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Lectures Notes Partial Differential Equations I Lectures Notes Partial Differential Equations I Winter Term 2018/19 Thomas Schmidt Version: May 1, 2020 Partial Differential Equations I: Table of Contents Table of Contents 1 1 Basics, examples, classification3 2 The Laplace equation and the Poisson equation 11 2.1 The fundamental solution............................... 12 Addendum on surface measures (and surface integration) ................ 13 2.2 Harmonic polynomials ................................ 15 2.3 Consequences of the divergence theorem ...................... 16 2.4 The mean value property and the maximum principle............... 18 Addendum on the technique of mollification........................ 25 2.5 Weakly harmonic functions and regularity of harmonic functions......... 31 2.6 Liouville and convergence theorems, Harnack's inequality............. 34 2.7 Generalized sub/superharmonic functions...................... 38 2.8 Green's representation formula and the Poisson integral.............. 42 2.9 Isolated singularities, analyticity, and reflection principles............. 56 2.10 Perron's method for the Dirichlet problem on general domains.......... 61 2.11 The Newton potential as a solution of the Poisson equation............ 67 2.12 On the eigenvalue problem for the Laplace operator................ 79 3 The heat equation (not typeset) 3.1 Consequences of the divergence theorem . 3.2 Fundamental solution and mean value property . 3.3 Maximum principles . 3.4 The initial-boundary value problem in space dimension 1 . 3.5 The initial value problem on the full space Rn ................... 3.6 The Green-Duhamel representation formula and the IBVP on domains in Rn .. 3.7 More on the heat equation . 4 On the wave equation (not typeset) 4.1 The initial value problem in space dimension 1 . 4.2 The initial value problem in higher space dimension . References 85 If you find mistakes in these notes and/or have any other comments, please communicate them to the author, either in person or at [email protected]. 1 2 TABLE OF CONTENTS 2 Chapter 1 Basics, examples, classification . In these notes we stick to the basic conventions N .= f1; 2; 3;:::g and N0 .= N [ f0g. Terminology (for general PDEs). A partial differential equation (PDE) is an equation for a function u in two or more (real) variables: 2 m−1 m F (x; u(x); Du(x); D u(x);:::; D u(x); D u(x)) = 0RM for all x 2 Ω(∗) or, in short-hand notation, 2 m−1 m F ( · ; u; Du; D u; : : : ; D u; D u) ≡ 0RM on Ω : Here we denote . • by m 2 N (if chosen minimal1) the order of the PDE (∗), • by Ω an arbitrary open set in Rn, • by u:Ω ! RN the unknown function (by Du(x) 2 L(Rn; RN ) = RN×n its first deriva- tive at the point x, regarded as linear mapping Rn ! RN or (N×n)-matrix, and more k k Rn RN generally by D u(x) 2 Lsym( ; ) its k-th derivative at x, regarded as symmetric k- linear mapping (Rn)k ! RN ), • by n 2 N the number of (independent) variables, here generally n ≥ 2, • by N 2 N the number of unknown (component) functions, • by M 2 N the number of (component) equations, RN Rn RN 2 Rn RN m−1 Rn RN m Rn RN RM • by F :Ω× ×L( ; )×Lsym( ; )×:::×Lsym ( ; )×Lsym( ; ) ! the given structure function of the PDE (∗). For N = 1 the unknown is a scalar/single function, otherwise a vector function. In case M = 1 we speak of a scalar/single PDE, otherwise of a system of M PDEs. Finally, we call u:Ω ! RN a solution of /to the PDE (∗) if (∗) holds for u. 1Minimality of m means that the PDE can see a difference in the m-th derivative only. In precise terms, this k k means that there exist functions u, v, and a point x0 2 Ω with D u(x0) = D v(x0) for k = 0; 1; 2; : : : ; m−1 but 2 m 2 m still with F (x0; u(x0); Du(x0); D u(x0);:::; D u(x0)) = 0 6= F (x0; v(x0); Dv(x0); D v(x0);:::; D v(x0)). 