation determines a unique function in the neighborhood of such that for , and , Practice Exercises : Show that the following functions satisfy conditions of the implicit function theorem in the neighborhood of (1) the indicated point. -
CALCULUS of VARIATIONS and TENSOR CALCULUS
CALCULUS OF VARIATIONS and TENSOR CALCULUS U. H. Gerlach September 22, 2019 Beta Edition 2 Contents 1 FUNDAMENTAL IDEAS 5 1.1 Multivariable Calculus as a Prelude to the Calculus of Variations. 5 1.2 Some Typical Problems in the Calculus of Variations. ...... 6 1.3 Methods for Solving Problems in Calculus of Variations. ....... 10 1.3.1 MethodofFiniteDifferences. 10 1.4 TheMethodofVariations. 13 1.4.1 Variants and Variations . 14 1.4.2 The Euler-Lagrange Equation . 17 1.4.3 Variational Derivative . 20 1.4.4 Euler’s Differential Equation . 21 1.5 SolvedExample.............................. 24 1.6 Integration of Euler’s Differential Equation. ...... 25 2 GENERALIZATIONS 33 2.1 Functional with Several Unknown Functions . 33 2.2 Extremum Problem with Side Conditions. 38 2.2.1 HeuristicSolution. 40 2.2.2 Solution via Constraint Manifold . 42 2.2.3 Variational Problems with Finite Constraints . 54 2.3 Variable End Point Problem . 55 2.3.1 Extremum Principle at a Moment of Time Symmetry . 57 2.4 Generic Variable Endpoint Problem . 60 2.4.1 General Variations in the Functional . 62 2.4.2 Transversality Conditions . 64 2.4.3 Junction Conditions . 66 2.5 ManyDegreesofFreedom . 68 2.6 Parametrization Invariant Problem . 70 2.6.1 Parametrization Invariance via Homogeneous Function .... 71 2.7 Variational Principle for a Geodesic . 72 2.8 EquationofGeodesicMotion . 76 2.9 Geodesics: TheirParametrization.. 77 3 4 CONTENTS 2.9.1 Parametrization Invariance. 77 2.9.2 Parametrization in Terms of Curve Length . 78 2.10 Physical Significance of the Equation for a Geodesic . ....... 80 2.10.1 Freefloatframe ........................ -
FUNCTIONAL ANALYSIS 1. Banach and Hilbert Spaces in What
FUNCTIONAL ANALYSIS PIOTR HAJLASZ 1. Banach and Hilbert spaces In what follows K will denote R of C. Definition. A normed space is a pair (X, k · k), where X is a linear space over K and k · k : X → [0, ∞) is a function, called a norm, such that (1) kx + yk ≤ kxk + kyk for all x, y ∈ X; (2) kαxk = |α|kxk for all x ∈ X and α ∈ K; (3) kxk = 0 if and only if x = 0. Since kx − yk ≤ kx − zk + kz − yk for all x, y, z ∈ X, d(x, y) = kx − yk defines a metric in a normed space. In what follows normed paces will always be regarded as metric spaces with respect to the metric d. A normed space is called a Banach space if it is complete with respect to the metric d. Definition. Let X be a linear space over K (=R or C). The inner product (scalar product) is a function h·, ·i : X × X → K such that (1) hx, xi ≥ 0; (2) hx, xi = 0 if and only if x = 0; (3) hαx, yi = αhx, yi; (4) hx1 + x2, yi = hx1, yi + hx2, yi; (5) hx, yi = hy, xi, for all x, x1, x2, y ∈ X and all α ∈ K. As an obvious corollary we obtain hx, y1 + y2i = hx, y1i + hx, y2i, hx, αyi = αhx, yi , Date: February 12, 2009. 1 2 PIOTR HAJLASZ for all x, y1, y2 ∈ X and α ∈ K. For a space with an inner product we define kxk = phx, xi . Lemma 1.1 (Schwarz inequality). -
José M. Vega-Guzmán
Jos´eM. Vega-Guzm´an Curriculum Vitae Contact Department of Mathematics Tel: (409) 880-8792 Information College of Arts and Sciences Fax: (409) 880-8794 Lamar University email: [email protected] 200-C Lucas Building URL: https://sites.google.com/site/profjmvg/ Beaumont, TX 77710 Education PhD (Applied Mathematics) 2008{2013 • Arizona State University, Tempe, AZ • Co-Advisors: Sergei K. Suslov and Carlos Castillo-Chavez • Dissertation Title: Solution Methods for Certain Evolution Equations Master in Natural Sciences (Mathematics) 2006{2008 • Arizona State University, Tempe, AZ B.S. (MathEd) 2000{2005 • Universidad de Puerto Rico, Cayey, PR • Advisor: Maria A. Avi~n´o Research Interests Nonlinear Wave Propagation. Quantum Mechanics. Integrability of Nonlinear Evolution Equations. Applications of ODE's and PDE's in the Natural Sciences. Mathematical Modeling. Publications 2017 Vega{Guzm´an,J.M.; Alqahtani, R.; Zhou, Q.; Mahmood, M.F.; Moshokoa, S.P.; Ullah, M.Z.; Biswas, A.; Belic, M. Optical solitons for Lakshmanan-Porsezian-Daniel model with spatio-temporal dispersion using the method of undetermined coefficients. Optik - International Journal for Light and Electron Optics. 144, 115{123. 2017 Wang, G.; Kara, A.H.; Vega{Guzm´an,J.M.; Biswas, A. Group analysis, nonlinear self-adjointness, conser- vation laws and soliton solutions for the mKdV systems. Nonlinear Analysis: Modeling and Control. 22(3), 334{346. 2017 Vega{Guzm´an,J.M.; Ullah, M.Z.; Asma, M.; Zhou, Q.; Biswas, A. Dispersive solitons in magneto-optic waveguides. Superlattices and Microstructures. 103, 161{170. 2017 Vega{Guzm´an,J.M.; Mahmood, M.F.; Zhou, Q.; Triki, H.; Arnous, A.H.; Biswas, A.; Moshokoa, S.P.; Belic, M. -
Chapter 8 Change of Variables, Parametrizations, Surface Integrals
