Partial differential equations in action from modelling to theory third edition

UNITEXT – La Matematica per il 3+2 Volume 99 Editor-in-chief A Quarteroni Series editors L Ambrosio P Biscari C Ciliberto M Ledoux W.J Runggaldier More information about this series at http://www.springer.com/series/5418 Sandro Salsa Partial Differential Equations in Action From Modelling to Theory Third Edition Sandro Salsa Dipartimento di Matematica Politecnico di Milano Milano, Italy ISSN 2038-5722 UNITEXT – La Matematica per il 3+2 ISBN 978-3-319-31237-8 DOI 10.1007/978-3-319-31238-5 ISSN 2038-5757 (electronic) ISBN 978-3-319-31238-5 (eBook) 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, reproduction on 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 (www.ptp-berlin.de) This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland To Anna, my wife Preface 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 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 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 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 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 the fundamental aspects of waves propagation are examined, leading to the classical formulas of d’Alembert, Kirchhoff and Poisson A simple model 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 and 11 In Chaps to 10 we 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 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 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 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 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, is 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 11 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 requires also Sect 6.10, while to cover Chap 10, also concepts and results from Sect 7.11 are needed Finally, Chap 11 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 Contents Introduction 1.1 Mathematical Modelling 1.2 Partial Differential Equations 1.3 Well Posed Problems 1.4 Basic Notations and Facts 1.5 Smooth and Lipschitz Domains 12 1.6 Integration by Parts Formulas 15 Diffusion 2.1 The Diffusion Equation 2.1.1 Introduction 2.1.2 The conduction of heat 2.1.3 Well posed problems (n = 1) 2.1.4 A solution by separation of variables 2.1.5 Problems in dimension n > 2.2 Uniqueness and Maximum Principles 2.2.1 Integral method 2.2.2 Maximum principles 2.3 The Fundamental Solution 2.3.1 Invariant transformations 2.3.2 The fundamental solution (n = 1) 2.3.3 The Dirac distribution 2.3.4 The fundamental solution (n > 1) 2.4 Symmetric Random Walk (n = 1) 2.4.1 Preliminary computations 2.4.2 The limit transition probability 2.4.3 From random walk to Brownian motion 2.5 Diffusion, Drift and Reaction 2.5.1 Random walk with drift 2.5.2 Pollution in a channel 2.5.3 Random walk with drift and reaction 17 17 17 18 20 23 32 34 34 36 39 39 41 43 47 48 49 52 54 58 58 60 63 B.1 Lebesgue Measure and Integral 669 Example B.10 A typical situation we often encounter in this book is the following Let f ∈ L1 (A) and, for ε > 0, set Aε = {|f| > ε} Then, we have f → Aε f as ε → A This follows from Theorem B.7 since, for every sequence εj → 0, we have |f| χAεj ≤ |f| and therefore f = Aεj A fχAεj → as ε → f A Let C0 (A) be the set of continuous functions in A, compactly supported in A An important fact is that any summable function may be approximated by a function in C0 (A) Theorem B.11 Let f ∈ L1 (A) Then, for every δ > 0, there exists a continuous function g ∈ C0 (A) such