Lecture Notes Methods of Mathematical Physics MATH 536

Lecture Notes Methods of Mathematical Physics MATH 536 Instructor: Ivan Avramidi Textbook: L. Debnath and P. Mikusinski, (Academic Press, 1999) New Mexico Institute of Mining and Technology Socorro, NM 87801 February 12, 2005

Author: Ivan Avramidi; File: appliedfunanal.tex; Date: May 24, 2005; Time: 13:58

Contents 1

2

Integral Operators 1.1 Existense Theorems . . . . . . . . . . . . . . . . . 1.1.1 Integral Equations . . . . . . . . . . . . . 1.1.2 Neumann Series and Fredholm Alternative 1.1.3 Homework . . . . . . . . . . . . . . . . . 1.2 Fredholm Integral Equations . . . . . . . . . . . . 1.2.1 Hilbert-Schmidt and Trace -Class Operators 1.2.2 Linear Fredholm Equations . . . . . . . . . 1.2.3 Nonlinear Fredholm Equations . . . . . . . 1.2.4 Method of Successive Approximations . . 1.2.5 Separable Kernel . . . . . . . . . . . . . . 1.2.6 Convolution Kernel . . . . . . . . . . . . . 1.2.7 Hilbert Transform . . . . . . . . . . . . . 1.2.8 Homework . . . . . . . . . . . . . . . . . 1.3 Volterra Integral Equations . . . . . . . . . . . . . 1.3.1 Volterra Equations of the Second Kind . . . 1.3.2 Volterra Equations of the First Kind . . . . 1.3.3 Abel’s Integral Equation . . . . . . . . . . 1.3.4 Riemann-Liouville Transform . . . . . . . 1.3.5 Homework . . . . . . . . . . . . . . . . . Ordinary Differential Operators 2.1 Ordinary Differential Operators . . . . . . . . 2.1.1 Homework . . . . . . . . . . . . . . 2.2 Sturm-Liouville Systems . . . . . . . . . . . 2.2.1 Homework . . . . . . . . . . . . . . 2.3 Green Functions . . . . . . . . . . . . . . . . 2.3.1 Operators with Constant Coefficients I

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1

CONTENTS 2.3.2 2.3.3 3

4

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Homework . . . . . . . . . . . . . . . . . . . . . . . . . 42

Distributions and Partial Differential Equations 3.1 Distributions . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Homework . . . . . . . . . . . . . . . . . . . 3.2 Fundamental Solutions of Partial Differential Equations 3.2.1 Boundary Value Problems . . . . . . . . . . . 3.2.2 Equations of Mathematical Physics . . . . . . 3.2.3 Green’s Formulas . . . . . . . . . . . . . . . . 3.2.4 Green Functions . . . . . . . . . . . . . . . . 3.2.5 Homework . . . . . . . . . . . . . . . . . . . 3.3 Weak Solutions of Elliptic Boundary Value Problems . 3.3.1 Homework . . . . . . . . . . . . . . . . . . . Wavelets and Optimization 4.1 Wavelets . . . . . . . . . . . . . . . . . . 4.1.1 Wavelet Transforms . . . . . . . 4.1.2 Homework . . . . . . . . . . . . 4.2 Calculus of Variations . . . . . . . . . . . 4.2.1 Gateaux and Fr´echet Differentials 4.2.2 Euler-Lagrange Equations . . . . 4.2.3 Homework . . . . . . . . . . . . 4.3 Dynamical Systems . . . . . . . . . . . . 4.3.1 Optimal Control . . . . . . . . . 4.3.2 Stability . . . . . . . . . . . . . . 4.3.3 Bifurcations . . . . . . . . . . . . 4.3.4 Homework . . . . . . . . . . . .

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Bibliography

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Notation

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2

CONTENTS

appliedfunanal.tex; May 24, 2005; 13:58; p. 2

Chapter 1 Integral Operators 1.1 1.1.1

Existense Theorems Integral Equations

• Let g be a given function, K(x, t) be a given function of two variables, and f be an unknown function. The Volterra equation of the first kind reads Z x g(x) = dt K(x, t) f (t) . a

• Volterra equation of the second kind is Z x g(x) = f (x) + dt K(x, t) f (t) . a

• The Fredholm equation of the first kind reads g(x) =

b

Z

dt K(x, t) f (t) .

a

• Fredholm equation of the second kind is g(x) = f (x) +

Z a

3

b

dt K(x, t) f (t) .

4

CHAPTER 1. INTEGRAL OPERATORS

1.1.2

Neumann Series and Fredholm Alternative

• Let T : H → H be a mapping on a vector space H. The fixed point set of T is the set of vectors x ∈ H such that T (x) = x0 . • If T is a linear operator, then its fixed point set is equal to the kernel of the operator (I − T ). • The mapping T is a contraction if there is 0 < α < 1 such that for any x, y ∈ H k T (x) − T (y) k≤ α k x − y k . Theorem 1.1.1 (Contraction Mapping Theorem) Let H be a Banach space and S be a closed subspace of H. Let T : S → S be a contraction mapping. Let x be an arbitrary vector in S and x0 = lim T n x . n→∞

•

Then x0 belongs to S and T (x0 ) = x0 . Moreover, this is the unique solution of the equaion T (x) = x, that is the unique fixed point of the mapping T . Proof: Banach Fixed Point Theorem. Theorem 1.1.2 Let H be a Banach space and T be a continuous mapping such that for some m ∈ N, T m is a contraction. Let x ∈ H and

•

x0 = lim T n x . n→∞

Then x0 is the unique fixed point of T . Proof: Easy.  appliedfunanal.tex; May 24, 2005; 13:58; p. 3

5

1.1. EXISTENSE THEOREMS

•

Theorem 1.1.3 Let H be a Banach space and A be a non-zero bounded linear operator on H. Let g ∈ H and α ∈ C be a complex number such that 1 . |α| < kAk Let T be a mapping on H defined by T ( f ) = αA f + g . Then T has a unique fixed point, that is the equation T ( f ) = f has a unique solution. Proof: Easy. 

• Note that T (f) = α A f + n

n

n

n X

αn An g .

k=0

• The fixed point f =

∞ X

αn An g ,

k=0

is the solution of the equation (I − αA) f = g . obtained by the geometric series (I − αA)

−1

=

∞ X

αn An ,

n=0

called the Neumann series. appliedfunanal.tex; May 24, 2005; 13:58; p. 4

6

CHAPTER 1. INTEGRAL OPERATORS Theorem 1.1.4 (Neumann Series) Let A be a bounded linear operator in a Banach space H. Then the resolvent R(λ) = (A − λI)−1 is a bounded operator for any λ ∈ C such that |λ| >k A k , •

that is, outside the circle or radius k A k. Moreover, R(λ) = −

∞ X

λ−(n+1) An

n=0

and k R(λ) k≤

1 . |λ|− k A k

Proof:

1. We have ∞ X

|λ−(n+1) | k An k< ∞ .

n=0

2. We check that

(A − λI)

∞ X n=0

∞  X    λ−(n+1) An =  λ−(n+1) An  (A − λI) = −I . n=0

3. Finally, k R(λ) k≤

∞ X n=0

|λ−(n+1) | k An k=

1 . |λ|− k A k 

appliedfunanal.tex; May 24, 2005; 13:58; p. 5

7

1.1. EXISTENSE THEOREMS Corollary 1.1.1 1. Let A be a bounded linear operator in a Banach space H, y ∈ H and α ∈ C be a complex number such that |α|
0 Z x 1 dy (x − y)α−1 f (y) . (Rα f )(x) = Γ(α) 0 n

One can show that Rα Rβ = Rα+β and R0 = Id . So, Rα is a complex power of the operator R (or ∂) Rα = Rα = (∂)−α . By analytic continuation we obtain an operator for any α. In particular, Rn = ∂n . Similar transforms, say on smooth functions of compact support, can be defined as follows Z x 1 (Bα f )(x) = dy (x − y)α−1 f (y) , Γ(α) −∞ Z ∞ 1 (Wα f )(x) = dy (y − x)α−1 f (y) , Γ(α) x Z 0 1 (Cα f )(x) = dy (y − x)α−1 f (y) , Γ(α) x which are related to each other.

