Partial Differential Equations in Action From Modelling to Theory

Partial Differential Equations in Action From Modelling to Theory

(Parte 1 de 5)

UNITEXT – La Matematica per il 3+2 Volume 9

Editor-in-chief A. Quarteroni

Series editors

L. Ambrosio P. Biscari C. Ciliberto M. Ledoux W.J. Runggaldier

More information about this series at More information about this series at

Sandro Salsa

Partial Differential Equations in Action

From Modelling to Theory

Third Edition

Sandro Salsa Dipartimento di Matematica Politecnico di Milano Milano, Italy

Library of Congress Control Number: 2016932390

© Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproductionon microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.

Cover illustration: Simona Colombo, Giochi di Grafica, Milano, Italy Typesetting with LATEX: PTP-Berlin, Protago TEX-Production GmbH, Germany (

This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland

To Anna, my wife To Anna, my wife


This book is designed as an advanced undergraduate or a first-year graduate course for students from various disciplines like applied mathematics, physics, engineering. It has evolved while teaching courses on partial differential equations (PDEs) during the last few years at the Politecnico di Milano.

The main purpose of these courses was twofold: on the one hand, to train the students to appreciate the interplay between theory and modelling in problems arising in the applied sciences, and on the other hand to give them a solid theoretical background for numerical methods, such as finite elements.

Accordingly, this textbook is divided into two parts, that we briefly describe below, writing in italics the relevant differences with the first edition, the second one being pretty similar.

The first part, Chaps. 2 to 5, has a rather elementary character with the goal of developing and studying basic problems from the macro-areas of diffusion, propagation and transport, waves and vibrations. I have tried to emphasize, whenever possible, ideas and connections with concrete aspects, in order to provide intuition and feeling for the subject.

For this part, a knowledge of advanced calculus and ordinary differential equations is required. Also, the repeated use of the method of separation of variables assumes some basic results from the theory of Fourier series, which are summarized in Appendix A.

Chapter 2 starts with the heat equation and some of its variants in which transport and reaction terms are incorporated. In addition to the classical topics, I emphasized the connections with simple stochastic processes, such as random walks and Brownian motion. This requires the knowledge of some elementary probability. It is my belief that it is worthwhile presenting this topic as early as possible, even at the price of giving up to a little bit of rigor in the presentation. An application to financial mathematics shows the interaction between probabilistic and deterministic modelling. The last two sections are devoted to two simple non linear models from flow in porous medium and population dynamics.

viii Preface

Chapter 3 mainly treats the Laplace/Poisson equation. The main properties of harmonic functions are presented once more emphasizing the probabilistic motivations. I have included Perron’s method of sub/super solution, due to is renewed importance as a solution technique for fully non linear equations. The second part of this chapter deals with representation formulas in terms of potentials. In particular, the basic properties of the single and double layer potentials are presented.

Chapter 4 is devoted to first order equations and in particular to first order scalar conservation laws. The methods of characteristics and the notions of shock and rarefaction waves are introduced through a simple model from traffic dynamics. An application to sedimentation theory illustrates the method for non convex/concave flux function. In the last part, the method of characteristics is extended to quasilinear and fully nonlinear equations in two variables.

In Chap. 5 the fundamental aspects of waves propagation are examined, leading to the classical formulas of d’Alembert, Kirchhoff and Poisson. As implem odel of Acoustic Thermography serves as an application of Huygens principle. In the final section, the classical model for surface waves in deep water illustrates the phenomenon of dispersion, with the help of the method of stationary phase.

The second part includes the two new Chaps. 9 and 1. In Chaps. 6t o1 0w e develope Hilbert spaces methods for the variational formulation and the analysis of mainly linear boundary and initial-boundary value problems. Given the abstract nature of these chapters, I have made an effort to provide intuition and motivation about the various concepts and results, sometimes running the risk of appearing a bit wordy. The understanding of these topics requires some basic knowledge of Lebesgue measure and integration, summarized in Appendix B.

Chapter 6 contains the tools from functional analysis in Hilbert spaces, necessary for a correct variational formulation of the most common boundary value problems. The main theme is the solvability of abstract variational problems, leading to the Lax-Milgram theorem and Fredholm’s alternative. Emphasis is given to the issues of compactness and weak convergence. Section 6.10 is devoted to the fixed point theorems of Banach and of Schauder and Leray-Schauder.