that f −g L1 (A) < δ The fundamental theorem of calculus extends to the Lebesgue integral in the following form: Theorem B.12 (Differentiation) Let f ∈ L1loc (R) Then d dx x f (t) dt = f (x) a.e x ∈ R a Finally, the integral of a summable function can be computed via iterated integrals in any order Precisely, let I1 = {x ∈Rn : −∞ ≤ < xi < bi ≤ ∞; i = 1, , n} and I2 = {y ∈Rm : −∞ ≤ aj < yj < bj ≤ ∞; j = 1, , m} Theorem B.13 (Fubini) Let f be summable in I = I1 × I2 ⊂ Rn × Rm Then f (x, ·) ∈ L1 (I2 ) for a.e x ∈ I1 , and f (·, y) ∈ L1 (I1 ) for a.e y ∈I2 I2 f (·, y) dy ∈ L1 (I1 ) and I1 f (x, ·) dx ∈ L1 (I2 ) The following formulas hold: f (x, y) dxdy = I dx I1 f (x, y) dy = I2 dx I2 f(x, y)dy I1 670 Appendix B Measures and Integrals Let f ∈ L1 (BR (p)) The following formula can be considered a version of Fubini’s Theorem in spherical coordinates: R f(x)d = ds BR (p) f(σ)dσ ∂Bs (p) B.1.5 Probability spaces, random variables and their integrals Let F be a σ-algebra in a set Ω A probability measure P on F is a measure in the sense of definition B.2, such that P (Ω) = and P : F → [0, 1] The triplet (Ω, F , P ) is called a probability space In this setting, the elements ω of Ω are sample points, while a set A ∈ F has to be interpreted as an event P (A) is the probability of (occurrence of) A A typical example is given by the triplet Ω = [0, 1] , F = M ∩ [0, 1] , P (A) = |A| which models a uniform random choice of a point in [0, 1] A 1-dimensional random variable in (Ω, F , P ) is a function X:Ω→R such that X is F −measurable, that is X −1 (C) ∈ F for each closed set C ⊆ R Example B.14 The number k of steps to the right after N steps in the random walk of Sect 2.4 is a random variable Here Ω is the set of walks of N steps By the same procedure used to define the Lebesgue integral we can define the integral of a random variable with respect to a probability measure We sketch the main steps N If X is simple, i.e X = j=1 sj χAj , we define N X dP = Ω sj P (Aj ) j=1 If X ≥ we set Y dP : Y ≤ X, Y simple X dP = sup Ω Ω B.1 Lebesgue Measure and Integral 671 Finally, if X = X + − X − we define X + dP − X dP = Ω Ω X − dP Ω provided at least one of the integral on the right hand side is finite In particular, if |X| dP < ∞, Ω then E (X) = X = X dP Ω is called the expected value (or mean value or expectation) of X, while (X − E (X)) dP Var (X) = Ω is called the variance of X Analogous definitions can be given componentwise for n-dimensional random variables X : Ω → Rn Appendix C Identities and Formulas C.1 Gradient, Divergence, Curl, Laplacian Let F be a smooth vector field and f a smooth real function, in R3 Orthogonal cartesian coordinates gradient: ∇f = ∂f ∂f ∂f i+ j+ k ∂x ∂y ∂z divergence (F =F1 i + F1 j + F3 k): div F = ∂ ∂ ∂ F1 + F2 + F3 ∂x ∂y ∂z Laplacian: Δf = ∂2f ∂2f ∂2f + + ∂x2 ∂y2 ∂z curl : i j k curl F = ∂x ∂y ∂z F1 F2 F3 Cylindrical coordinates x = r cos θ, y = r sin θ, z = z (r > 0, ≤ θ ≤ 2π) er = cos θi + sin θj, eθ = − sin θi + cos θj, ∇f = ∂f ∂f ∂f er + eθ + ez ∂r r ∂θ ∂z ez = k gradient: © Springer International Publishing Switzerland 2016 S Salsa, Partial Differential Equations in Action From Modelling to Theory, 3rd Ed., UNITEXT – La Matematica per il 3+2 99, DOI 10.1007/978-3-319-31238-5_C 674 Appendix C Identities and Formulas divergence (F = Fr er + Fθ eθ + Fz k): div F = ∂ ∂ ∂ (rFr ) + Fθ + Fz r ∂r r ∂θ ∂z Laplacian: Δf = ∂ r ∂r r ∂f ∂r + ∂2f ∂2f ∂2f ∂f ∂2f ∂2f + = + + 2 + 2 2 r ∂θ ∂z ∂r r ∂r r ∂θ ∂z curl : curl F = e re e r θ z ∂r ∂θ ∂z r F rF F r θ z