1.3.5

Homework

• Exercises: 5.12[9,16]

appliedfunanal.tex; May 24, 2005; 13:58; p. 21

Chapter 2 Ordinary Differential Operators 2.1

Ordinary Differential Operators

• Let I = (a, b), where a can be −∞ and b can be +∞. Let a2 ∈ C 2 (I) be a positive twice-differentiable function, a1 ∈ C 1 (I) be a differentiable function and a0 ∈ C(I) be a continuous function. For simplicity we can just assume that all coefficients are smooth, that is are functions from C ∞ (I) and for any x ∈ I .

a2 (x) , 0,

A second-order ordinary differential operator L is an operator acting on twice differentiable functions by L = a2 ∂2x + a1 ∂ x + a0 . Again, we could restrict first only to smooth functions, that is, restrict first the domain of L to C ∞ (I), or, even, to smooth functions of compact support C0∞ (I). • The boundary data of a function f are the values of the function f and its derivative f 0 at the points a and b. • Let (αi j ) and (βi j ), where i, j = 1, 2, be two constant matrices and (bi ) be a constant 2-vector. Assume that the 4-vectors (α11 , α12 , β11 , β12 ) and (α21 , α22 , β21 , β22 ) are linearly independent. 23

24

CHAPTER 2. ORDINARY DIFFERENTIAL OPERATORS • The boundary conditions are B1 ( f ) = c1 ,

B2 ( f ) = c2 .

• The homogeneous boundary conditions are B1 ( f ) = 0,

B2 ( f ) = 0 .

• The boundary operators are defined by B1 ( f ) = α11 f (a) + α12 f 0 (a) + β11 f (b) + β12 f 0 (b), B2 ( f ) = α21 f (a) + α22 f 0 (a) + β21 f (b) + β22 f 0 (b) . • Let Γ be the boundary data map defined by ! f (a) (Γ f )(a) = , (Γ f )(b) = f 0 (a)

! f (b) , f 0 (b)

• Let A(a) be the matrix (αi j ) and A(b) be the matrix (βi j ). Then the boundary conditions are (AΓ f )(a) + (AΓ f )(b) = C , where C is the column vector C = (ci ). • The separated (or local) boundary operators have the form B1 ( f ) = α11 f (a) + α12 f 0 (a), B2 ( f ) = β21 f (b) + β22 f 0 (b) . • The periodic boundary conditions have the form B1 ( f ) = f (a) − f (b) = 0, B2 ( f ) = f 0 (a) − f 0 (b) = 0 .

•

Definition 2.1.1 Let I = [a, b] and L be a differential operator of order n in L2 (I). The domain D(L) of the differential operator L is the set of all functions whose k-th derivative is square integrable, that is L f ∈ L2 (I), and which satisfy the homogeneous boundary conditions Bi ( f ) = 0, where i = 1, . . . , n − 1. appliedfunanal.tex; May 24, 2005; 13:58; p. 22

2.1. ORDINARY DIFFERENTIAL OPERATORS

•

25

Definition 2.1.2 Let I = [a, b] and L be a differential operator of order n in L2 (I). An operator L∗ is the adjoint of the operator L if for any f ∈ D(L) and any g ∈ D(L∗ ) (L f, g) = ( f, L∗ g) .

• Let L2 (I) have the standard inner product with the weight function equal to 1. The adjoint of the operator L = a2 ∂2x + a1 ∂ x + a0 . is given by L∗ = ∂2x a2 − ∂ x a1 + a0 = a2 ∂2x + (2a02 − a1 )∂ x + (a002 − a01 + a0 ) . • Let f and g be differentiable on [a, b]. The bilinear concomitant of f and g is J( f, g) = a2 [(∂ x f )g − f ∂ x g] + (a1 − a02 ) f g . • For a f ∈ D(L) and any g ∈ L2 (I) we have (L f, g) = ( f, L∗ g) + J( f, g)|ba . • The domain D(L∗ ) of the adjoint L∗ of the operator L is the set of all twicedifferentiable functions g in L2 (I) such that J( f, g)|ba = 0 for any f = D(L). That is the functions that satisfy the adjoint homogeneous boundary conditions B∗1 (g) = B∗2 (g) = 0 . • The operator L = a2 ∂2x + a1 ∂ x + a0 . is formally self-adjoint if a02 = a1 . It can be written in the form L = ∂ x a2 ∂ x + a0 . • The operator L is self-adjoint if it is formally self-adjoint and D(L) = D(L∗ ). appliedfunanal.tex; May 24, 2005; 13:58; p. 23

26

CHAPTER 2. ORDINARY DIFFERENTIAL OPERATORS • For a formally self-adjoint operator L the concomitant has the form J( f, g) = a2 [(∂ x f )g − f ∂ x g] . • Any operator of the form L = a2 ∂2x + a1 ∂ x + a0 . can be made formally self-adjoint by multiplying it by the integrating factor "Z # a1 1 dx exp µ= . a2 a2 • Let ω be a weight function that is positive except possibly at isolated points where ω(x) = 0. • We define the Hilbert space L2 (I, ω) by the inner product Z b ( f, g) = dx ω(x) f (x)¯g(x) . a

• The adjoint of the operator L = a2 ∂2x + a1 ∂ x + a0 with respect to the weighted inner product is L∗ = ω−1 ∂2x ωa2 − ω−1 ∂ x ωa1 + a0 ! ω0 2 0 ∂x = a2 ∂ x + 2a2 − a1 + 2a2 ω " # ω0 ω00 00 0 0 + a2 − a1 + a0 + (2a2 − a1 ) + a2 ω. ω ω • So, the operator L is formally self-adjoint if a1 =

a02

ω0 + a2 . ω

• A formally self-adjoint operator can be written in the form L=

1 ∂ x ωp∂ x + q . ω appliedfunanal.tex; May 24, 2005; 13:58; p. 24

27

2.1. ORDINARY DIFFERENTIAL OPERATORS • The spectrum of the operator L is determined by the equations L fn = λ n fn ,

B1 ( fn ) = B2 ( fn ) = 0,

k fn k= 1 .

• Examples. 1. The operator L = i∂ x acting on smooth functions of compact support on R, C0∞ (R), is symmetric and essentially self-adjoint on L2 (R). That is L∗ is self-adjoint and L ⊂ L∗ . 2. The operator L = i∂ x acting on smooth functions of compact support on R+ , C0∞ (R+ ), where R+ = (0, ∞), is symmetric but does not have any self-adjoint extensions to L2 (R+ ). 3. The operator L = i∂ x acting on smooth functions of compact support on [0, 1] is symmetric. It has a self-adjoint extension to L2 ([0, 1]) defined by the boundary conditions f (0) = f (1) = 0 . The general self-adjoint extension is defined by the boundary conditions f (1) = eiϕ f (0) , where ϕ is an angle. 4. Let I = [0, 1] and a2 (x) , 0 on I. The operator L = a2 ∂2x + a1 ∂ x + a0 acting on smooth functions on [0, 1] with the boundary conditions f (0) = 0,

f 0 (1) = 0 ,

is self-adjoint. appliedfunanal.tex; May 24, 2005; 13:58; p. 25

28

CHAPTER 2. ORDINARY DIFFERENTIAL OPERATORS 5. The operator L = −∂2x acting on smooth functions on [0, 1] with the boundary conditions f (0) = 0 ,

f 0 (0) = 0 ,

is not self-adjoint. 6. The operator L = −∂2x acting on smooth functions on R+ with compact support is symmetric. It has a symmetric extension defined by the boundary conditions f (0) = 0 , which is essentially self-adjoint on L2 (R+ ). The most general self-adjoint extension of the operator L is defined by the boundary conditions cos ϕ f (0) + sin ϕ f 0 (0) = 0 , where ϕ is an arbitrary angle. 7. Legendre, Associated Legendre, Chebyshev, Jacobi, Laguerre, Associated Laguerre, Bessel, Hermite, operators.

