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  • JUHA KINNUNEN

    Partial Differential Equations

    Department of Mathematics and Systems Analysis, Aalto University

    2017

  • Contents

    1 INTRODUCTION 1

    2 FOURIER SERIES AND PDES 52.1 Periodic functions* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 The Lp space on [,]* . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 The Fourier series* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 The best square approximation* . . . . . . . . . . . . . . . . . . . . . 192.5 The Fourier series on a general interval* . . . . . . . . . . . . . . . . . 242.6 The real form of the Fourier series* . . . . . . . . . . . . . . . . . . . . 242.7 The Fourier series and differentiation* . . . . . . . . . . . . . . . . . . 272.8 The Dirichlet kernel* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.9 Convolutions* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.10 A local result for the convergence of Fourier series* . . . . . . . . . . 322.11 The Laplace equation in the unit disc . . . . . . . . . . . . . . . . . . 342.12 The heat equation in one-dimension . . . . . . . . . . . . . . . . . . . 452.13 The wave equation in one-dimension . . . . . . . . . . . . . . . . . . 522.14 Approximations of the identity

    in [,]* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.15 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

    3 FOURIER TRANSFORM AND PDES 633.1 The Lp-space on Rn* . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.2 The Fourier transform* . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.3 The Fourier transform and differentiation* . . . . . . . . . . . . . . . . 673.4 The Fourier transform of the Gaussian* . . . . . . . . . . . . . . . . . . 713.5 The Fourier inversion formula* . . . . . . . . . . . . . . . . . . . . . . . 723.6 The Fourier transformation and convolution . . . . . . . . . . . . . . 763.7 Plancherels formula* . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.8 Approximations of the identity in Rn* . . . . . . . . . . . . . . . . . . . 793.9 The Laplace equation in the upper half-space . . . . . . . . . . . . 813.10 The heat equation in the upper half-space . . . . . . . . . . . . . . . 863.11 The wave equation in the upper half-space . . . . . . . . . . . . . . 913.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

  • CONTENTS ii

    4 LAPLACE EQUATION 934.1 Gauss-Green theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.2 PDEs and physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.3 Boundary values and physics . . . . . . . . . . . . . . . . . . . . . . . 974.4 Fundamental solution of the Laplace equation . . . . . . . . . . . . 1014.5 The Poisson equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.6 Greens function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.7 Greens function for the upper half-space* . . . . . . . . . . . . . . . 1154.8 Greens function for the ball* . . . . . . . . . . . . . . . . . . . . . . . 1174.9 Mean value formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.10 Maximum principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.11 Harnacks inequality* . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.12 Energy methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.13 Weak solutions* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324.14 The Laplace equation in other coordinates* . . . . . . . . . . . . . . 1334.15 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

    5 HEAT EQUATION 1405.1 Physical interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.2 The fundamental solution . . . . . . . . . . . . . . . . . . . . . . . . . 1425.3 The nonhomogeneous problem . . . . . . . . . . . . . . . . . . . . . 1435.4 Separation of variables in Rn . . . . . . . . . . . . . . . . . . . . . . . 1465.5 Maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1505.6 Energy methods for the heat equation . . . . . . . . . . . . . . . . . 1555.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

    6 WAVE EQUATION 1586.1 Physical interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1596.2 The one-dimensional wave equation . . . . . . . . . . . . . . . . . . 1596.3 Euler-Poisson-Darboux equation . . . . . . . . . . . . . . . . . . . . . 1666.4 The three-dimensional wave equation . . . . . . . . . . . . . . . . . 1696.5 The two-dimensional wave equation . . . . . . . . . . . . . . . . . . 1736.6 The nonhomogeneous problem . . . . . . . . . . . . . . . . . . . . . 1776.7 Energy methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1796.8 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1816.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

    7 NOTATION AND TOOLS 183

  • Partial differential equations are not only extremely impor-tant in applications of mathematics in physical, geometricand probabilistic phenomena, but they also are of theoreticinterest.

    1Introduction

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

    A partial differential equation is an equation involving an unknown functionof two ore more variables and its partial derivatives. Although PDE are general-izations of ordinary differential equations (ODE), for most PDE problems it is notpossible to write down explicit formulas for solutions that are common in the ODEtheory. This means that there is greater emphasis on qualitative features. Thereis no general method to solve all PDE, however, some methods have turned outto be more useful than other. We study special cases, in which explicit solutionsand representation formulas are available and focus on features that are presentin more general situations. Qualitative aspects are also important in numericalsolutions of PDE. Without existence, uniqueness and stability results numericalmethods may give inaccurate or completely wrong solutions.

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

    (1) Laplaces equation

    u = f , u = u(x), u =n

    j=1

    2ux2j

    ,

    (2) the heat equationut

    u = f , u = u(x, t),

    1

  • CHAPTER 1. INTRODUCTION 2

    (3) and the wave equation

    2ut2

    u = f u = u(x, t).

    Here we have set all physical constants equal to one. Physically, solutions ofLaplaces equation correspond to steady states for time evolutions such as heatflow or wave motion, with f corresponding to external driving forces such asheat sources or wave generators. A solution u = u(x) to Laplaces equation gives,for example, the temperature at the point x and a solution u = u(x, t) tothe heat equation gives the temperature at the point x at the moment oftime t. A solution u = u(x, t) to the wave equation gives the displacement ofa body at the point x in the moment of time t. We shall later discuss theorigin and interpretation of these PDEs in more detail. If f = 0 (the function,which is identically zero), the PDE is called homogeneous, otherwise it is saidto be inhomogeneous. All PDEs above are linear, which means that any linearcombination 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 verifying the PDE ina class of functions, which possibly satisfy certain auxiliary conditions. A PDEtypically has many solutions, but there may be only one solution satisfying specificboundary or initial value conditions conditions. These conditions are motivatedby the physics and describe the physical state at a given moment or/and on theboundary of the domain. For Laplaces equation we can describe, for example,the temperature on the boundary . For the heat equation we can, in addition,describe the initial temperature and for the wave equation the initial velocity at agiven moment of time. By finding a solution to a PDE we mean that we obtainexplicit representation formulas for solutions or deduce general properties thathold true for all solutions. A PDE problem is well posed, if

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

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

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

    These are all desirable features when we talk about solving a PDE. The lastcondition is particularly important in physical problems, since we would like thatour (unique) solution changes little when the conditions specifying the problemchange little.

    There is at least one more important aspect in solving PDE. We have not yetspecified what does it mean that a function actually is a solution to a PDE. Weshall consider classical solutions, which means that all partial derivatives whichappear in the PDE exist and are continuous. In this case, we can verify by adirect computation that a function solves the PDE. However, the PDE can be so

  • CHAPTER 1. INTRODUCTION 3

    strong that it forces the solution to be smoother than assumed in the beginning.A PDE may also have physically relevant weak solutions with l