Spherical coordinates x = r cos θ sin ψ, y = r sin θ sin ψ, z = r cos ψ (r > 0, ≤ θ ≤ 2π, ≤ ψ ≤ π) er = cos θ sin ψi + sin θ sin ψj + cos ψk eθ = − sin θi + cos θj eψ = cos θ cos ψi + sin θ cos ψj − sin ψk gradient: ∇f = ∂f ∂f ∂f er + eθ + eψ ∂r r sin ψ ∂θ r ∂ψ divergence (F = Fr er + Fθ eθ + Fψ eψ ): div F = ∂ ∂ ∂ Fr + Fr + Fθ + Fψ + cot ψFψ ∂r r r sin ψ ∂θ ∂ψ radial part spherical part Laplacian: Δf = ∂f ∂2f + + ∂r r ∂r r radial part ∂2f ∂ 2f ∂f + + cot ψ (sin ψ)2 ∂θ2 ∂ψ2 ∂ψ spherical part (Laplace-Beltrami operator) curl : rot F = er reψ r sin ψeθ ∂ ∂θ r ∂ψ r sin ψ Fr rFψ r sin ψFz C.2 Formulas 675 C.2 Formulas Gauss’ formulas In Rn , n ≥ 2, let: • • • • Ω be a bounded smooth domain and and ν the outward unit normal on ∂Ω u, v be vector fields of class C Ω ϕ, ψ be real functions of class C Ω dσ be the area element on ∂Ω div u dx = ∂Ω u · ν dσ ∇ϕ dx = ∂Ω ϕν dσ Ω Δϕ dx = ∂Ω ∇ϕ · ν dσ = Ω (Divergence Theorem) Ω ∂Ω ∂ν ϕ dσ ψ div F dx = ∂Ω ψF · ν dσ − Ω ∇ψ · F dx ψΔϕ dx = ∂Ω ψ∂ν ϕ dσ − Ω ∇ϕ · ∇ψ dx Ω (ψΔϕ − ϕΔψ) dx = ∂Ω (ψ∂ν ϕ − ϕ∂ν ψ) dσ Ω curl u dx = − ∂Ω u × ν dσ Ω Ω Ω u· curl v dx = Ω v· curl u dx − ∂Ω (Integration by parts) (Green’s identity I) (Green’s identity II) (u × v) · ν dσ Identities div curl u = curl ∇ϕ = div (ϕu) = ϕ div u + ∇ϕ · u curl (ϕu) = ϕ curl u + ∇ϕ × u curl (u × v) = (v · ∇) u − (u · ∇) v + (div v) u − (div u) v div (u × v) = curlu · v−curlv · u ∇ (u · v) = u× curl v + v× curl u + (u · ∇) v + (v · ∇) u (u · ∇) u = curlu × u + 12 ∇ |u| curl curl u = ∇(div u) − Δu References Partial Differential Equations [1] DiBenedetto, E.: Partial Differential Equations, Birkhäuser, Boston, 1995 [2] Friedman, A.: Partial Differential Equations of parabolic Type, Prentice-Hall, Englewood Cliffs, 1964 [3] Gilbarg, D and Trudinger, N.: Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin Heidelberg, 1998 [4] Grisvard, P.: Elliptic Problems in nonsmooth domains, Pitman, Boston, 1985 [5] Guenter, R.B and Lee, J.W.: Partial Differential Equations of Mathematical Physics and Integral Equations, Dover Publications, Inc., New York, 1998 [6] Helms, O.: Introduction to Potential Theory, Krieger Publishing Company, New York, 1975 [7] John, F.: Partial Differential Equations, 4th ed., Springer-Verlag, New York, 1982 [8] Kellog, O.: Foundations of Potential Theory, Dover, New York, 1954 [9] Galdi, G.: Introduction to the Mathematical Theory of Navier-Stokes Equations, vols I and II, Springer-Verlag, New York, 1994 [10] Lieberman, G.M.: Second Order Parabolic Partial Differential Equations, World Scientific, Singapore, 1996 [11] Lions, J.L., Magenes, E.: Nonhomogeneous Boundary Value Problems and Applications, ols 1, 2, Springer-Verlag, New York, 1972 [12] McOwen, R.: Partial Differential Equations: Methods and Applications, Prentice-Hall, New Jersey, 1996 [13] Olver, P.J.: Introduction to Partial Differential Equations, Springer International Publishing Switzerland, 2014 [14] Protter, M and Weinberger, H.: Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, 1984 [15] Renardy, M and Rogers, R.C.: An Introduction to Partial Differential Equations, Springer-Verlag, New York, 1993 [16] Rauch, J.: Partial Differential Equations, Springer-Verlag, Heidelberg, 1992 © Springer International Publishing Switzerland 2016 S Salsa, Partial Differential Equations in Action From Modelling to Theory, 3rd Ed., UNITEXT – La Matematica per il 3+2 99, DOI 10.1007/978-3-319-31238-5 678 References [17] Salsa, S and Verzini, G.: Partial Differential Equation in Action Complements and