Chapter 7 is divided into two parts. The first one is a brief introduction to the theory of distributions (or generalized functions) of L. Schwartz. In the second one, the most used Sobolev spaces and their basic properties are discussed.

Chapter 8 is devoted to the variational formulation of linear elliptic boundary value problems and their solvability. The development starts with Poisson’s equation and ends with general second order equations in divergence form.

In Chap. 9 I have gathered a number of applications of the variational theory of elliptic equations, in particular to elastostatics and to the stationary Navier-Stokes equations. Also, an application to a simple control problem is discussed.

The issue in Chap. 10, which has been almost completely remodeled,i s the variational formulation of evolution problems, in particular of initial-boundary value problems for second order parabolic operators in divergence form and for the wave equation.

Preface ix

Chapter 1 contains a brief introduction to the basic concepts of the theory of systems of first order conservation laws, in one spatial dimension. In particular we extend from the scalar case of Chap 4, the notions of characteristics, shocks, rarefaction waves, contact discontinuity and entropy condition. The main focus is the solution of the Riemann problem.

At the end of each chapter, a number of exercises is included. Some of them can be solved by a routine application of the theory or of the methods developed in the text. Other problems are intended as a completion of some arguments or proofs in the text. Also, there are problems in which the student is required to be more autonomous. The most demanding problems are supplied with answers or hints.

Other (completely solved) exercises can be found in [17], the natural companion of this book by S. Salsa, G. Verzini, Springer 2015.

The order of presentation of the material is clearly a consequence of my

prejudices. However, the exposition if flexible enough to allow substantial changes without compromising the comprehension and to facilitate a selection of topics for a one or two semester course.

In the first part, the chapters are, in practice, mutually independent, with the exception of Subsection 3.3.1 and Sect. 3.4, which presume the knowledge of Sect. 2.6.

In the second part, more attention has to be paid to the order of the arguments.

The material in Sects. 6.1–6.9 and in Sect. 7.1–7.4 and 7.7–7.10 is necessary for understanding the topics in Chap. 8–10. Moreover, Chap. 9 requires also Sect. 6.10, while to cover Chap. 10, also concepts and results from Sect. 7.1 are needed. Finally, Chap. 1 uses Subsections 4.4.2, 4.4.3 and 4.6.1.

Acknowledgments. While writing this book, during the first edition, I benefitted from comments and suggestions of many collegues and students.

Among my collegues, I express my gratitude to Luca Dedé, Fausto Ferrari, Carlo

Pagani, Kevin Payne, Alfio Quarteroni, Fausto Saleri, Carlo Sgarra, Alessandro Veneziani, Gianmaria A. Verzini and, in particular to Cristina Cerutti, Leonede De Michele and Peter Laurence.

Among the students who have sat through my course on PDEs, I would like to thank Luca Bertagna, Michele Coti-Zelati, Alessandro Conca, Alessio Fumagalli, Loredana Gaudio, Matteo Lesinigo, Andrea Manzoni and Lorenzo Tamellini.

Fo the last two editions, I am particularly indebted to Leonede de Michele,

Ugo Gianazza and Gianmaria Verzini for their interest, criticism and contribution. Many thanks go to Michele Di Cristo, Giovanni Molica-Bisci, Nicola Parolini Attilio Rao and Francesco Tulone for their comments and the time we spent in precious (for me) discussions. Finally, I like to express my appreciation to Francesca Bonadei and Francesca Ferrari of Springer Italia, for their constant collaboration and support.