Chapter 8 Change of Variables, Parametrizations, Surface Integrals x0. The transformation formula In evaluating any integral, if the integral depends on an auxiliary function of the variables involved, it is often a good idea to change variables and try to simplify the integral. The formula which allows one to pass from the original integral to the new one is called the transformation formula (or change of variables formula). It should be noted that certain conditions need to be met before one can achieve this, and we begin by reviewing the one variable situation. Let D be an open interval, say (a; b); in R , and let ' : D! R be a 1-1 , C1 mapping (function) such that '0 6= 0 on D: Put D¤ = '(D): By the hypothesis on '; it's either increasing or decreasing everywhere on D: In the former case D¤ = ('(a);'(b)); and in the latter case, D¤ = ('(b);'(a)): Now suppose we have to evaluate the integral Zb I = f('(u))'0(u) du; a for a nice function f: Now put x = '(u); so that dx = '0(u) du: This change of variable allows us to express the integral as Z'(b) Z I = f(x) dx = sgn('0) f(x) dx; '(a) D¤ where sgn('0) denotes the sign of '0 on D: We then get the transformation formula Z Z f('(u))j'0(u)j du = f(x) dx D D¤ This generalizes to higher dimensions as follows: Theorem Let D be a bounded open set in Rn;' : D! Rn a C1, 1-1 mapping whose Jacobian determinant det(D') is everywhere non-vanishing on D; D¤ = '(D); and f an integrable function on D¤: Then we have the transformation formula Z Z Z Z ¢ ¢ ¢ f('(u))j det D'(u)j du1::: dun = ¢ ¢ ¢ f(x) dx1::: dxn: D D¤ 1 Of course, when n = 1; det D'(u) is simply '0(u); and we recover the old formula. -
Some Inequalities in the Theory of Functions^)
SOME INEQUALITIES IN THE THEORY OF FUNCTIONS^) BY ZEEV NEHARI 1. Introduction. Many of the inequalities of function theory and potential theory may be reduced to statements regarding the properties of harmonic domain functions with vanishing or constant boundary values, that is, func- tions which can be obtained from the Green's function by means of elementary processes. For the derivation of these inequalities a large number of different techniques and procedures have been used. It is the aim of this paper to show that many of the known inequalities of this type, and also others which are new, can be obtained as simple consequences of the classical minimum prop- erty of the Dirichlet integral. In addition to the resulting simplification, this method has the further advantage of being capable of generalization to a wide class of linear partial differential equations of elliptic type in two or more variables. The idea of using the positive-definite character of an integral as the point of departure for the derivation of function-theoretic inequalities is, of course, not new and it has been successfully used for this purpose by a num- ber of authors [l; 2; 8; 9; 16]. What the present paper attempts is to give a more or less systematic survey of the type of inequality obtainable in this way. 2. Monotonie functionals. 1. The domains we shall consider will be as- sumed to be bounded by a finite number of closed analytic curves and they will be embedded in a given closed Riemann surface R of finite genus. The symbol 5(a) will be used to denote a "singularity function" with the follow- ing properties: 5(a) is real, harmonic, and single-valued on R, with the excep- tion of a finite number of points at which 5(a) has specified singularities. -
A Short Introduction to the Quantum Formalism[S]
A short introduction to the quantum formalism[s] François David Institut de Physique Théorique CNRS, URA 2306, F-91191 Gif-sur-Yvette, France CEA, IPhT, F-91191 Gif-sur-Yvette, France [email protected] These notes are an elaboration on: (i) a short course that I gave at the IPhT-Saclay in May- June 2012; (ii) a previous letter [Dav11] on reversibility in quantum mechanics. They present an introductory, but hopefully coherent, view of the main formalizations of quantum mechanics, of their interrelations and of their common physical underpinnings: causality, reversibility and locality/separability. The approaches covered are mainly: (ii) the canonical formalism; (ii) the algebraic formalism; (iii) the quantum logic formulation. Other subjects: quantum information approaches, quantum correlations, contextuality and non-locality issues, quantum measurements, interpretations and alternate theories, quantum gravity, are only very briefly and superficially discussed. Most of the material is not new, but is presented in an original, homogeneous and hopefully not technical or abstract way. I try to define simply all the mathematical concepts used and to justify them physically. These notes should be accessible to young physicists (graduate level) with a good knowledge of the standard formalism of quantum mechanics, and some interest for theoretical physics (and mathematics). These notes do not cover the historical and philosophical aspects of quantum physics. arXiv:1211.5627v1 [math-ph] 24 Nov 2012 Preprint IPhT t12/042 ii CONTENTS Contents 1 Introduction 1-1 1.1 Motivation . 1-1 1.2 Organization . 1-2 1.3 What this course is not! . 1-3 1.4 Acknowledgements . 1-3 2 Reminders 2-1 2.1 Classical mechanics . -