Exercises, Springer International Publishing Switzerland, 2015 [18] Smoller, J.: Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983 [19] Strauss, W.: Partial Differential Equation: An Introduction, Wiley, 1992 [20] Widder, D.V.: The Heat Equation, Academic Press, New York, 1975 Mathematical Modelling [21] Acheson, A.J.: Elementary Fluid Dynamics, Clarendon Press, Oxford, 1990 [22] Billingham, J and King, A.C.: Wave Motion, Cambridge University Press, 2000 [23] Courant, R and Hilbert, D.: Methods of Mathematical Phisics, vols and 2, Wiley, New York, 1953 [24] Dautray, R and Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology, vols 1–5, Springer-Verlag, Berlin Heidelberg, 1985 [25] Lin, C.C and Segel, L.A.: Mathematics Applied to Deterministic Problems in the Natural Sciences, SIAM Classics in Applied Mathematics, 4th ed., 1995 [26] Murray, J.D.: Mathematical Biology, vols and 2, Springer-Verlag, Berlin Heidelberg, 2001 [27] Rhee, H., Aris, R., and Amundson, N.: First Order Partial Differential Equations, vola and 2, Dover, New York, 1986 [28] Scherzer, O., Grasmair, M., Grosshauer, H.; Haltmeier, M., and Lenzen, F.: Variational Methods in Imaging, Applied Mathematical Sciences 167, Springer, New York, 2008 [29] Segel, L.A.: Mathematics Applied to Continuum Mechanics, Dover Publications, Inc., New York, 1987 [30] Whitham, G.B.: Linear and Nonlinear Waves, Wiley-Interscience, 1974 ODEs, Analysis and Functional Analysis [31] Adams, R.: Sobolev Spaces, Academic Press, New York, 1975 [32] Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2010 [33] Coddington, E.A and Levinson, N.: Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955 [34] Gelfand, I.M and Shilov, E.: Generalized Functions, vol 1: Properties and Operations, Academic Press, 1964 [35] Maz’ya, V.G.: Sobolev Spaces, Springer-Verlag, Berlin Heidelberg, 1985 [36] Rudin, W.: Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, 1976 [37] Schwartz, L.: Théorie des Distributions, Hermann, Paris, 1966 [38] Taylor, A.E.: Introduction to Functional Analysis, John Wiley & Sons, 1958 [39] Yoshida, K.: Functional Analysis, 3rd ed., Springer-Verlag, New York, 1971 References 679 [40] Ziemer, W.: Weakly Differentiable Functions, Springer-Verlag, Berlin Heidelberg, 1989 [41] Zygmund, R and Wheeden, R.: Measure and Integral, Marcel Dekker, 1977 Numerical Analysis [42] Dautray, R and Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology, vols and 6, Springer-Verlag, Berlin Heidelberg, 1985 [43] Quarteroni, A.: Numerical Models for Differential Problems, MS&A, SpringerVerlag Italia, Milan, 2014 [44] Quarteroni, A and Valli, A.: Numerical Approximation of Partial Differential Equations, Springer-Verlag, Berlin Heidelberg, 1994 [45] Godlewski, E and Raviart, P.A.: Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer-Verlag, New York, 1996 Stochastic Processes and Finance [46] Baxter, M and Rennie, A.: Financial Calculus: An Introduction to Derivative Pricing, Cambridge University Press, 1996 [47] Øksendal, B.K.: Stochastic Differential Equations: An Introduction with Applications, 4th ed., Springer-Verlag, Berlin Heidelberg, 1995 [48] Wilmott, P., Howison, S., and Dewinne, J.: The Mathematics of Financial Derivatives A Student Introduction, Cambridge University Press, 1996 Index A Absorbing barriers 111 Adjoint of a bilinear form 402 Adjoint problem 574 Advection 180 Alternative – for the Dirichlet problem 526 – for the Neumann problem 529 Angular frequency 260 Arbitrage 92 B Barenblatt solutions 103 Barrier 143 