Milan, April 2016 Sandro Salsa

1 Introduction1
1.1 Mathematical Modelling1
1.2 Partial Differential Equations2
1.3 Well Posed Problems5
1.4 Basic Notations and Facts7
1.5 Smooth and Lipschitz Domains12
1.6 Integration by Parts Formulas15
2 Diffusion17
2.1 The Diffusion Equation17
2.1.1 Introduction17
2.1.2 The conduction of heat18
2.1.3 Well posed problems (n =1 )20
2.1.4 A solution by separation of variables23
2.1.5 Problems in dimension n> 132
2.2 Uniqueness and Maximum Principles34
2.2.1 Integral method34
2.2.2 Maximum principles36
2.3 The Fundamental Solution39
2.3.1 Invariant transformations39
2.3.2 The fundamental solution (n =1 )41
2.3.3 The Dirac distribution43
2.3.4 The fundamental solution (n> 1)47
2.4 Symmetric Random Walk (n =1 )48
2.4.1 Preliminary computations49
2.4.2 The limit transition probability52
2.4.3 From random walk to Brownian motion54
2.5 Diffusion, Drift and Reaction58
2.5.1 Random walk with drift58
2.5.2 Pollution in a channel60

Contents 2.5.3 Random walk with drift and reaction . .... .... .... .... ... 63

2.5.4 Critical dimension in a simple population dynamics64
2.6 Multidimensional Random Walk6
2.6.1 The symmetric case6
2.6.2 Walks with drift and reaction70
2.7 An Example of Reaction–Diffusion in Dimension n =371
2.8 The Global Cauchy Problem (n =1 )76
2.8.1 The homogeneous case76
2.8.2 Existence of a solution78
2.8.3 The nonhomogeneous case. Duhamel’s method79
2.8.4 Global maximum principles and uniqueness82
2.8.5 The proof of the existence theorem 2.1285
2.9 An Application to Finance8
2.9.1 European options8
2.9.2 An evolution model for the price S89
2.9.3 The Black-Scholes equation91
2.9.4 The solutions95
2.9.5 Hedging and self-financing strategy100
2.10 Some Nonlinear Aspects102
2.10.1 Nonlinear diffusion. The porous medium equation102
2.10.2 Nonlinear reaction. Fischer’s equation105
3 The Laplace Equation115
3.1 Introduction115
3.2 Well Posed Problems. Uniqueness116
3.3 Harmonic Functions118
3.3.1 Discrete harmonic functions118
3.3.2 Mean value properties122
3.3.3 Maximum principles124
3.3.4 The Hopf principle126
3.3.5 The Dirichlet problem in a disc. Poisson’s formula127
3.3.6 Harnack’s inequality and Liouville’s theorem131
3.3.7 Analyticity of harmonic functions133
3.4 A probabilistic solution of the Dirichlet problem135
3.5 Sub/Superharmonic Functions. The Perron Method140
3.5.1 Sub/superharmonic functions140
3.5.2 The method142
3.5.3 Boundary behavior143
3.6 Fundamental Solution and Newtonian Potential147
3.6.1 The fundamental solution147
3.6.2 The Newtonian potential148
3.7 The Green Function155

xii Contents 3.6.3 A divergence-curl system. Helmholtz decomposition 3.7.1 An integral identity . .... .... .... .... .... .... .... .... ... 155

3.7.2 Green’s function157
3.7.3 Green’s representation formula160
3.7.4 The Neumann function161
3.8 Uniqueness in Unbounded Domains163
3.8.1 Exterior problems163
3.9 Surface Potentials166
3.9.1 The double and single layer potentials166
3.9.2 The integral equations of potential theory171
4 Scalar Conservation Laws and First Order Equations179
4.1 Introduction179
4.2 Linear Transport Equation180
4.2.1 Pollution in a channel180
4.2.2 Distributed source182
4.2.3 Extinction and localized source183
4.2.4 Inflow and outflow characteristics. A stability estimate185
4.3 Traffic Dynamics187
4.3.1 A macroscopic model187
4.3.2 The method of characteristics189
4.3.3 The green light problem191
4.3.4 Traffic jam ahead196
4.4 Weak (or Integral) Solutions199
4.4.1 The method of characteristics revisited199
4.4.2 Definition of weak solution202
4.5 An Entropy Condition209
4.6 The Riemann problem212
4.6.1 Convex/concave flux function212
4.6.2 Vanishing viscosity method214
4.6.3 The viscous Burgers equation218
4.6.4 Flux function with inflection points220
4.7 An Application to a Sedimentation Problem224
4.8 The Method of Characteristics for Quasilinear Equations230
4.8.1 Characteristics230
4.8.2 The Cauchy problem232
4.8.3 Lagrange method of first integrals239
4.8.4 Underground flow241
4.9 General First Order Equations244
4.9.1 Characteristic strips244
4.9.2 The Cauchy Problem246