3 4 CHAPTER 1. Basics, examples, classification We emphasize that the word `partial' in the term `partial differential equation' signifies α α1 α2 αn Nn the occurrence of partial derivatives @ u = @1 @2 :::@n u with α 2 0 (which are the k components of D u with k = jαj = α1+α2+ ::: +αn ≤ m) and is used to distinguish these equations from ordinary differential equations (ODEs) for functions of a single variable. Terminology (for linear PDEs). The PDE (∗) above is termed linear if the structure function F is an affine function of the u, Du, D2u,..., Dm−1u, Dmu variables, that is if it takes the form N X X ij α j aα (x)@ ui(x) = f (x) for all x 2 Ω and j = 1; 2;:::;M−1;M (∗∗) i=1 jα|≤m ij j with coefficients aα :Ω ! R and inhomogeneities f :Ω ! R. The PDE (∗∗) has con- ij stant coefficients if all coefficients aα are constant functions; and it is homogeneous if all inhomogeneities f j vanish. While linear PDEs are clearly a very basic type, a lot of advanced PDE theory has nowadays been developed for cases which are not truly linear but linear in the highest-order derivatives at least. Though not all authors use the same terminology for such equations, there is some agreement that quasilinear PDEs are equations of the type N X X ij 2 m−1 α j 2 m−1 Aα ( · ; u; Du; D u; : : : ; D u)@ ui = G ( · ; u; Du; D u; : : : ; D u) i=1 jαj=m and semilinear PDEs are equations of the somewhat more special type N X X ij α j 2 m−1 aα ( · )@ ui = G ( · ; u; Du; D u; : : : ; D u) : i=1 jαj=m PDEs which are not even quasilinear are generally known as fully non-linear PDEs. n Examples (of PDEs and PDE systems). Consider an open set Ω ⊂ R with n 2 N≥2. (1) The Cauchy-Riemann system @u @v @u @v = ; = − @x @y @y @x is a first-order (m=1) linear system of M=2 PDEs for N=2 functions u; v :Ω ! RN in n=2 variables. It corresponds to the case of the structure function F (z; w; `) = (`11−`22; `12+`21) for (z; w; `) 2 Ω×R2×R2×2 in (∗). If we identify R2 with C, solutions (u; v):Ω ! R2 of the Cauchy-Riemann system turn out to be precisely the holomorphic functions h:Ω ! C. Thus, the Cauchy-Riemann system can be viewed as the underlying PDE system in complex analysis (at least in case of a single complex variable). The Cauchy integral and the Poisson integral yield explicit formulas for solutions. 4 5 (2) The Laplace equation div(ru) ≡ 0 and the Poisson equation div(ru) = f with non-vanishing f :Ω ! R, respectively, are scalar (M=1) second-order (m=2) linear PDEs for a single (N=1) function u:Ω ! R in an arbitrary number n of variables. On the left-hand side these equations involve the Laplace operator ∆, defined by n @2u @2u @2u @2u X ∆u .= div(ru) = + + ::: + + = @2u = trace(r2u) @x2 @x2 @x2 @x2 i 1 2 n−1 n i=1 2 2 Rn×n (where D u(x) is represented, in this scalar case, by the Hessian r u(x) 2 sym ). The Pois- son equation corresponds to the case of the structure function F (x; u; `; q) = trace(q)−f(x) R Rn Rn×n for (x; u; `; q) 2 Ω× × × sym in (∗), and clearly the choice f ≡ 0 yields the Laplace equation. Solutions of the Laplace equation are known as harmonic functions and will be of central interest in this lecture. For n=2 there is strong connection to (1), as harmonic functions of two variables arise as the real and imaginary parts of holomorphic functions. The Poisson equation on Ω = R3 serves as a model equation in electrostatics, which deter- mines the electric potential u corresponding to the charge distribution f. (3) (Linear) Transport Equations take the form @u + b ·r u + cu ≡ 0 ; @t x where the variables (t; x) 2 Ω ⊂ R×Rn−1 are split into a single `time' variable t and (n−1) `space' variables x. Here, the non-vanishing time-dependent vector field b:Ω ! Rn−1 and the coefficient c:Ω ! R are considered as given, and the equations are scalar (M=1) first-order (m=1) linear PDEs for a single (N=1) function u:Ω ! R in an arbitrary num- ber n of variables. They correspond to the case of the structure function F (z; u; `) = 0 n 0 n−1 `0+b(z)·` +c(z)u for (z; u; `) 2 Ω×R×R , ` = (`0; ` ) 2 R×R in (∗). The case of constant b and c is discussed in the exercise class. Linear transport equations in n = 1+3 time-space variables model the mass transport in a velocity field b. (4) The Heat Equation or Diffusion Equation @u − ∆ u ≡ 0 @t x and the Wave Equation (for n = 2 sometimes called Equation of the Vibrating String) @2u − ∆ u ≡ 0 ; @t2 x respectively, involve once more the time-space split variables (t; x) 2 Ω ⊂ R×Rn−1. These equations are scalar (M=1) second-order (m=2) linear PDEs for a single (N=1) function 5 6 CHAPTER 1. Basics, examples, classification u:Ω ! R in an arbitrary number n of variables, and they correspond to the case of Pn−1 Pn−1 the structure functions F (z; u; `; q) = `0− i=1 qii and F (z; u; `; q) = q00− i=1 qii for R Rn Rn×n 0 R Rn−1 (z; u; `; q) 2 Ω× × × sym , ` = (`0; ` ) 2 × , q = (qij)i;j=0;1;2;:::;n−1 in (∗).
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