Partial Differential Equations: A Comprehensive Guide to Theory and Applications, Slides of Differential Equations

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Partial Differential Equations

Lecture Notes

Erich Miersemann

Department of Mathematics

Leipzig University

Version October, 2012

• 1 Introduction
  • 1.1 Examples
  • 1.2 Equations from variational problems
    • 1.2.1 Ordinary differential equations
    • 1.2.2 Partial differential equations
  • 1.3 Exercises
• 2 Equations of first order
  • 2.1 Linear equations
  • 2.2 Quasilinear equations
    • 2.2.1 A linearization method
    • 2.2.2 Initial value problem of Cauchy
  • 2.3 Nonlinear equations in two variables
    • 2.3.1 Initial value problem of Cauchy
  • 2.4 Nonlinear equations in Rn
  • 2.5 Hamilton-Jacobi theory
  • 2.6 Exercises
• 3 Classification
  • 3.1 Linear equations of second order
    • 3.1.1 Normal form in two variables
  • 3.2 Quasilinear equations of second order
    • 3.2.1 Quasilinear elliptic equations
  • 3.3 Systems of first order
    • 3.3.1 Examples
  • 3.4 Systems of second order
    • 3.4.1 Examples
  • 3.5 Theorem of Cauchy-Kovalevskaya
    • 3.5.1 Appendix: Real analytic functions
  • 3.6 Exercises 4 CONTENTS
• 4 Hyperbolic equations
  • 4.1 One-dimensional wave equation
  • 4.2 Higher dimensions
    • 4.2.1 Case n=3
    • 4.2.2 Case n =
  • 4.3 Inhomogeneous equation
  • 4.4 A method of Riemann
  • 4.5 Initial-boundary value problems
    • 4.5.1 Oscillation of a string
    • 4.5.2 Oscillation of a membrane
    • 4.5.3 Inhomogeneous wave equations
  • 4.6 Exercises
• 5 Fourier transform
  • 5.1 Definition, properties
    • 5.1.1 Pseudodifferential operators
  • 5.2 Exercises
• 6 Parabolic equations
  • 6.1 Poisson’s formula
  • 6.2 Inhomogeneous heat equation
  • 6.3 Maximum principle
  • 6.4 Initial-boundary value problem
    • 6.4.1 Fourier’s method
    • 6.4.2 Uniqueness
  • 6.5 Black-Scholes equation
  • 6.6 Exercises
• 7 Elliptic equations of second order
  • 7.1 Fundamental solution
  • 7.2 Representation formula
    • 7.2.1 Conclusions from the representation formula
  • 7.3 Boundary value problems
    • 7.3.1 Dirichlet problem
    • 7.3.2 Neumann problem
    • 7.3.3 Mixed boundary value problem
  • 7.4 Green’s function for
    • 7.4.1 Green’s function for a ball
• CONTENTS - 7.4.2 Green’s function and conformal mapping
  • 7.5 Inhomogeneous equation
  • 7.6 Exercises

Preface

These lecture notes are intented as a straightforward introduction to partial differential equations which can serve as a textbook for undergraduate and beginning graduate students. For 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. Courant and D. Hilbert[4] and D. Gilbarg and N. S. Trudinger [9]. Some material of these lecture notes was taken from some of these books.

8 CONTENTS

10 CHAPTER 1. INTRODUCTION

x

y

x

y

0

0

Figure 1.1: Initial value problem

for all (x, y 1 ), (x, y 2 ).

Then there exists a unique solution y ∈ C^1 (x 0 −α, x 0 +α) of the above initial value problem, where α = min(b/K, a).

The linear ordinary differential equation

y(n)^ + an− 1 (x)y(n−1)^ +... a 1 (x)y′^ + a 0 (x)y = 0,

where aj are continuous functions, has exactly n linearly independent solu- tions. In contrast to this property the partial differential uxx +uyy = 0 in R^2 has infinitely many linearly independent solutions in the linear space C^2 (R^2 ).

The ordinary differential equation of second order

y′′(x) = f (x, y(x), y′(x))

has in general a family of solutions with two free parameters. Thus, it is naturally to consider the associated initial value problem

y′′(x) = f (x, y(x), y′(x)) y(x 0 ) = y 0 , y′(x 0 ) = y 1 ,

where y 0 and y 1 are given, or to consider the boundary value problem

y′′(x) = f (x, y(x), y′(x)) y(x 0 ) = y 0 , y(x 1 ) = y 1.