Contents xiii 4.4.3 Piecewise smooth functions and the Rankine-Hugoniot Problems .... .... .... .... .... .... .... .... .... .... .... .... .... ... 251

5W aves and Vibrations259
5.1 General Concepts259
5.1.1 Types of waves259
5.1.2 Group velocity and dispersion relation261
5.2 Transversal Waves in a String264
5.2.1 The model264
5.2.2 Energy266
5.3 The One-dimensional Wave Equation267
5.3.1 Initial and boundary conditions267
5.3.2 Separation of variables269
5.4 The d’Alembert Formula275
5.4.1 The homogeneous equation275
5.4.2 Generalized solutions and propagation of singularities279
5.4.3 The fundamental solution282
5.4.4 Nonhomogeneous equation. Duhamel’s method285
5.4.5 Dissipation and dispersion286
5.5 Second Order Linear Equations288
5.5.1 Classification288
5.5.2 Characteristics and canonical form291
5.6 The Multi-dimensional Wave Equation (n> 1)296
5.6.1 Special solutions296
5.6.2 Well posed problems. Uniqueness298
5.7 Two Classical Models302
5.7.1 Small vibrations of an elastic membrane302
5.7.2 Small amplitude sound waves306
5.8 The Global Cauchy Problem310
5.8.2 The Kirchhoff formula313
5.8.3 The Cauchy problem in dimension 2316
5.9 The Cauchy Problem with Distributed Sources318
5.9.1 Retarded potentials (n =3 )318
5.9.2 Radiation from a moving point source320
5.10 An Application to Thermoacoustic Tomography324
5.1 Linear Water Waves328
5.1.1 A model for surface waves328
5.1.2 Dimensionless formulation and linearization332
5.1.3 Deep water waves334
5.1.4 Interpretation of the solution336
5.1.5 Asymptotic behavior338
5.1.6 The method of stationary phase340
6 Elements of Functional Analysis347
6.1 Motivations347
6.2 Norms and Banach Spaces353
6.3 Hilbert Spaces358
6.4 Projections and Bases363
6.4.1 Projections363
6.4.2 Bases367
6.5 Linear Operators and Duality373
6.5.1 Linear operators373
6.5.2 Functionals and dual space377
6.5.3 The adjoint of a bounded operator379
6.6 Abstract Variational Problems382
6.6.1 Bilinear forms and the Lax-Milgram Theorem382
6.6.2 Minimization of quadratic functionals387
6.6.3 Approximation and Galerkin method388
6.7 Compactness and Weak Convergence391
6.7.1 Compactness391
6.7.2 Compactness in C(Ω) and in Lp(Ω)392
6.7.3 Weak convergence and compactness393
6.7.4 Compact operators397
6.8 The Fredholm Alternative399
6.8.1 Hilbert triplets399
6.8.2 Solvability for abstract variational problems402
6.8.3 Fredholm’s alternative405
6.9 Spectral Theory for Symmetric Bilinear Forms407
6.9.1 Spectrum of a matrix407
6.9.2 Separation of variables revisited407
6.9.3 Spectrum of a compact self-adjoint operator408
6.9.4 Application to abstract variational problems411
6.10 Fixed Points Theorems416
6.10.1 The Contraction Mapping Theorem417
6.10.2 The Schauder Theorem418
6.10.3 The Leray-Schauder Theorem420
7 Distributions and Sobolev Spaces427
7.1 Distributions. Preliminary Ideas427
7.2 Test Functions and Mollifiers429
7.3 Distributions433
7.4 Calculus438
7.4.1 The derivative in the sense of distributions438
7.4.2 Gradient, divergence, Laplacian440
7.5 Operations with Ditributions443
7.5.1 Multiplication. Leibniz rule443
7.5.3 Division448
7.5.4 Convolution449
7.5.5 Tensor or direct product451
7.6 Tempered Distributions and Fourier Transform454
7.6.1 Tempered distributions454
7.6.2 Fourier transform in S′457
7.6.3 Fourier transform in L2460
7.7 Sobolev Spaces461
7.7.1 An abstract construction461
7.7.2 The space H1 (Ω)462
7.7.3 The space H10 (Ω)466
7.7.4 The dual of H10(Ω)467
7.7.5 The spaces Hm (Ω), m> 1470
7.7.6 Calculus rules471
7.7.7 Fourier transform and Sobolev spaces473
7.8 Approximations by Smooth Functions and Extensions474