2.1.1

Homework

• Exercises: 5.12[17,19,21,] • Find the spectrum (eigenvalues and the eigenfunctions) of the operators L = −∂2x on the interval [a, b] with the following boundary conditions 1. Dirichlet f (a) = f (b) = 0 , 2. Neumann f 0 (a) = f 0 (b) = 0 , 3. Zaremba f (a) = f 0 (b) = 0 , 4. Periodic f (a) = f (b) . appliedfunanal.tex; May 24, 2005; 13:58; p. 26

29

2.2. STURM-LIOUVILLE SYSTEMS

2.2

Sturm-Liouville Systems

• Let I = [a, b] be an interval, ω be a smooth real-valued positive function on I, i.e. ω(x) > 0 for any x ∈ I, be a smooth real-valued positive function on I, i.e. p(x) > 0 for any x ∈ I, and q be a smooth real-valued function on I. • Let L be a differential operator acting on smooth functions, f ∈ C ∞ (I), on I defined by L = −ω−1 ∂ x ωp∂ x + q . • Let B be the boundary operator defined by B f = (u f + v f 0 )|∂I , where u and v are some real-valued functions of the boundary of I, consisting of two points, ∂I = {a, b}, which are not simultaneously zero, that is u2 + v2 > 0 . • Let L2 (I, ω) be the Hilbert space with the weight function ω and k · k be the corresponding norm. A regular Sturm-Lioville system is the boundary value problem Lf = λf , k f k= 1 , with the homogeneous boundary conditions Bf = 0. • If the function p is positive only on an open interval (a, b) but vanishes at one or both endpoints of the interval [a, b], then the boundary condition B f at that point is replaced by the condition that f must be bounded at that point. Such a boundary value problem is called a singular Sturm-Lioville system. • If the functions p, ω, and q are periodic, that is p(a) = p(b), ω(a) = ω(b) and q(a) = q(b), and the boundary conditions are periodic, that is, f (a) = f (b),

f 0 (a) = f 0 (b) ,

then the boundary value problem is called a periodic Sturm-Lioville system. appliedfunanal.tex; May 24, 2005; 13:58; p. 27

30

CHAPTER 2. ORDINARY DIFFERENTIAL OPERATORS • Examples. • The domain of L D(L) = { f ∈ L2 (I, ω)| f 00 ∈ L2 (I, ω), B f |∂I = 0} is the space of all complex valued functions f on I for which f 00 is in L2 (I, ω) and which satisfy the boundary conditions.

•

Theorem 2.2.1 (Lagrange Indentity) . For any f, g ∈ D(L) there holds f Lg − (L f )g = −ω−1 ∂ x [ωp( f ∂ x g − (∂ x f )g)] . Proof: Computation.  Theorem 2.2.2 (Abel’s Formula) . Let f and g be two eigenfunctions of the operator L corresponding to the same eigenvalue λ, L f = λ f,

•

Lg = λg .

Then pW( f, g) = const , where ( f, g) is the Wronskian defined by W( f, g) = f ∂ x g − (∂ x f )g . Proof: Integrate Lagrange identity from a to x and use the boundary conditions. 

•

Theorem 2.2.3 The eigenvalues of a regular Sturm-Lioville system are simple, that is, have multiplicity one. Proof: Use Abel formula and the boundary conditions. 

•

Theorem 2.2.4 L is a self-adjoint operator on L2 (I, ω). Proof: Integrate Lagrange identity from a to b and use the boundary conditions.  appliedfunanal.tex; May 24, 2005; 13:58; p. 28

31

2.2. STURM-LIOUVILLE SYSTEMS •

Theorem 2.2.5 The eigenvalues of a Sturm-Lioville system are real. Proof: Easy. 

• Remark. The eigenvalues of a regular Sturm-Liouville system form an infinite sequence, λn , such that λn → ∞. (Will prove later). •

Theorem 2.2.6 The eigenfunctions corresponding to distinct eigenvalues of a Sturm-Lioville system are orthogonal in L2 (I, ω). Proof: Easy. 

2.2.1

Homework

• Exercises: 5.12[18,19,21,22,23,24,25]

appliedfunanal.tex; May 24, 2005; 13:58; p. 29

32

CHAPTER 2. ORDINARY DIFFERENTIAL OPERATORS

2.3

Green Functions

• Let L be a differential operator acting on functions on I = [a, b] satisfying given boundary conditions. Then the solution of the boundary value problem Lϕ = f, Bϕ = 0 has the form ϕ = L−1 f , or, in integral form, ϕ(x) =

b

Z

dy G(x, y) f (y) ,

a

where G(x, y) is the Green function of the operator L. It is the kernel of the operator L−1 . Theorem 2.3.1 Let I = [a, b] and p, q ∈ C ∞ (I) be smooth real-valued functions on I such that p(x) > 0 for any x ∈ I. Let L be an operator acting on smooth functions on I defined by L = −∂p∂ + q . Let B be the boundary operator defined by Bϕ = (uϕ + vϕ0 )|∂I , where u and v are real-valued functions on ∂I = {a, b} such that • u2 + v2 > 0 . Let ϕ1 and ϕ2 be the non-zero solutions of the equation Lϕ = 0 with the boundary conditions at one point, that is u(a)ϕ1 (a) + v(a)ϕ01 (a) = 0 , u(b)ϕ2 (b) + v(b)ϕ02 (b) = 0 . appliedfunanal.tex; May 24, 2005; 13:58; p. 30

33

2.3. GREEN FUNCTIONS Theorem 2.3.2 Suppose that the operator L with the boundary conditions Bϕ = 0 does not have zero eigenvalue. Let W(ϕ1 , ϕ2 ) be the Wronskian of the solutions ϕ1 , ϕ2 , W(ϕ1 , ϕ2 ) = ϕ1 ϕ02 − ϕ01 ϕ2 and C be the constant defined by C = −pW(ϕ1 , ϕ2 ) . Let f ∈ C(I) be a continuous function on I. Then the Sturm-Liouville boundary value problem Lϕ = f ,

Bϕ = 0 ,

has a unique solution ϕ(x) =

b

Z

dy G(x, y)ϕ(y) , a

where the Green function G(x, y) is defined by  1    ϕ1 (x)ϕ2 (y), for x < y     C G(x, y) =    1     ϕ2 (x)ϕ1 (y), for x ≥ y . C • Remark. Let θ be a function on I defined by     0, for x < y θ(x, y) =    1, for x ≥ y . Then the Green function G(x, y) can be written in the form G(x, y) =

1 1 θ(x, y)ϕ1 (x)ϕ2 (y) + θ(y, x)ϕ2 (x)ϕ1 (y) . C C

• Proof: Use the method of variation of parameters.  appliedfunanal.tex; May 24, 2005; 13:58; p. 31

34

CHAPTER 2. ORDINARY DIFFERENTIAL OPERATORS Theorem 2.3.3 Let L be as described above. The operator L−1 : L2 (I) → C(I)

•

is a self-adjoint compact operator. • Proof: 1. L−1 is compact because it is an integral operator with a continuous kernel. 2. L−1 is self-adjoint because its kernel is self-adjoint G(x, y) = G(y, x) . 3. The range of the operator L−1 is C(I) since the Green function G(x, y) is continuous in the first argument.  • Remark. If the operator L has smooth coefficients, then the operator L−1 is a smoothing operator L−1 : L2 (I) → C ∞ (I) . •

Theorem 2.3.4 Let L be the operator with the boundary conditions defined above. Then the operator L−1 does not have a zero eigenvalue.

