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  • JUHA KINNUNEN

    Partial Differential Equations

    Department of Mathematics and Systems Analysis, Aalto University

    2019

  • Contents

    1 INTRODUCTION 1

    2 FOURIER SERIES AND PDES 5 2.1 Periodic functions* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 The Lp space on [−π,π]* . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 The Fourier series* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 The best square approximation* . . . . . . . . . . . . . . . . . . . . . 18 2.5 The Fourier series on a general interval* . . . . . . . . . . . . . . . . . 24 2.6 The real form of the Fourier series* . . . . . . . . . . . . . . . . . . . . 24 2.7 The Fourier series and differentiation* . . . . . . . . . . . . . . . . . . 27 2.8 The Dirichlet kernel* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.9 Convolutions* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.10 A local result for the convergence of Fourier series* . . . . . . . . . . 32 2.11 The Laplace equation in the unit disc . . . . . . . . . . . . . . . . . . 34 2.12 The heat equation in one-dimension . . . . . . . . . . . . . . . . . . . 46 2.13 The wave equation in one-dimension . . . . . . . . . . . . . . . . . . 53 2.14 Approximations of the identity

    in [−π,π]* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.15 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

    3 FOURIER TRANSFORM AND PDES 64 3.1 The Lp-space on Rn* . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.2 The Fourier transform* . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.3 The Fourier transform and differentiation* . . . . . . . . . . . . . . . . 68 3.4 The Fourier transform of the Gaussian* . . . . . . . . . . . . . . . . . . 72 3.5 The Fourier inversion formula* . . . . . . . . . . . . . . . . . . . . . . . 73 3.6 The Fourier transformation and convolution . . . . . . . . . . . . . . 77 3.7 Plancherel’s formula* . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.8 Approximations of the identity in Rn* . . . . . . . . . . . . . . . . . . . 80 3.9 The Laplace equation in the upper half-space . . . . . . . . . . . . 82 3.10 The heat equation in the upper half-space . . . . . . . . . . . . . . . 89 3.11 The wave equation in the upper half-space . . . . . . . . . . . . . . 93 3.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

  • CONTENTS ii

    4 LAPLACE EQUATION 96 4.1 Gauss-Green theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.2 PDEs and physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.3 Boundary values and physics . . . . . . . . . . . . . . . . . . . . . . . 100 4.4 Fundamental solution of the Laplace equation . . . . . . . . . . . . 105 4.5 The Poisson equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.6 The Green’s function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.7 The Green’s function for the upper half-space* . . . . . . . . . . . . 121 4.8 The Green’s function for a ball* . . . . . . . . . . . . . . . . . . . . . . 123 4.9 Mean value formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.10 Maximum principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.11 Harnack’s inequality* . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.12 Energy methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.13 Weak solutions* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.14 The Laplace equation in other coordinates* . . . . . . . . . . . . . . 139 4.15 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

    5 HEAT EQUATION 147 5.1 Physical interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.2 The fundamental solution . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.3 The nonhomogeneous problem . . . . . . . . . . . . . . . . . . . . . 150 5.4 Separation of variables in Rn . . . . . . . . . . . . . . . . . . . . . . . 153 5.5 Maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.6 Energy methods for the heat equation . . . . . . . . . . . . . . . . . 162 5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

    6 WAVE EQUATION 165 6.1 Physical interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.2 The one-dimensional wave equation . . . . . . . . . . . . . . . . . . 166 6.3 The Euler-Poisson-Darboux equation . . . . . . . . . . . . . . . . . . . 173 6.4 The three-dimensional wave equation . . . . . . . . . . . . . . . . . 176 6.5 The two-dimensional wave equation . . . . . . . . . . . . . . . . . . 181 6.6 The nonhomogeneous problem . . . . . . . . . . . . . . . . . . . . . 185 6.7 Energy methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 6.8 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 6.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

    7 NOTATION AND TOOLS 191

  • Partial differential equations are not only extremely impor- tant in applications of mathematics in physical, geometric and probabilistic phenomena, but they also are of theoretic interest.

    1 Introduction

    These notes are meant to be an elementary introduction to partial differential equations (PDEs) for undergraduate students in mathematics, the natural sciences and engineering. They assume only advanced multidimensional differential calculus including partial derivatives, integrals and the Gauss-Green formulas. The sections denoted by * consist of additional material, which is essential in understanding the rest of the material, but can omitted or glanced through quickly in the first reading.

    A partial differential equation is an equation involving an unknown function of two ore more variables and its partial derivatives. Although PDEs are general- izations of ordinary differential equations (ODEs), for most PDE problems it is not possible to write down explicit formulas for solutions that are common in the ODE theory. This means that there is greater emphasis on qualitative features. There is no general method to solve PDEs, however, some methods have turned out to be more useful than other. We study special cases, in which explicit solutions and representation formulas are available, and focus on features that are present in more general situations. Qualitative aspects are also important in numerical solutions of PDE. Without existence, uniqueness and stability results numerical methods may give inaccurate or completely wrong solutions.

    Let x ∈Ω, where Ω is an open subset of Rn and t ∈R. In these notes we study

    (1) Laplace’s equation

    ∆u = f , u = u(x), ∆u = n∑

    j=1

    Ç2u Çx2j

    ,

    (2) the heat equation Çu Çt

    −∆u = f , u = u(x, t),

    1

  • CHAPTER 1. INTRODUCTION 2

    (3) and the wave equation

    Ç2u Çt2

    −∆u = f u = u(x, t).

    Here we have set all physical constants equal to one. Physically, solutions of Laplace’s equation correspond to steady states or equilibria for time evolutions in heat distribution or wave motion, with f corresponding to external driving forces such as heat sources or wave generators. A solution u = u(x) to Laplace’s equation gives, for example, the temperature at the point x ∈Ω and a solution u = u(x, t) to the heat equation gives the temperature at the point x ∈ Ω at the moment of time t. A solution u = u(x, t) to the wave equation gives the displacement of a body at the point x ∈ Ω at the moment of time t. We shall later discuss the physical interpretation of these PDEs in more detail. If f = 0 (the function, which is identically zero), the PDE is called homogeneous, otherwise it is said to be inhomogeneous. All homogeneous versions of the PDEs above are linear, which means that any linear combination of solutions is a solution. More precisely, if u1 and u2 are solutions, then au1 +bu2, a, b ∈R, is a solution of the corresponding equation as well.

    By solving a PDE we mean that we find all functions u satisfying the PDE in a class of functions, which possibly satisfy certain auxiliary conditions. A PDE typically has many solutions, but there may be only one solution satisfying specific boundary or initial value conditions. These conditions are motivated by the physics and describe the physical state at a given moment or/and on the boundary of the domain. For Laplace’s equation we can describe, for example, the temperature on the boundary ÇΩ. For the heat equation we can, in addition, describe the initial temperature and for the wave equation the initial velocity at a given moment of time. By finding a solution to a PDE we mean that we obtain explicit representation formulas for solutions or deduce general properties that hold true for all solutions. A PDE problem is well posed, if

    (1) (existence) the problem has a solution,

    (2) (uniqueness) there exists only one solution and

    (3) (stability) the solution depends continuously on the data given in the problem.

    These are all desirable features when we talk about solving a PDE. The last condition is particularly important in physical problems, since we would like that our (unique) solution changes little when the conditions specifying the problem change little.

    There is at least one more important aspect in solving PDE. We have not yet specified what does it mean that a function actually is a solution to a PDE. We shall consider classical solutions, which means that all parti