Introduction to the Modern Calculus of Variations
MA4G6 Lecture Notes Introduction to the Modern Calculus of Variations Filip Rindler Spring Term 2015 Filip Rindler Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom [email protected] http://www.warwick.ac.uk/filiprindler Copyright ©2015 Filip Rindler. Version 1.1. Preface These lecture notes, written for the MA4G6 Calculus of Variations course at the University of Warwick, intend to give a modern introduction to the Calculus of Variations. I have tried to cover different aspects of the field and to explain how they fit into the “big picture”. This is not an encyclopedic work; many important results are omitted and sometimes I only present a special case of a more general theorem. I have, however, tried to strike a balance between a pure introduction and a text that can be used for later revision of forgotten material. The presentation is based around a few principles: • The presentation is quite “modern” in that I use several techniques which are perhaps not usually found in an introductory text or that have only recently been developed. • For most results, I try to use “reasonable” assumptions, not necessarily minimal ones. • When presented with a choice of how to prove a result, I have usually preferred the (in my opinion) most conceptually clear approach over more “elementary” ones. For example, I use Young measures in many instances, even though this comes at the expense of a higher initial burden of abstract theory. • Wherever possible, I first present an abstract result for general functionals defined on Banach spaces to illustrate the general structure of a certain result. -
Finite Elements for the Treatment of the Inextensibility Constraint
Comput Mech DOI 10.1007/s00466-017-1437-9 ORIGINAL PAPER Fiber-reinforced materials: finite elements for the treatment of the inextensibility constraint Ferdinando Auricchio1 · Giulia Scalet1 · Peter Wriggers2 Received: 5 April 2017 / Accepted: 29 May 2017 © Springer-Verlag GmbH Germany 2017 Abstract The present paper proposes a numerical frame- 1 Introduction work for the analysis of problems involving fiber-reinforced anisotropic materials. Specifically, isotropic linear elastic The use of fiber-reinforcements for the development of novel solids, reinforced by a single family of inextensible fibers, and efficient materials can be traced back to the 1960s are considered. The kinematic constraint equation of inex- and, to date, it is an active research topic. Such materi- tensibility in the fiber direction leads to the presence of als are composed of a matrix, reinforced by one or more an undetermined fiber stress in the constitutive equations. families of fibers which are systematically arranged in the To avoid locking-phenomena in the numerical solution due matrix itself. The interest towards fiber-reinforced mate- to the presence of the constraint, mixed finite elements rials stems from the mechanical properties of the fibers, based on the Lagrange multiplier, perturbed Lagrangian, and which offer a significant increase in structural efficiency. penalty method are proposed. Several boundary-value prob- Particularly, these materials possess high resistance and lems under plane strain conditions are solved and numerical stiffness, despite the low specific weight, together with an results are compared to analytical solutions, whenever the anisotropic mechanical behavior determined by the fiber derivation is possible. The performed simulations allow to distribution. -
Functional Integration on Paracompact Manifods Pierre Grange, E
Functional Integration on Paracompact Manifods Pierre Grange, E. Werner To cite this version: Pierre Grange, E. Werner. Functional Integration on Paracompact Manifods: Functional Integra- tion on Manifold. Theoretical and Mathematical Physics, Consultants bureau, 2018, pp.1-29. hal- 01942764 HAL Id: hal-01942764 https://hal.archives-ouvertes.fr/hal-01942764 Submitted on 3 Dec 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Functional Integration on Paracompact Manifolds Pierre Grangé Laboratoire Univers et Particules, Université Montpellier II, CNRS/IN2P3, Place E. Bataillon F-34095 Montpellier Cedex 05, France E-mail: [email protected] Ernst Werner Institut fu¨r Theoretische Physik, Universita¨t Regensburg, Universita¨tstrasse 31, D-93053 Regensburg, Germany E-mail: [email protected] ........................................................................ Abstract. In 1948 Feynman introduced functional integration. Long ago the problematic aspect of measures in the space of fields was overcome with the introduction of volume elements in Probability Space, leading to stochastic formulations. More recently Cartier and DeWitt-Morette (CDWM) focused on the definition of a proper integration measure and established a rigorous mathematical formulation of functional integration. CDWM’s central observation relates to the distributional nature of fields, for it leads to the identification of distribution functionals with Schwartz space test functions as density measures.