• Proof: Calculation. 

•

Theorem 2.3.5 Let L be the operator with the boundary conditions defined above. Let {(λn , ϕn )} be the eigenvalues and the eigenfunctions of the operator L and {(µn , ψn )} be the eigenvalues and the eigenfunctions of the operator L−1 . Then µn =

1 , λn

ψn = ϕn .

• Proof: Obvious.  • Properties of the spectrum. Let L be as described above. Then: appliedfunanal.tex; May 24, 2005; 13:58; p. 32

35

2.3. GREEN FUNCTIONS 1. The operator L is self-adjoint. 2. The spectrum of the operator L is real. 3. The spectrum of the operator L is bounded below. 4. The spectrum is discrete. 5. All eigenvalues are simple, that is, have multiplicity one.

6. The eigenvalues form an unbounded monotonically increasing sequence. 7. There are at most finitely many negative eigenvalues. 8. The eigenfunctions form an orthonormal basis in L2 ([a, b]). 9. The resolvent of the operator L is a Hilbert-Schmidt operator. • The spectrum λn of the operator L is defined by Lϕn = λn ϕn ,

k ϕn k= 1 .

• Then (ϕn , ϕm ) = δnm and

∞ X

ϕn (x)ϕ¯ n (x0 ) = δ(x − x0 ) ,

n=1

where δ(x−x0 ) is the kernel of the identity operator (the Dirac delta-function). • Let λ ∈ C be in the resolvent set of the operator L, that is the operator (L − λI)−1 be a bounded operator. • The resolvent satisfies the equation (L x − λI)G(λ; x, x0 ) = δ(x − x0 ) , the self-adjointness condition ¯ x0 x) , G(λ; x, x0 ) = G(λ; and the boundary conditions in both variables. • The resolvent is given by G(λ; x, x0 ) =

∞ X n=1

1 ϕn (x)ϕ¯ n (x0 ) . λn − λ appliedfunanal.tex; May 24, 2005; 13:58; p. 33

36

CHAPTER 2. ORDINARY DIFFERENTIAL OPERATORS • Let C be a contour in the complex plane going around the spectrum in the counterclockwise direction from ∞ + iε to ∞ − iε. Then a function f (L) of the operator L can be defined in terms of the resolvent by the Cauchy formula Z 1 dλ f (λ)G(λ) . f (L) = − 2πi C The kernel of this operator is Z 1 f (L)(x, x ) = − dλ f (λ)G(λ; x, x0 ) 2πi C ∞ X f (λn )ϕn (x)ϕ¯ n (x0 ) . = 0

n=1

• In some cases, depending on the function f one can define the L2 -trace by Z

TrL2 f (L) =

b

dx f (L)(x, x) a ∞ X

=

f (λn ) .

n=1

• In particular, for t > 0 we define the heat kernel Z 1 U(t; x, x ) = − dλ e−tλG(λ; x, x0 ) 2πi C ∞ X = e−tλn ϕn (x)ϕ¯ n (x0 ) 0

n=1

and its trace Θ(t) = Tr L2 exp(−tL) Z b = dx U(t; x, x) a

=

∞ X

e−tλn .

n=1

appliedfunanal.tex; May 24, 2005; 13:58; p. 34

37

2.3. GREEN FUNCTIONS

• Let µ ∈ R be a real number such that µ < λ1 and s ∈ C be a complex number such that Re s > 21 . Then we define the kernel of the zeta-function Z 1 1 s 0 G (µ; x, x ) = − dλ G(λ; x, x0 ) 2πi C (λ − µ) s ∞ X 1 = ϕn (x)ϕ¯ n (x0 ) , s (λn − µ) n=1 and the zeta-function ζ(s, µ) = Tr L2 (L − µI)−s Z b = dx G s (µ; x, x) a

=

∞ X n=1

1 . (λn − µ) s

• Determinant of the differential operator Det L2 (L − µI) = exp[−∂ s ζ(0, µ)] . • Expression of the resolvent and the zeta function in terms of the heat kernel Z ∞ 0 G(λ; x, x ) = dt etλ U(t; x, x0 ) 0

and

1 G (µ; x, x ) = Γ(s)

Z

∞

0

s

dt t s−1 etλ U(t; x, x0 ) .

0

• Asymptotics of the resolvent and the heat kertnel. As x → x0 and λ → −∞ 0

0

G(λ; x, x ) ∼ |x − x |

∞ X n X

0 n−k b(n) (x − x0 )n k (x )(−λ)

n=0 k=0 ∞ X ∞ X −1/2 0 n−k +(−λ) c(n) (x k (x )(−λ) n=0 k=0

− x0 )n ,

As t → 0 and x → x0 # ∞ ∞ (x − x0 )2 X X n (k) 0 t an (x )(x − x0 )k . exp − 4t n=0 k=0 "

0

−1/2

U(t; x, x ) ∼ (4πt)

appliedfunanal.tex; May 24, 2005; 13:58; p. 35

38

CHAPTER 2. ORDINARY DIFFERENTIAL OPERATORS

2.3.1

Operators with Constant Coefficients

• Let us consider the space C0∞ (R) of all smooth functions of compact support (which is dense in L2 (R)). Let ∂ be the operator of differentiation. • A differential operator of order m with constant coefficients is a polynomial in the operator ∂ of degree m with constant coefficients, that is, L = Pm (∂) =

m X

ak (∂)k ,

k=0

where ak are some constants and am , 0. • Let

" K(x) = F −1

1 Pm (iω)

!# (x) .

• The Green function of the operator L is G(x, y) = K(x − y) . • Let f ∈ L2 (R). Then the solution of the equation Lϕ = f , is given by ϕ=K∗ f that is ϕ(x) = (2π)

Z

∞

−1/2

dy K(x − y) f (y) .

−∞

2.3.2

Examples

• Let L be the operator L = −∂2 on I ⊆ R with different boundary conditions. Let λ ∈ C and G(λ; x, y) be the resolvent kernel of the operator √ L, that is, the kernel of the operator −1 (L − λI) . Let Re λ < 0 and µ = −λ such that Re µ > 0. appliedfunanal.tex; May 24, 2005; 13:58; p. 36

39

2.3. GREEN FUNCTIONS 1. Dirichlet operator. Let I = [a, b] with the boundary conditions ϕ(a) = ϕ(b) = 0 . Eigenfunctions r ϕn (x) =

x − a  2 sin nπ , b−a b−a

Eigenvalues

n = 1, 2, 3, . . . .

 π 2 λn = n2 b−a

The lowest eigenvalue λmin

 π 2 = . b−a

Resolvent (   1 G(λ; x, y) = −θ(x − y) sinh µ(x − y) µ    ) sinh µ(b − y) sinh µ(x − a) +   sinh µ(b − a) (   1 = exp −µ|x − y| 2µ     cosh µ(x − y) exp −µ(b − a) +   sinh µ(b − a)  ) cosh µ(x + y − b − a) −   sinh µ(b − a) 2. Neumann operator. Let I = [a, b] with the boundary conditions ϕ0 (a) = ϕ0 (b) = 0 . Eigenfunctions r ϕn (x) =

x − a  2 cos nπ , b−a b−a

n = 0, 1, 2, . . . .

appliedfunanal.tex; May 24, 2005; 13:58; p. 37

40

CHAPTER 2. ORDINARY DIFFERENTIAL OPERATORS Eigenvalues λn =

 π 2 n2 b−a

The lowest eigenvalue λmin = 0 . Resolvent (   1 G(λ; x, y) = −θ(x − y) sinh µ(x − y) µ    ) cosh µ(b − y) cosh µ(x − a)   + sinh µ(b − a) (   1 = exp −µ|x − y| 2µ     cosh µ(x − y) exp −µ(b − a) +   sinh µ(b − a)  ) cosh µ(x + y − b − a) +   sinh µ(b − a) 3. Zaremba operator. Let I = [a, b] with the boundary conditions ϕ0 (a) = ϕ(b) = 0 . Eigenfunctions r ϕn (x) =

" ! # 2 x−a 1 cos n+ π , b−a b−a 2

n = 0, 1, 2, . . . .

