Partial Differential Equations - uni-leipzig.de

1 Partial Differential EquationsLecture NotesErich MiersemannDepartment of MathematicsLeipzig UniversityVersion October, 20122 Contents1 Examples .. Equations from variational problems .. Ordinary differential Equations .. Partial differential Equations .. Exercises .. 222 Equations of first Linear Equations .. Quasilinear Equations .. A linearization method .. Initial value problem of Cauchy .. Nonlinear Equations in two variables .. Initial value problem of Cauchy .. Nonlinear Equations inRn.. Hamilton-Jacobi theory .. Exercises .. 593 Linear Equations of second order .. Normal form in two variables .. Quasilinear Equations of second order .. Quasilinear elliptic Equations .. Systems of first order.

2 Examples .. Systems of second order .. Examples .. Theorem of Cauchy-Kovalevskaya .. Appendix: Real analytic functions .. Exercises .. 1014 Hyperbolic One-dimensional wave equation .. Higher dimensions .. Case n=3 .. Casen= 2 .. Inhomogeneous equation .. A method of Riemann .. Initial-boundary value problems .. Oscillation of a string .. Oscillation of a membrane .. Inhomogeneous wave Equations .. Exercises .. 1365 Fourier Definition, properties .. Pseudodifferential operators .. Exercises .. 1496 Parabolic Poisson s formula .. Inhomogeneous heat equation .. Maximum principle .. Initial-boundary value problem .. Fourier s method .. Uniqueness .. Black-Scholes equation.

3 Exercises .. 1707 Elliptic Equations of second Fundamental solution .. Representation formula .. Conclusions from the representation formula .. Boundary value problems .. Dirichlet problem .. Neumann problem .. Mixed boundary value problem .. Green s function for4.. Green s function for a ball .. Green s function and conformal mapping .. Inhomogeneous equation .. Exercises .. 1956 CONTENTSP refaceThese lecture notes are intented as a straightforward introduction to partialdifferential Equations which can serve as a textbook for undergraduate andbeginning graduate additional reading we recommend following books: W. I. Smirnov [21],I. G. Petrowski [17], P. R. Garabedian [8], W. A. Strauss [23], F. John [10],L. C. Evans [5] and R.

4 Courant and D. Hilbert[4] and D. Gilbargand N. [9]. Some material of these lecture notes was taken from some ofthese 1 IntroductionOrdinary and Partial differential Equations occur in many applications. Anordinary differential equation is a special case of a Partial differential equa-tion but the behaviour of solutions is quite different in general. It is muchmore complicated in the case of Partial differential Equations caused by thefact that the functions for which we are looking at are functions of morethan one independent (x, y(x), y (x), .. , y(n)) = 0is anordinary differential equationof n-th order for the unknown functiony(x), whereFis important problem for ordinary differential Equations is theinitialvalue problemy (x) =f(x, y(x))y(x0) =y0,wherefis a given real function of two variablesx, yandx0, y0are givenreal of (i)f(x, y)is continuous in a rectangleQ={(x, y) R2:|x x0|< a,|y y0|< b}.

5 (ii) There is a constantKsuch that|f(x, y)| Kfor all(x, y) Q.(ii) Lipschitz condition: There is a constantLsuch that|f(x, y2) f(x, y1)| L|y2 y1|910 chapter 1. INTRODUCTION xyxy00 Figure : Initial value problemfor all(x, y1),(x, y2).Then there exists a unique solutiony C1(x0 , x0+ )of the above initialvalue problem, where = min(b/K, a).The linear ordinary differential equationy(n)+an 1(x)y(n 1)+.. a1(x)y +a0(x)y= 0,whereajare continuous functions, has exactlynlinearly independent solu-tions. In contrast to this property the Partial differentialuxx+uyy= 0 inR2has infinitely many linearly independent solutions in the linear spaceC2(R2).The ordinary differential equation of second ordery (x) =f(x, y(x), y (x))has in general a family of solutions with two free parameters.

6 Thus, it isnaturally to consider the associatedinitial value problemy (x) =f(x, y(x), y (x))y(x0) =y0, y (x0) =y1,wherey0andy1are given, or to consider theboundary value problemy (x) =f(x, y(x), y (x))y(x0) =y0, y(x1) = and boundary value problems play an important role also in thetheory of Partial differential Equations . Apartial differential EXAMPLES11yy0xxy10x1 Figure : Boundary value problemthe unknown functionu(x, y) is for exampleF(x, y, u, ux, uy, uxx, uxy, uyy) = 0,where the functionFis given. This equation is of second equation is said to be ofn-th orderif the highest derivative whichoccurs is of equation is said to belinearif the unknown function and its deriva-tives are linear inF. For example,a(x, y)ux+b(x, y)uy+c(x, y)u=f(x, y),where the functionsa, b, candfare given, is a linear equation of equation is said to bequasilinearif it is linear in the highest deriva-tives.