Initial and boundary value problems play an important role also in the theory of partial differential equations. A partial differential equation for

1.1. EXAMPLES 11

y

y 0

x x

y 1

0 1 x

Figure 1.2: Boundary value problem

the unknown function u(x, y) is for example

F (x, y, u, ux, uy, uxx, uxy, uyy) = 0,

where the function F is given. This equation is of second order. An equation is said to be of n-th order if the highest derivative which occurs is of order n. An equation is said to be linear if the unknown function and its deriva- tives are linear in F. For example,

a(x, y)ux + b(x, y)uy + c(x, y)u = f (x, y),

where the functions a, b, c and f are given, is a linear equation of first order. An equation is said to be quasilinear if it is linear in the highest deriva- tives. For example,

a(x, y, u, ux, uy)uxx + b(x, y, u, ux, uy)uxy + c(x, y, u, ux, uy)uyy = 0

is a quasilinear equation of second order.

1.1 Examples

1. uy = 0, where u = u(x, y). All functions u = w(x) are solutions.
2. ux = uy, where u = u(x, y). A change of coordinates transforms this equation into an equation of the first example. Set ξ = x + y, η = x − y, then

u(x, y) = u

ξ + η 2

ξ − η 2

=: v(ξ, η).

1.1. EXAMPLES 13

for given C^1 -functions M, N. Then we seek a C^1 -function μ(x, y) such that μM dx + μN dy is a total differential, i. e., that (μM )y = (μN )x is satisfied. This is a linear partial differential equation of first order for μ:

M μy − N μx = μ(Nx − My).

5. Two C^1 -functions u(x, y) and v(x, y) are said to be functionally dependent if

det

ux uy vx vy

which is a linear partial differential equation of first order for u if v is a given C^1 -function. A large class of solutions is given by

u = H(v(x, y)),

where H is an arbitrary C^1 -function.

6. Cauchy-Riemann equations. Set f (z) = u(x, y)+iv(x, y), where z = x+iy and u, v are given C^1 (Ω)-functions. Here is Ω a domain in R^2. If the function f (z) is differentiable with respect to the complex variable z then u, v satisfy the Cauchy-Riemann equations

ux = vy, uy = −vx.

It is known from the theory of functions of one complex variable that the real part u and the imaginary part v of a differentiable function f (z) are solutions of the Laplace equation

4 u = 0, 4 v = 0,

where 4 u = uxx + uyy.

7. The Newton potential

u =

x^2 + y^2 + z^2

is a solution of the Laplace equation in R^3 \ (0, 0 , 0), i. e., of

uxx + uyy + uzz = 0.

14 CHAPTER 1. INTRODUCTION

8. Heat equation. Let u(x, t) be the temperature of a point x ∈ Ω at time t, where Ω ⊂ R^3 is a domain. Then u(x, t) satisfies in Ω × [0, ∞) the heat equation ut = k 4 u,

where 4 u = ux 1 x 1 +ux 2 x 2 +ux 3 x 3 and k is a positive constant. The condition

u(x, 0) = u 0 (x), x ∈ Ω,

where u 0 (x) is given, is an initial condition associated to the above heat equation. The condition

u(x, t) = h(x, t), x ∈ ∂Ω, t ≥ 0 ,

where h(x, t) is given is a boundary condition for the heat equation. If h(x, t) = g(x), that is, h is independent of t, then one expects that the solution u(x, t) tends to a function v(x) if t → ∞. Moreover, it turns out that v is the solution of the boundary value problem for the Laplace equation

4 v = 0 in Ω v = g(x) on ∂Ω.

9. Wave equation. The wave equation

y

u(x,t ) 1 u(x,t )

2

l^ x

Figure 1.4: Oscillating string

utt = c^24 u,

where u = u(x, t), c is a positive constant, describes oscillations of mem- branes or of three dimensional domains, for example. In the one-dimensional case utt = c^2 uxx

describes oscillations of a string.

16 CHAPTER 1. INTRODUCTION

y

y 0

y 1

a b x

Figure 1.5: Admissible variations

Basic lemma in the calculus of variations. Let h ∈ C(a, b) and ∫ (^) b

a

h(x)φ(x) dx = 0

for all φ ∈ C 01 (a, b). Then h(x) ≡ 0 on (a, b).