Bernoulli’s equation 330 Bessel function 73, 297 Bilinear form 382 Bond number 333 Boundary conditions 21 – Dirichlet 21, 33 – mixed 22, 33 – Neumann 22, 33 – Robin 22, 33 Breaking time 201 Brownian motion 55 Brownian path 55 Burgers, viscous 216 C Canonical form 293, 295 Canonical isometry 379 Characteristic 181, 230, 620, 621 – parallelogram 277 – strip 246 – system 245 Chebyshev polynomials 370 Closure Comparison 38 Compatibility conditions 403, 405 Condition – compatibility 118 – E 220 – entropy 638 – Rankine-Hugoniot 631 Conjugate exponent 11, 358 Conormal derivative 527 Contact discontinuity 626, 632, 640 Continuous isomorphism 385 Convection 61 Convergence – least squares 660 – uniform 11 – weak 394 Convolution 430, 449 Cost functional 571 Critical mass 75 Critical survival value 66 Curve – rarefaction 633 – shock 640 D d’Alembert formula 276 d-harmonic function 121 Darcy’s law 102 Diffusion 18 Diffusion coefficient 54 Dirac comb 436 Dirac measure 44 Direct product 452 Direct sum 364 Dirichlet eigenfunctions 517 © Springer International Publishing Switzerland 2016 S Salsa, Partial Differential Equations in Action From Modelling to Theory, 3rd Ed., UNITEXT – La Matematica per il 3+2 99, DOI 10.1007/978-3-319-31238-5 682 Index Dirichlet Principle 546 Dispersion 287 – relation 261, 287, 335 Dissipation – external/internal 286 Distribution 434 – composition 445 – division 448 Distributional derivative 438 Domain – C^1, C^k 12 – Lipschitz 14 – of dependence 278, 315 – smooth 12 Drift 60, 89 Duhamel method 285 E Eigenfunction 370 – of a bilinear form 411 Eigenspace 407, 409, 411 Eigenvalue 370, 409 – of a bilinear form 411 Eigenvector 409 Elastic restoring force 111 Elliptic equation 505 Entropy condition 209, 210 Equal area rule 202 Equation – backward 94 – backward heat 39 – Bessel 73 – Bessel’s 372 – biharmonic 555 – Black-Scholes 3, 93 – Bukley-Leverett 243 – Burgers – Chebyshev 370 – diffusion 2, 17 – Eiconal – eikonal 249 – elastostatics 557 – elliptic 289 – Fisher – fully nonlinear – Hermite’s 371 – hyperbolic 289 – Klein-Gordon 287 – Laplace – Legendre’s 371 – linear elasticity – linear, nonlinear – Maxwell – minimal surface – Navier 557 – Navier Stokes – Navier-Stokes 153, 561 – Navier-Stokes, stationary 566 – parabolic 289, 581 – parametric Bessel’s (of order p) 372 – partial differential – Poisson 3, 115 – porous media 103 – Porous medium – quasilinear – reduced wave 178 – Schrodinger – semilinear – stationary Fisher 553 – stochastic differential 89 – Sturm-Liouville 370 – transport – Tricomi 289 – uniformly parabolic 582 – vibrating plate – wave Equicontinuity 392 Equipartition of energy 342 Escape probability 137 Essential support 429 Essential supremum 358 Euler equation 388 European options 88 Expectation 58, 70 Expiry date 88 Extension operator 477 Exterior Dirichlet problem 163 Exterior domain 164 Exterior Robin problem 165, 177 F Fick’s law 61 Final payoff 94 First exit time 135 First integral 238, 240 First variation 388 Flux function 179 Focussing effect 345 Forward cone 313 Fourier coefficients 367 Fourier law 20 Fourier series 28 Fourier transform 454, 473 Fourier-Bessel series 74, 373 Frequency 260 Froude number 333 Index Function – Bessel’s of first kind and order p 373 – characteristic 10 – compactly supported 10 – complementary error 220 – continuous 10 – d-harmonic 119 – essentially bounded 358 – Green’s 157 – Hölder continuous 357 – harmonic 18, 115 – Heaviside 44 – piecewise continuous 205 – summable 357 – test 48, 429 – weigth 370 