7 For example,a(x, y, u, ux, uy)uxx+b(x, y, u, ux, uy)uxy+c(x, y, u, ux, uy)uyy= 0is a quasilinear equation of second 0, whereu=u(x, y). All functionsu=w(x) are , whereu=u(x, y). A change of coordinates transforms thisequation into an equation of the first example. Set =x+y, =x y,thenu(x, y) =u( + 2, 2)=:v( , ).12 chapter 1. INTRODUCTIONA ssumeu C1, thenv =12(ux uy).Ifux=uy, thenv = 0 and vice versa, thusv=w( ) are solutions forarbitraryC1-functionsw( ). Consequently, we have a large class of solutionsof the original Partial differential equation:u=w(x+y) with necessary and sufficient condition such that for givenC1-functionsM, Nthe integral P1P0M(x, y)dx+N(x, y)dyis independent of the curve which connects the pointsP0withP1in a simplyconnected domain R2is the Partial differential equation (condition ofintegrability)My=Nxin.

8 Yx PP01 Figure : Independence of the pathThis is one equation for two functions. A large class of solutions is givenbyM= x, N= y, where (x, y) is an arbitraryC2-function. It followsfrom Gauss theorem that these are allC1-solutions of the above of an integrating multiplier for an ordinary differential the ordinary differential equationM(x, y)dx+N(x, y)dy= EXAMPLES13for givenC1-functionsM, N. Then we seek aC1-function (x, y) such that M dx+ N dyis a total differential, i. e., that ( M)y= ( N)xis is a linear Partial differential equation of first order for :M y N x= (Nx My). (x, y) andv(x, y) are said to befunctionally dependentifdet(uxuyvxvy)= 0,which is a linear Partial differential equation of first order foruifvis a givenC1-function. A large class of solutions is given byu=H(v(x, y)),whereHis (z) =u(x, y)+iv(x, y), wherez=x+iyandu, vare givenC1( )-functions.

9 Here is a domain inR2. If the functionf(z) is differentiable with respect to the complex variablezthenu, vsatisfythe Cauchy-Riemann equationsux=vy, uy= is known from the theory of functions of one complex variable that thereal partuand the imaginary partvof a differentiable functionf(z) aresolutions of theLaplace equation4u= 0,4v= 0,where4u=uxx+ potentialu=1 x2+y2+z2is a solution of the Laplace equation inR3\(0,0,0), i. e., ofuxx+uyy+uzz= 1. (x, t) be the temperature of a pointx at timet, where R3is a domain. Thenu(x, t) satisfies in [0, ) theheatequationut=k4u,where4u=ux1x1+ux2x 2+ux3x3andkis a positive constant. The conditionu(x,0) =u0(x), x ,whereu0(x) is given, is aninitial conditionassociated to the above heatequation. The conditionu(x, t) =h(x, t), x , t 0,whereh(x, t) is given is aboundary conditionfor the heat (x, t) =g(x), that is,his independent oft, then one expects that thesolutionu(x, t) tends to a functionv(x) ift.]

10 Moreover, it turns outthatvis the solution of theboundary value problemfor the Laplace equation4v= 0 in v=g(x) on . wave equationyu(x,t )u(x,t )12xlFigure : Oscillating stringutt=c24u,whereu=u(x, t),cis a positive constant, describes oscillations of mem-branes or of three dimensional domains, for example. In the one-dimensionalcaseutt=c2uxxdescribes oscillations of a Equations FROM VARIATIONAL PROBLEMS15 Associatedinitial conditionsareu(x,0) =u0(x), ut(x,0) =u1(x),whereu0, u1are given functions. Thus the initial position and the initialvelocity are the string is finite one describes additionallyboundary conditions, forexampleu(0, t) = 0, u(l, t) = 0 for allt Equations from variational problemsA large class of ordinary and Partial differential Equations arise from varia-tional Ordinary differential equationsSetE(v) = baf(x, v(x), v (x))dxand for givenua, ub RV={v C2[a, b] :v(a) =ua, v(b) =ub},where < a < b < andfis sufficiently regular.