Eigenvalues !2  π 2 1 λn = n+ b−a 2 The lowest eigenvalue λmin =

 π 2 1 . b−a 4 appliedfunanal.tex; May 24, 2005; 13:58; p. 38

41

2.3. GREEN FUNCTIONS Resolvent (   1 −θ(x − y) sinh µ(x − y) G(λ; x, y) = µ    ) sinh µ(b − y) cosh µ(x − a) +   cosh µ(b − a) (   1 = exp −µ|x − y| 2µ     cosh µ(x − y) exp −µ(b − a) −   cosh µ(b − a)  ) sinh µ(x + y − b − a) −   cosh µ(b − a)

4. Laplacian on the real line. Let I = R with the boundedness condition at ±∞. Spectrum is [0, ∞) with no discrete eigenvalues. Resolvent   1 exp −µ|x − y| G(λ; x, y) = 2µ 5. Dirichlet operator on half-line. Let I = R+ with the boundedness condition at ∞ and Dirichlet condition at zero ϕ(0) = 0 . Spectrum is [0, ∞) with no discrete eigenvalues. Resolvent G(λ; x, y) =

    1  exp −µ|x − y| − exp −µ(x + y) . 2µ

6. Neumann operator on half-line. Let I = R+ with the boundedness condition at ∞ and Neumann condition at zero ϕ0 (0) = 0 . Spectrum is [0, ∞) with no discrete eigenvalues. Resolvent G(λ; x, y) =

    1  exp −µ|x − y| + exp −µ(x + y) . 2µ appliedfunanal.tex; May 24, 2005; 13:58; p. 39

42

CHAPTER 2. ORDINARY DIFFERENTIAL OPERATORS

2.3.3

Homework

• Exercises:

appliedfunanal.tex; May 24, 2005; 13:58; p. 40

Chapter 3 Distributions and Partial Differential Equations 3.1

Distributions

• Let x = (xµ ) = (x1 , . . . , xn ) be Cartesian coordinates in Rn . • We consider the space C ∞ (Rn ) of smooth functions on Rn . • The operators

∂ ∂xµ are partial differential operators acting on smooth functions. D µ = ∂µ =

• Let α = (α1 , . . . , αn ),

αi ≥ 0 ,

be a multiindex with αi be nonnegative integers and |α| = α1 + · · · αn be the norm (length) of the multiindex α. • We denote by α

D =

Dα1 1

· · · Dαn n

∂|α| = ∂(x1 )α1 · · · ∂(xn )αn

a partial differential operator of order |α|. 43

44CHAPTER 3. DISTRIBUTIONS AND PARTIAL DIFFERENTIAL EQUATIONS • Let aα = aα1 ...αn be smooth functions on Rn . Then the operator X L= aα (x)Dα |α|≤m

is a linear partial differential operator of order m. • Alternatively, any differential operator of order m can be written in the form L=

m X

aµ1 ...µk ∂µ1 · · · ∂µk ,

k=0

where aµ1 ...µk are smooth functions over Rn and a summation over repeated indices is understood. • The formal adjoint of the operator L with respect to the L2 inner product with unit weight is X L∗ = (−1)|α| Dα aα . |α|≤m

• Let ξ = (ξµ ) = (ξ1 , . . . , ξn ) be a vector in Rn and ξα = ξ1α1 · · · ξnαn . The symbol of the operator L is a function on Rn × Rn defined by X σ(L; x, ξ) = aα (x)ξα . |α|≤m

• The principal part of the operator L is X Lp = aα (x)Dα . |α|=m

• The principal symbol of the operator L is a function on Rn × Rn defined by X σ p (L; x, ξ) = aα (x)ξα . |α|=m

appliedfunanal.tex; May 24, 2005; 13:58; p. 41

45

3.1. DISTRIBUTIONS

• Idea of distributions. Let g be a given function. Consider the differential equation Lf = g. Let us multiply it by a “nice” function ϕ. Then (L f, ϕ) = (g, ϕ) or ( f, L∗ ϕ) = (g, ϕ) . A function f may satisfy this equation without being m times differentiable! • Notice that if f ∈ C(Rn ) is continuous and ϕ ∈ C0∞ (R) is smooth with compact support, then the equation ( f, ϕ) = 0 implies f = 0. •

Definition 3.1.1 (Test Functions.) A test function is a smooth function with compact support in Rn . The space of the test functions is denoted by D(Rn ). That is D(Rn ) = C0∞ (Rn ).

• Example. Let |x| =

(x1 )2 + · · · + (xn )2 . Then !   1    , if |x| < 1  exp |x|2 − 1 ϕ(x) =      0, if |x| ≥ 1 p

is a test function. Theorem 3.1.1 The space D(Rn ) is a vector space. Let ϕ, ψ ∈ D(Rn ), f ∈ C ∞ (Rn ) and A : Rn → Rn be an affine transformation (that is a nondegenerate linear transformation). Then •

1. f ϕ ∈ D(Rn ), that is D(Rn ) is a module over C ∞ (Rn ), 2. ϕ ◦ A ∈ D(Rn ), that is D(Rn ) is invariant under the action of the group of affine transformations, 3. ϕ ∗ ψ ∈ D(Rn ), that is D(Rn ) is closed under the convolution. Proof: Exercise.  appliedfunanal.tex; May 24, 2005; 13:58; p. 42

46CHAPTER 3. DISTRIBUTIONS AND PARTIAL DIFFERENTIAL EQUATIONS Definition 3.1.2 (Convergence of Test Functions.) Let ϕ ∈ D(Rn ) be a test function and (ϕn ) be a sequence of test functions. Then the sequence (ϕn ) converges to ϕ in D(Rn ) if: 1. there is a compact set B ⊂ Rn (independent of n) such that all ϕn vanish outside B, that is •

∞ [

supp ϕn ⊂ B ,

n=1

2. for any α

Dα ϕn → Dα ϕ

uniformly on Rn . • Notation. Convergence in D(Rn ) is denoted by D

ϕn → ϕ • Examples. Theorem 3.1.2 Let (ϕn ) and (ψn ) be sequences in D(Rn ) and ϕ, psi ∈ D(Rn ) be such that D

ϕn → ϕ

and

D

ψn → ψ .

let a, b ∈ C, f ∈ C ∞ (Rn ), A be an affine transformation of Rn , and α be a multiindex. Then: •

D

1. aϕn + bψn → aϕ + bψ, D

2. f ϕn → f ϕ, D

3. ϕn ◦ A → ϕ ◦ A, D

4. Dα ϕn → Dα ϕ. Proof: Exercise.  appliedfunanal.tex; May 24, 2005; 13:58; p. 43

47

3.1. DISTRIBUTIONS Definition 3.1.3 A distribution (or a generalized function) F on Rn is a continuous linear functional on D(Rn ). That is, a distribution F on Rn is a mapping F : Rn → C such that •

1. F(aϕ + bψ) = aF(ϕ) + bF(ψ) for any a, b ∈ C and ϕ, ψ ∈ D(Rn ), 2. F(ϕn ) → F(ϕ) in C for any sequence (ϕn ) in D(Rn ) converging to D

ϕ ∈ D(Rn ) in D(Rn ), i.e. ϕn → ϕ. • Notation. The space of all distributions is denoted by D0 (Rn ). The action of the distribution F on a test function ϕ is denoted by F(ϕ) = hF, ϕi . • Every locally integrable function f can be identified with a distribution F by hF, ϕ, i = (ϕ, f¯) . Definition 3.1.4 (Regular and Singular Distributions.) A distribution F ∈ D0 (Rn ) is called regular if there is a locally integrable function f such that for any test function ϕ ∈ D(Rn ), • hF, ϕi = (ϕ, f¯) . A distribution which is not regular is called singular. • Heaviside distribution. Let Ω ⊂ Rn be an open set in Rn and χΩ be the characteristic function of Ω. The Heaviside distribution H is a regular distribution defined by Z hF, ϕi = (ϕ, χΩ ) = ϕ. Ω