Functional 377 Fundamental solution 43, 48, 148, 282, 311 G Gas dynamics 617 Gaussian law 56, 68 Genuinely nonlinear 634 Global Cauchy problem 23, 34, 76 – nonhomogeneous 80 Gram-Schmidt process 369 Greatest lower bound Group velocity 261 H Harmonic lifting 141 Harmonic measure 138 Harnack’s inequality 131 Heisenberg Uncertainty Principle – for the first eigenvalue 421 Helmholtz decomposition formula 151 Hermite polynomials 371 Hilbert triplet 401 Hooke’s law 556 Hopf’s maximum principle 126 Hopf-Cole transformation 218 Hugoniot line 626 I Identity – Green’s (first and second) 15 – strong Parseval’s 460 – weak Parseval’s 458 Inequality – Hölder 358 Infimum Inflow/outflow boundary 239 683 Inflow/outflow characteristics 185 Inner product space 359 Integral surface 230 integration by parts 15 Interior shere condition 126 Invasion problem 113 Inward heat flux 33 Isometry – isometric 360 Ito’s formula 90 K Kernel 374 Kinematic condition 331 Kinetic energy 266 L Lagrange multiplier 565 Lattice 66, 118 Least squares 28 Least upper bound Lebesgue spine 145 Legendre polynomials 371 Light cone 249 Linearly degenerate 642 Liouville Theorem 132 little o 11 Local chart 12 Local wave speed 190 Localization 477 Logarithmic potential 150 Logistic growth 105 Lognormal density 91 M Mach number 308 Markov properties 57, 69 Mass conservation 60 Material derivative 154 Maximum principle 83, 120 – weak 36, 532, 601 Mean value property 123 Method 23 – Duhamel 81 – electrostatic images 157 – Galerkin’s 388 – of characteristics 189 – of descent 316 – of Faedo-Galerkin 591, 605 – of stationary phase 263 – reflection 477 – separation of variables 23, 26, 269, 304, 407, 520 684 Index – time reversal 325 – vanishing viscosity 214 Metric space 353 Minimax property (of the eigenvalues) – for the first eigenvalue 416, 426 Mollifier 430 Monotone iteration scheme 551 Multidimensional symmetric random walk 66 Multiplicity (of an eigenvalue) 409 N Neumann eigenfunctions 519 Neumann function 163 Norm – Integral of order p 357 – least squares 355 – maximum 355 – maximum of order k 356 Normal probability density 43 Normed space 353 Numerical sets O Omeomorphism 418 Open covering 477 Operator – adjoint 380 – bounded,continuous 374 – compact 397 – discrete Laplace 119 – linear 373 – mean value 118 Optimal control 572 Optimal state 572 Orthonormal basis 367 P Parabolic – boundary 23, 34 Parabolic dilations 40 Parallelogram law 360 Partition of unity 478 Perron method 142 Phase speed 260 Poincaré’s inequality 466, 488 Point – boundary – interior – limit Point source solution – two dimensional 345 Poisson formula 131 Potential 115 – double layer 166 – energy 267 – Newtonian 149 – retarded 318, 345 – single layer 170 Principal Dirichlet eigenvalue 518 Principle of virtual work 560 Probability – measure 670 – space 670 Problem – abstract parabolic 586 – abstract variational 382 – Characteristic Cauchy 343 – eigenvalue 27 – Goursat 343 – ill posed (heat equation) 343 – inverse 325 – Riemann 623 – well posed 7, 21 Projected characteristics 238 Projection – on closed convex sets 424 Put-call parity 97 Q Qantum mechanics harmonic oscillator 422 R Random variable 54 Random walk 49 – with drift 58 Range 374 – of influence 278, 313 Rankine-Hugoniot condition 198, 205 Rarefaction/simple waves 194 Rayleigh quotient 414, 518 Reaction 63 Reflecting barriers 111 Regular point 144 Resolvent 407 – of a bilinear form 411 – of a bounded operator 408 Retarded potential 318 retrocone 301 Reynolds number 562 Riemann invariant 