7.8.1 Local approximations474
7.8.2 Extensions and global approximations475
7.9 Traces479
7.9.1 Traces of functions in H1(Ω)479
7.9.2 Traces of functions in Hm(Ω)483
7.9.3 Trace spaces484
7.10 Compactness and Embeddings487
7.10.1 Rellich’s theorem487
7.10.2 Poincaré’s inequalities488
7.10.3 Sobolev inequality in Rn490
7.10.4 Bounded domains492
7.1 Spaces Involving Time494
7.1.1 Functions with values into Hilbert spaces494
7.1.2 Sobolev spaces involving time497
8 Variational Formulation of Elliptic Problems505
8.1 Elliptic Equations505
8.2 Notions of Solutions507
8.3 Problems for the Poisson Equation509
8.3.1 Dirichlet problem509
8.3.2 Neumann, Robin and mixed problems512
8.3.3 Eigenvalues and eigenfunctions of the Laplace operator517
8.3.4 An asymptotic stability result519
8.4 General Equations in Divergence Form521
8.4.1 Basic assumptions521
8.4.2 Dirichlet problem522
8.4.3 Neumann problem527
8.5 Weak Maximum Principles531
8.6 Regularity536
9 Further Applications551
9.1 A Monotone Iteration Scheme for Semilinear Equations551
9.2 Equilibrium of a Plate554
9.3 The Linear Elastostatic System556
9.4 The Stokes System561
9.5 The Stationary Navier Stokes Equations566
9.5.1 Weak formulation and existence of a solution566
9.5.2 Uniqueness569
9.6 A Control Problem571
9.6.1 Structure of the problem571
9.6.2 Existence and uniqueness of an optimal pair572
9.6.3 Lagrange multipliers and optimality conditions574
9.6.4 An iterative algorithm575
10 Weak Formulation of Evolution Problems581
10.1 Parabolic Equations581
10.2 The Cauchy-Dirichlet Problem for the Heat Equation583
10.3 Abstract Parabolic Problems586
10.3.1 Formulation586
10.3.2 Energy estimates. Uniqueness and stability589
10.3.3 The Faedo-Galerkin approximations591
10.3.4 Existence592
10.4 Parabolic PDEs593
10.4.1 Problems for the heat equation593
10.4.2 General Equations596
10.4.3 Regularity598
10.5 Weak Maximum Principles600
10.6 The Wave Equation602
10.6.1 Hyperbolic Equations602
10.6.2 The Cauchy-Dirichlet problem603
10.6.3 The method of Faedo-Galerkin605
10.6.4 Solution of the approximate problem606
10.6.5 Energy estimates607
10.6.6 Existence, uniqueness and stability609
1 Systems of Conservation Laws615
1.1 Introduction615
1.2 Linear Hyperbolic Systems620
1.2.2 Classical solutions of the Cauchy problem621
Riemann problem623
1.3 Quasilinear Conservation Laws627
1.3.1 Characteristics and Riemann invariants627
1.4 The Riemann Problem631
1.4.4 Solution of the Riemann problem by a single k-shock640
1.4.5 The linearly degenerate case642
1.4.6 Local solution of the Riemann problem643
1.5 The Riemann Problem for the p-system644
1.5.1 Shock waves644
1.5.2 Rarefaction waves646
1.5.3 The solution in the general case649
Appendix A. Fourier Series657
A.1 Fourier Coefficients657
A.2 Expansion in Fourier Series660
Appendix B. Measures and Integrals663
B.1 Lebesgue Measure and Integral663
B.1.1 A counting problem663
B.1.2 Measures and measurable functions665
B.1.3 The Lebesgue integral667
B.1.4 Some fundamental theorems668
Appendix C. Identities and Formulas673
C.1 Gradient, Divergence, Curl, Laplacian673
C.2 Formulas675

xviii Contents 1.2.3 Homogeneous systems with constant coefficients. The 1.3.2 Weak (or integral) solutions and the Rankine-Hugoniot 1.4.1 Rarefaction curves and waves. Genuinely nonlinear systems .633 1.4.2 Solution of the Riemann problem by a single rarefaction 1.4.3 Lax entropy condition. Shock waves and contact B.1.5 Probability spaces, random variables and their Index . .. .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... 681