If Ω = (0, ∞) × · · · × (0, ∞), then the distribution Z ∞ Z ∞ hH, ϕi = ··· dx1 · · · dxn ϕ(x) . 0

0

• Dirac distribution δ is a singular distribution defined by hδ, ϕi = ϕ(0) . appliedfunanal.tex; May 24, 2005; 13:58; p. 44

48CHAPTER 3. DISTRIBUTIONS AND PARTIAL DIFFERENTIAL EQUATIONS • Let A be an operator defined on test functions, i.e. on D(Rn ). We can extend it to distributions, i.e. to D0 (Rn ),by hAF, ϕi = hF, A∗ ϕi . Definition 3.1.5 (Derivative of Distributions) The derivative ∂µ F of a distribution F ∈ D0 (Rn ) is defined by h∂µ F, ϕi = hF, (−∂µ )ϕi . •

More generally,

hDα F, ϕi = hF, (−1)|α| Dα ϕi ,

and for any differential operator L hLF, ϕi = hF, L∗ ϕi . •

Theorem 3.1.3 Let F be a distribution and L be a differential operator with smooth coefficients. Then LF is a distribution. Proof: Prove the linearity and continuity of Dα F. 

• Example.

hDα δ, ϕi = (−1)|α| Dα ϕ(0) .

Definition 3.1.6 (Weak Distributional Convergence.) A sequence of distributions (Fn ) in D0 (Rn ) converges to a distribution F ∈ D0 (Rn ) if for every ϕ ∈ D(Rn ) • hFn , ϕi → hF, ϕi . This convergence is called weak distributional convergence. • Examples. 1. Let f, fn ∈ C(Rn ) and fn → f uniformly on any compact subset of Rn . Let F, Fn be the corresponding regular distributions. Then Fn → F. 2. Let f, fn ∈ L1 (Rn ) and fn → f in L1 (Rn ). Let F, Fn be the corresponding regular distributions. Then Fn → F. appliedfunanal.tex; May 24, 2005; 13:58; p. 45

49

3.1. DISTRIBUTIONS

• Terminology. If a sequence of regular distributions Fn corresponding to a sequence of functions fn converges to a distribution F, then we say that the sequence of functions fn converges in the distributional sense (or converges distributionally) to the distribution F. • Example. Sequences of functions converging distributionally to the Dirac distribution. Let n ∈ N. 1. fn (x) = 2.

1 n . π 1 + n2 x 2

n 2 2 fn (x) = √ e−n x . π

3. fn (x) =

•

sin(nx) . πx

Theorem 3.1.4 Let (Fn ) be a sequence in D0 (Rn ) converging to F ∈ D0 (Rn ), that is Fn → F. Then for any multiindex α Dα F n → Dα F . Proof: Note that hDα Fn , ϕi → (−1)|α| hF, Dα ϕi = hDα F, ϕi . 

•

Definition 3.1.7 (Antiderivative of Distributions.) Let F, G ∈ D0 (R) be distributions on R. Then G is an anti-derivative of F if G0 = F.

•

Theorem 3.1.5 Every distribution on R has an anti-derivative. Proof: 1. Let ϕ0 ∈ D(R) be a test function such that Z ϕ0 = 1 .

appliedfunanal.tex; May 24, 2005; 13:58; p. 46

50CHAPTER 3. DISTRIBUTIONS AND PARTIAL DIFFERENTIAL EQUATIONS 2. Let ϕ ∈ D(R) and K=

Z

ϕ.

Then there is a decomposition ϕ = Kϕ0 + ϕ1 , where

Z

3. Let ψ(x) =

ϕ1 = 0 . Z

x

dt ϕ1 (t) .

−∞

Let F, G ∈ D0 (R) such that hG, ϕi = KC0 − hF, ψi , where C0 is a constant. Then G0 = F .  •

Theorem 3.1.6 Let F ∈ 0 (R) be a distribution on R. If F 0 = 0, then F is a constant function. Proof: 1. Let ϕ = Kϕ0 + ϕ1 ∈ D(R) , where

Z

Z

ϕ0 = 1 ,

and K= Let ψ(x) =

Z

Z

ϕ1 = 0 ,

ϕ.

x

dt ϕ1 (t) .

−∞

appliedfunanal.tex; May 24, 2005; 13:58; p. 47

51

3.1. DISTRIBUTIONS 2. Then hF, ϕ1 i = hF, ψ0 i = −hF 0 , ψi = 0 . 3. Therefore, hF, ϕi = KhF, ϕ0 i =

Z

Cϕ ,

where C = hF, ϕ0 i. 4. Thus, F is a regular distribution generated by the constant function C. 

•

Definition 3.1.8 (The Space D(Ω)) . Test Functions on an Open Set Ω of Rn . Let Ω ⊂ Rn be an open set in Rn . The space D(Ω) of test functions is the space C0∞ (Ω) of smooth functions with compact support contained in Ω. Definition 3.1.9 (Convergence in D(Ω)) . Let ϕ ∈ D(Ω) be a test function and (ϕn ) be a sequence of test functions on Ω. Then the sequence (ϕn ) converges to ϕ in D(Ω), denoted ϕn → ϕ in D(Ω), if: 1. there is a compact set B ⊂ Ω (independent of n) such that all ϕn vanish outside B, that is

•

∞ [

supp ϕn ⊂ B ,

n=1

2. for any α

Dα ϕn → Dα ϕ

uniformly on Ω.

•

3.1.1

Definition 3.1.10 (The Space D0 (Ω)) . Distributions on an Open Set Ω of Rn . The space D0 (Ω) of distributions is the space of all continuous linear functionals on D(Ω).

Homework

• Exercises:

appliedfunanal.tex; May 24, 2005; 13:58; p. 48

52CHAPTER 3. DISTRIBUTIONS AND PARTIAL DIFFERENTIAL EQUATIONS

3.2 3.2.1

Fundamental Solutions of Partial Differential Equations Boundary Value Problems Definition 3.2.1 Let m ∈ N and let L : C ∞ (Rn ) → C ∞ (Rn ) be a differentiable operator of order m with smooth coefficients acting on smooth functions on Rn . 1. Let f ∈ C m (Rn ) be a m times differentiable function on Rn . Then a function u ∈ C m (Rn ) such that (Lu)(x) = f (x) ,

for all x ∈ Rn

is the classical solution of the equation Lu = f . •

2. Let f be a function or a distribution on Rn . Then a function u on Rn such that (u, L∗ ϕ) = ( f, ϕ) for every test function ϕ ∈ D(Rn ) is the weak solution of the equation Lu = f . 3. Let f ∈ D0 (Rn ) be a distribution on Rn . Then a distribution u ∈ D0 (Rn ) such that Lu = f is the distributional solution of the equation Lu = f .

• Examples.

•

Definition 3.2.2 Let L be a differential operator. A fundamental solution G of L is a distributional solution of the equation LG = δ .

• A fundamental solution is not unique. • Let f be a function, G be a fundamental solution of the operator L and u = f ∗ G. Then u satisfies the equation Lu = f . appliedfunanal.tex; May 24, 2005; 13:58; p. 49

3.2. FUNDAMENTAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS53 • The Heaviside distribution is the fundamental solution of the operator L = ∂x . • Let Ω ⊂ Rn be a bounded open set in Rn with a smooth boundary ∂Ω. Let N be the outward pointing unit normal to Ω and ∇N be the normal derivative. Let f be a function on Rn . The homogeneous boundary value problem is the equation Lu = f in Ω , with the boundary conditions Bu ∂Ω = 0 , where B is the boundary operator. One considers the following boundary operators: Dirichlet : Neumann : Robin :

B = 1, B = ∇N , B = ∇N + g ,

where g is a smooth function on ∂Ω. •

3.2.2

Definition 3.2.3 A Green function G of the boundary value problem (L, B) is a fundamental solution of the operator L that satisfies the homogeneous boundary conditions.