628 Riemann problem 212 Rodrigues’ formula 371 Index S Schwarz inequality 360 Schwarz reflection principle 175 Self-financing portfolio 92, 100 Selfadjoint operator 381 Sequence – Cauchy 354 – fundamental 354 Set – bounded – closed – compact – compactly contained – connected – convex – dense – open – precompact 391 – sequentially closed – sequentially compact 8, 391 Shock – curve 198 – speed 198 – wave 198 Similarity, self-similar solutions 41 Sobolev exponent 491 Solution – classical 507 – distributional 507 – integral 205 – self-similar 103 – steady state 25 – strong 507 – unit source 46 – variational 507 – viscosity 507 – weak 205 Sommerfeld condition 178 Space – separable 367 Space-like curve 250 Spectral decomposition – of a matrix 407 – of an operator 411 Spectrum 407 – continous 409 – of a bilinear form 411 – of a bounded operator 408 – point 409 – residual 409 Spherical waves 261 Stability estimate 386 Standing wave 271 Stationary phase (method of) 340 Steepest descent 575 Stiffness matrix 389 Stochastic process 55, 68 Stokes System – equazione biarmonica 562 Stopping time 57, 135 Strike price 88 Strip condition 247 Strong Huygens’ principle 313, 315 Sub/superharmonic function 141 Sub/supersolution 36 Superposition principle 17, 77, 268 Support 10 – of a distribution 437 Surface – of the unit sphere (ωn ) – tension 328, 330 Symbol o(h) 53 Symbol “big O” 63 System – hyperbolic 616 – p-system 619 T Tempered distribution 456 Tensor – deformation 556 – stress 153, 556 Tensor product 452 Term by term – differentiation 12 – integration 12 Theorem – Ascoli-Arzelà 392 – Contraction mapping 417 – Dominated Convergence 668 – Fubini 669 – Lax-MIlgram 383 – Leray-Shauder 420 – Monotone Convergence 668 – projection 364 – Rellich 487 – Riesz’s representation 378 – Riesz-Fréchet-Kolmogoroff 393 – Schauder 419 time-like curve 250 Topology 7, 354 – euclidean – relative Trace 479 – inequality 486 Traffic in a tunnel 253 685 686 Index Transition function 69 Transition layer 216 Transition probability 57, 121 Transmission conditions 547 Travelling wave 182, 190, 215 trivial extension 477 Tychonov class 83 U Uniform ellipticity 521 Unit impulse 45 Upper,lower limit V Value function 88 Variational formulation – Dirichlet problem 510, 523 – Mixed problem 516, 530 – Neumann problem 513, 528 – Robin problem 515 Variational inequality – on closed convex sets 424 Variational principle – for the first eigenvalue 414 – for the k-th eigenvalue 415 Volatility 89 W Wave – capillarity 337 – cylindrical 296 – gravity 336 – harmonic 259 – incoming/outgoing 298 – linear 328 – linear gravity 346 – monochromatic/harmonic 296 – number 260 – packet 262 – plane 261, 296 – rarefaction, p-system 646, 647 – shock 640 – shock, p-system 645, 646 – simple 633 – spherical 297 – standing 260 – travelling 259 Weak coerciveness 525 Weak formulation – Cauchy-Dirichlet problem 584 – Cauchy-Robin/Neumann problem 594– 596 – Initial-Dirichlet problem (wave eq.) 604 Weakly coercive (bilinear form) 402, 596 Weierstrass test 11, 29 Y Young modulus 342 ... http://www.springer.com/series/5418 Sandro Salsa Partial Differential Equations in Action From Modelling to Theory Third Edition Sandro Salsa Dipartimento di Matematica Politecnico di Milano Milano, Italy... 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,... order equations in divergence form In Chap I have gathered a number of applications of the variational theory of elliptic equations, in particular to elastostatics and to the stationary Navier-Stokes