Chapter 1 Introduction

1.1 Mathematical Modelling

Mathematical modelling plays a big role in the description of a large part of phenomena in the applied sciences and in several aspects of technical and industrial activity.

By a “mathematical model” we mean a set of equations and/or other mathematical relations capable of capturing the essential features of a complex natural or artificial system, in order to describe, forecast and control its evolution. The applied sciences are not confined to the classical ones; in addition to physics and chemistry, the practice of mathematical modelling heavily affects disciplines like finance, biology, ecology, medicine, sociology.

In the industrial activity (e.g. for aerospace or naval projects, nuclear reactors, combustion problems, production and distribution of electricity, traffic control, etc.) the mathematical modelling, involving first the analysis and the numerical simulation and followed by experimental tests, has become a common procedure, necessary for innovation, and also motivated by economic factors. It is clear that all of this is made possible by the enormous computational power now available.

In general, the construction of a mathematical model is based on two main ingredients:

general laws and constitutive relations.

In this book we shall deal with general laws coming from continuum mechanics and appearing as conservation or balance laws (e.g. of mass, energy, linear momentum, etc.).

The constitutive relations are of an experimental nature and strongly depend on the features of the phenomena under examination. Examples are the Fourier law of heat conduction, the Fick’s law for the diffusion of a substance or the way the speed of a driver depends on the density of cars ahead.

The outcome of the combination of the two ingredients is usually a partial differential equation or a system of them.

© Springer International Publishing Switzerland 2016 S. Salsa, Partial Differential Equations in Action. From Modelling to Theory,3 rd Ed., UNITEXT – La Matematica per il 3+2 9, DOI 10.1007/978-3-319-31238-5_1

2 1 Introduction 1.2 Partial Differential Equations

F (x1,,x n,u ,u x ,... ,u x ,u x x ,u x x ... ,u x x ,u x x x ,... )=0 (1.1)
where the unknown u = u(x1,xn) is a function of n variables and ux ,... ,
ux x ,are its partial derivatives. The highest order of differentiation occurring

A partial differential equation is a relation of the following type: in the equation is the order of the equation.

A first important distinction is between linear and nonlinear equations. Equation (1.1) is linear if F is linear with respect to u and all its derivatives, otherwise it is nonlinear. A second distinction concerns the types of nonlinearity. We distinguish:

• Semilinear equations when F is nonlinear only with respect to u but is linear with respect to all its derivatives, with coefficients depending only on x.

• Quasi-linear equations when F is linear with respect to the highest order derivatives of u, with coefficients depending only on x, u and lower order derivatives.

• Fully nonlinear equations when F is nonlinear with respect to the highest order derivatives of u.

The theory of linear equations can be considered sufficiently well developed and consolidated, at least for what concerns the most important questions. On the contrary, the nonlinearities present such a rich variety of aspects and complications that a general theory does not appear to be conceivable. The existing results and the new investigations focus on more or less specific cases, especially interesting in the applied sciences.

To give the reader an idea of the wide range of applications we present a series of examples, suggesting one of the possible interpretations. Most of them are considered at various level of deepness in this book. In the examples, x represents a space variable (usually in dimension n =1 ,2,3)a nd t is a time variable.

We start with linear equations. In particular, equations (1.2)–(1.5) are fundamental and their theory constitutes a starting point for many other equations.

1. Transport equation (first order):

It describes for instance the transport of a solid polluting substance along a channel; here u is the concentration of the substance and v is the stream speed. We consider the one-dimensional version of (1.2) in Sect. 4.2.

(Parte 1 de 5)