Equations of Mathematical Physics

• Let x ∈ Rn be the space coordinates and t ∈ R be the time. • Let gµν be the components of the metric tensor in some local coordinates. The Laplace Operator (or Laplacian) is a second-order partial differential operator acting on smooth functions of the form ∆ = g−1/2 ∂µ g1/2 gµν ∂ν . In Cartesian coordinates in Rn the Laplacian has the form µν

∆ = δ ∂µ ∂ν =

n X

∂2µ .

µ=1

appliedfunanal.tex; May 24, 2005; 13:58; p. 50

54CHAPTER 3. DISTRIBUTIONS AND PARTIAL DIFFERENTIAL EQUATIONS • The D’Alambert Operator (or D’Alambertian) is a second-order partial differential operator  = −∂2t + ∆ . • Laplace Equation ∆u = 0 . • Poisson Equation ∆u = f . • Wave Equation −u = (∂2t − ∆)u = f . Homogeneous wave equation u = (∂2t − ∆)u = 0 . • Heat Equation (∂t − ∆)u = f . Homogeneous heat equation (∂t − ∆)u = 0 . • Telegrapher Equation (∂2t + 2∂t − ∆ + q)u = 0 , where q is a constant. • Helmholtz Equation (k2 + ∆)u = f , Homogeneous Helmholtz equation (k2 + ∆)u = 0 , where k is a constant. • Biharmonic Wave Equation. (∂2t + ∆2 )u = 0 . appliedfunanal.tex; May 24, 2005; 13:58; p. 51

3.2. FUNDAMENTAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS55 • Biharmonic Equation. ∆2 u = 0 . • Schr¨odinger Equation. (i∂t − ∆ + V)u = 0 , where V is a function on Rn . • Stationary Schr¨odinger Equation. (−∆ + V − E)u = 0 , where E is a constant. • Klein-Gordon Equation. (− + m2 )u = (∂2t − ∆ + m2 )u = 0 , where m is a constant.

3.2.3

Green’s Formulas

• Let Ω be a bounded open set in Rn with a smooth boundary ∂Ω. Let ∂Ω be parametrized by xµ = xµ ( xˆ) , where xˆi , i = 1, . . . , n − 1, are the parameters (local coordinates on ∂Ω). • Let gµν be the metric in Ω (equal to δµν in Cartesian coordinates). The induced metric gˆ i j on ∂Ω is defined by ∂xµ ∂xν gˆ i j ( xˆ) = gµν (x( xˆ)) i j . ∂ xˆ ∂ xˆ • The volume elements in Ω and ∂Ω are defined by p p dvol(x) = dx g(x) , dvol( xˆ) = d xˆ gˆ ( xˆ) where g = det gµν ,

gˆ = det gi j . appliedfunanal.tex; May 24, 2005; 13:58; p. 52

56CHAPTER 3. DISTRIBUTIONS AND PARTIAL DIFFERENTIAL EQUATIONS • The vectors eµi =

∂xµ ∂ xˆi

are tangent to ∂Ω and the vector n−1 N ν = gνµ εµ1 ···µn−1 µ eµ11 · · · eµn−1

defines the normal to ∂Ω. The unit normal is defined by nµ = p

Nµ gαβ N α N β

.

• The normal derivative is defined by ∇N = nµ ∇µ . • Let (u, v) =

Z dxu(x)¯v(x) Ω

denote the L2 inner product in Ω and Z dvol( xˆ)u( xˆ)¯v( xˆ) hu, vi = ∂Ω

be the L2 inner product on ∂Ω. • Green’s First Identity. (u, ∆v) = −(∇u, ∇v) + hu, ∇N vi . In particular, (u, ∆u) = −(∇u, ∇u) + hu, ∇N ui . • If the boundary conditions are such that the boundary contribution vanishes, then −(u, ∆u) = (∇u, ∇u) > 0 , so that, (−∆) is a non-negative operator. • Green’s Second Identity. (u, ∆v) − (∆u, v) = hu, ∇N vi − h∇N u, vi . appliedfunanal.tex; May 24, 2005; 13:58; p. 53

3.2. FUNDAMENTAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS57 • More generally, let L be the operator L = ∆ + aµ ∂µ + b . Then (Lu, v) − (u, L∗ v) = h∇N u, vi − hu, ∇N vi + haN u, vi , where aN = aµ nµ . • In particular case a = 0, we obtain a self-adjoint operator (Schr¨odinger operator) L = ∆ + b. • For a self-adjoint operator L = ∆ + b we have (Lu, v) + (∇u, ∇v) − (bu, v) = h∇N u, vi . • The bilinear form E(u, v) = (∇u, ∇v) − (bu, v) is called the Dirichlet integral of the operator L (the energy functional). If b > 0, then for any u , 0 E(u, u) > 0 . • Let L=

X

aα Dα .

|α|≤m

Then (Lu, v) − (u, L∗ v) = hJu, vi − hu, J ∗ vi , where J is a differential operator of order m − 1.

3.2.4

Green Functions

• The Green function of the Laplace operator L = −∆ in Rn . It can be obtained by the Fourier transform Z 1 −n G(x, y) = (2π) d p eip(x−y) 2 , |p| Rn where px = pµ xµ and |p| =

p δµν pν pµ . appliedfunanal.tex; May 24, 2005; 13:58; p. 54

58CHAPTER 3. DISTRIBUTIONS AND PARTIAL DIFFERENTIAL EQUATIONS For n ≥ 3 we compute G(x, y) = (4π)

−n/2

Γ

n 2

 −1

2n−2 . |x − y|(n−2)

For n = 3 one obtains

1 1 . 4π |x − y| For n = 2 one needs to use a regularization to get G(x, y) =

G(x, y) = −

1 ln |x − y|2 . 4π

• The Green function of the Helmholtz operator L = −∆ + m2 in Rn . It can be obtained by the Fourier transform Z 1 −n G(x, y) = (2π) , d p eip(x−y) 2 |p| + m2 Rn Further, by using

Z ∞ 1 2 2 = ds e−s(|p| +m ) 2 2 |p| + m 0 and computing the Gaussian integral over p # " Z |x − y|2 −n ip(x−y)−s|p|2 −n/2 (2π) dp e = (4πs) exp − 4s Rn

we obtain G(x, y) =

Z

∞

|x − y|2 exp −sm − 4s "

−n/2

ds (4πs) 0

#

2

• The Green function of the Helmholtz operator L = ∆ + k2 in R3 with the radiation asymptotic condition lim r(∂r + ik)u = 0

r→∞

is obtained by the direct solution of the radial Laplace equation ! 2 1 2 2 ∂r + ∂r + k G = δ(r) . r 4πr2 It has the form G(x, y) = −

1 e−ik|x−y| . 4π |x − y| appliedfunanal.tex; May 24, 2005; 13:58; p. 55

3.2. FUNDAMENTAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS59 • The Green function of the heat operator ∂t − ∆ in Rn . It can be obtained by the Fourier transform Z 0 0 2 0 0 −n G(t, x; t , x ) = (2π) d p eip(x−x )−(t−t )|p| . Rn

It has the form # |x − y|2 exp − . 4(t − t0 ) "

G(t, x; t , x ) = (4π(t − t )) 0

0

0

−n/2

The solution of the problem (∂t − ∆)u = q with the initial condition u|t=0 = f , is given by Z Z t Z 0 0 0 0 u(t, x) = dx G(t, x; 0, x ) f (x ) + dt dx0 G(t, x; t0 , x0 )q(t0 , x0 ) . Rn

Rn

0

• The Green function of the wave operator −∂2t +∆ in Rn . It can be obtained by the Fourier transform ) ( Z sin(t|p|) 0 0 −n ip(x−x0 ) + B cos(t|p|) . G(t, x; t , x ) = (2π) dp e A |p| Rn where A and B are constants determined from the supplementary conditions. The solution of the problem (−∂2t + ∆)u = −p with the initial conditions u|t=0 = 0 ,

∂t u|t=0 = 0 ,

and the asymptotic condition lim u = 0

|x|→0

is given by u(t, x) =

t

Z

Z dt

0

0

dx0 G(t, x; t0 , x0 )p(t0 , x0 ) .