Proof. Assume h(x 0 ) > 0 for an x 0 ∈ (a, b), then there is a δ > 0 such that (x 0 − δ, x 0 + δ) ⊂ (a, b) and h(x) ≥ h(x 0 )/2 on (x 0 − δ, x 0 + δ). Set

φ(x) =

δ^2 − |x − x 0 |^2

if x ∈ (x 0 − δ, x 0 + δ) 0 if x ∈ (a, b) \ [x 0 − δ, x 0 + δ]

Thus φ ∈ C 01 (a, b) and ∫ (^) b

a

h(x)φ(x) dx ≥ h(x 0 ) 2

∫ (^) x 0 +δ

x 0 −δ

φ(x) dx > 0 ,

which is a contradiction to the assumption of the lemma. 2

1.2.2 Partial differential equations

The same procedure as above applied to the following multiple integral leads to a second-order quasilinear partial differential equation. Set

E(v) =

Ω

F (x, v, ∇v) dx,

1.2. EQUATIONS FROM VARIATIONAL PROBLEMS 17

where Ω ⊂ Rn^ is a domain, x = (x 1 ,... , xn), v = v(x) : Ω 7 → R, and ∇v = (vx 1 ,... , vxn ). Assume that the function F is sufficiently regular in its arguments. For a given function h, defined on ∂Ω, set

V = {v ∈ C^2 (Ω) : v = h on ∂Ω}.

Euler equation. Let u ∈ V be a solution of (P), then

∑^ n

i=

∂xi Fuxi − Fu = 0

in Ω.

Proof. Exercise. Hint: Extend the above fundamental lemma of the calculus of variations to the case of multiple integrals. The interval (x 0 − δ, x 0 + δ) in the definition of φ must be replaced by a ball with center at x 0 and radius δ.

Example: Dirichlet integral

In two dimensions the Dirichlet integral is given by

D(v) =

Ω

v^2 x + v^2 y

dxdy

and the associated Euler equation is the Laplace equation 4 u = 0 in Ω. Thus, there is natural relationship between the boundary value problem

4 u = 0 in Ω, u = h on ∂Ω

and the variational problem

min v∈V D(v).

But these problems are not equivalent in general. It can happen that the boundary value problem has a solution but the variational problem has no solution, see for an example Courant and Hilbert [4], Vol. 1, p. 155, where h is a continuous function and the associated solution u of the boundary value problem has no finite Dirichlet integral. The problems are equivalent, provided the given boundary value function h is in the class H^1 /^2 (∂Ω), see Lions and Magenes [14].

1.2. EQUATIONS FROM VARIATIONAL PROBLEMS 19

n ≤ 7, see Simons [19]. If n ≥ 8, then there exists also other solutions which define cones, see Bombieri, De Giorgi and Giusti [3]. The linearized minimal surface equation over u ≡ 0 is the Laplace equa- tion 4 u = 0. In R^2 linear functions are solutions but also many other functions in contrast to the minimal surface equation. This striking differ- ence is caused by the strong nonlinearity of the minimal surface equation. More general minimal surfaces are described by using parametric rep- resentations. An example is shown in Figure 1.7^1. See [18], pp. 62, for example, for rotationally symmetric minimal surfaces.

Figure 1.7: Rotationally symmetric minimal surface

Neumann type boundary value problems

Set V = C^1 (Ω) and

E(v) =

Ω

F (x, v, ∇v) dx −

∂Ω

g(x, v) ds,

where F and g are given sufficiently regular functions and Ω ⊂ Rn^ is a bounded and sufficiently regular domain. Assume u is a minimizer of E(v) in V , that is u ∈ V : E(u) ≤ E(v) for all v ∈ V, (^1) An experiment from Beutelspacher’s Mathematikum, Wissenschaftsjahr 2008, Leipzig

20 CHAPTER 1. INTRODUCTION

then

∫

Ω

( ∑n

i=

Fuxi (x, u, ∇u)φxi + Fu(x, u, ∇u)φ

dx

∂Ω

gu(x, u)φ ds = 0

for all φ ∈ C^1 (Ω). Assume additionally u ∈ C^2 (Ω), then u is a solution of the Neumann type boundary value problem

∑^ n

i=

∂xi Fuxi − Fu = 0 in Ω

∑^ n

i=

Fuxi νi − gu = 0 on ∂Ω,

where ν = (ν 1 ,... , νn) is the exterior unit normal at the boundary ∂Ω. This follows after integration by parts from the basic lemma of the calculus of variations.

Example: Laplace equation

Set E(v) =

Ω

|∇v|^2 dx −

∂Ω

h(x)v ds,

then the associated boundary value problem is

4 u = 0 in Ω ∂u ∂ν = h on ∂Ω.

Example: Capillary equation

Let Ω ⊂ R^2 and set

E(v) =

Ω

1 + |∇v|^2 dx + κ 2

Ω

v^2 dx − cos γ

∂Ω

v ds.

Here κ is a positive constant (capillarity constant) and γ is the (constant) boundary contact angle, i. e., the angle between the container wall and