Rn

appliedfunanal.tex; May 24, 2005; 13:58; p. 56

60CHAPTER 3. DISTRIBUTIONS AND PARTIAL DIFFERENTIAL EQUATIONS • The Green function of the Klein-Gordon operator −∂2t + ∆ − m2 in Rn with the initial conditions u|t=0 = ∂t u|t=0 = 0 , and the boundary conditions lim = 0 .

|x|→∞

It can be obtained by the Fourier transform p ( Z sin(t |p|2 + m2 ) 0 0 −n ip(x−x0 ) G(t, x; t , x ) = (2π) dp e A p Rn |p|2 + m2 ) p +B cos(t |p|2 + m2 ) , where A and B are constants determined from the supplementary conditions. The solution of the problem (−∂2t + ∆ − m2 )u = −p with the initial conditions u|t=0 = f ,

∂t u|t=0 = g ,

can be written as u(t, x) =

t

Z

Z dt

Rn

0

+

dx0 G(t, x; t0 , x0 )p(t0 , x0 )

0

Z

  dx0 G(t, x; 0, x0 )g(x0 ) − ∂t0 G(t, x; 0, x0 ) f (x0 ) .

Rn

Retarded Green function Advanced Green function Symmetric Green function Feynmann Green function Schwinger function Pauli-Jordan function Hadamard function appliedfunanal.tex; May 24, 2005; 13:58; p. 57

3.2. FUNDAMENTAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS61

3.2.5

Homework

• Exercises: 6.6[27,28,29,32]

appliedfunanal.tex; May 24, 2005; 13:58; p. 58

62CHAPTER 3. DISTRIBUTIONS AND PARTIAL DIFFERENTIAL EQUATIONS

3.3

Weak Solutions of Elliptic Boundary Value Problems

• Let Ω ⊂ Rn be a bounded open set in Rn with a smooth boundary ∂Ω. Let L = −∆ be the Laplacian acting on smooth functions on Ω and f ∈ C ( Ω) be a continuous function. We consider the Dirichlet boundary value problem: Lu = f

in Ω

with the boundary condition u|∂Ω = 0 . ¯ • A classical solution of this boundary value problem is a function u ∈ C 2 (Ω) that satisfies the equation at every point x ∈ Ω. • Let ϕ ∈ D(Ω). Then for any classical solution we have (∇u, ∇ϕ) = ( f, ϕ) . • If f is not continuous then there is no classical solution. • Let H01 (Ω) be the subspace of the Sobolev space H 1 (Ω) consisting of functions satisfying Dirichlet boundary conditions. The space H01 (Ω) is the closure of D(Ω) in H 1 (Ω). • For L2 functions the derivatives are understood in the generalized sense. Then for any u ∈ H 1 (Ω0 ), it follows ∇u ∈ L2 (Ω). • Let f ∈ L2 (Ω), then a u ∈ H01 (Ω) satisfying the equation (∇u, ∇ϕ) = ( f, ϕ), for every ϕ ∈ H01 (Ω) is a weak solution of the boundary value problem. This is the variational formulation (or weak formulation) of the boundary value problem. appliedfunanal.tex; May 24, 2005; 13:58; p. 59

3.3. WEAK SOLUTIONS OF ELLIPTIC BOUNDARY VALUE PROBLEMS63 Theorem 3.3.1 Let Ω ⊂ Rn be a bounded open set with a boundary ∂Ω. Let H01 (Ω) be the Sobolev space of functions u such that ∇u ∈ L2 (Ω) and u|∂Ω = 0. Let f ∈ L2 (Ω) and J : H01 (Ω) → R be a functional on H01 (Ω) defined by 1 J(v) = (∇v, ∇v) − ( f, v) . 2 Then: • 1. For any v ∈ H01 (Ω) there is a unique u ∈ H01 (Ω) satisfying the equation (∇u, ∇v) = ( f, v) . 2. A function u ∈ H01 (Ω) is a solution of the equation (∇u, ∇v) = ( f, v), where v ∈ H01 (Ω), if and only if u is a function at which the minimum of the functional J is attained. Proof: 1. Let a be a bilinear form on H01 (Ω) defined by a(u, v) = (∇u, ∇v) . 2. Let k u k21 = (∇u, ∇u) + (u, u) . 3. We have the Friedrichs’ first inequality: there is constants α > 0, such that for any u ∈ H01 (Ω) (u, u) ≤ α(∇u, ∇u) ≤ α k u k21 . 4. Therefore, a(u, u) = (∇u, ∇u) ≥ K k u k21 , where K = min{1/2, 1/(2α)}. 5. We also have a(u, u) ≤k u k21 . 6. Thus, a is a bounded symmetric and coercive (elliptic) bilinear form on H01 (Ω). 7. By the Lax-Milgram theorem there exists a solution in H01 (Ω) of the equation a(u, v) = ( f, v) . appliedfunanal.tex; May 24, 2005; 13:58; p. 60

64CHAPTER 3. DISTRIBUTIONS AND PARTIAL DIFFERENTIAL EQUATIONS  • Neumann Boundary Value Problem. Let b > 0 be a positive constant and L = −∆ + b. The Neumann boundary value problem is the problem Lu = f

in Ω

with the boundary condition ∇N u|∂Ω = 0 . Every classical solution u ∈ H 1 (Ω) satisfies the equation (∇u, ∇v) + (bu, v) = ( f, v) for every v ∈ H 1 (Ω). • Let f ∈ L2 (Ω). A weak solution of the Neumann boundary value problem is a u ∈ H 1 (Ω) satisfying the equation (∇u, ∇v) + (bu, v) = ( f, v) for every v ∈ H 1 (Ω). • Let a be the bilinear form on H 1 (Ω) defined by a(u, v) = (∇u, ∇v) + (bu, v) . • We estimate a(u, v) ≤ M k u k k v k , where M = max{1, b}. • So, a is continuous and coercive bilinear form in H 1 (Ω). • So, by Lax-Milgram theorem there exists a unique solution u ∈ H 1 (Ω) satisfying the equation a(u, v) = ( f, v) for all v ∈ H 1 (Ω). appliedfunanal.tex; May 24, 2005; 13:58; p. 61

3.3. WEAK SOLUTIONS OF ELLIPTIC BOUNDARY VALUE PROBLEMS65 • The weak solution of the Neumann boundary value problem minimizes on H 1 (Ω) the functional 1 J(v) = (∇v, ∇v) + (bv, v) − ( f, v) . 2 • Example. ¯ be bounded • General Elliptic Boundary Value Problem. Let aµν , q ∈ C 1 (Ω) differentiable functions such that aµν = aνµ . We will assume for simplicity that the matrix aµν is positive definite and q ≥ 0. Let L be a second-order operator of the form L = −∂µ aµν ∂ν + q . We consider the Dirichlet boundary value problem (L, B) for this operator Lu = f

in Ω

with the boundary condition u|∂Ω = 0 . • The operator L is called uniformly elliptic if there is a constant C such that for any x ∈ Ω, ξ ∈ Rn , ξ , 0, |aµν (x)ξµ ξν | ≥ C|ξ|2 , where |ξ| =

p

δµν ξµ ξν .

• Let f ∈ L2 (Ω). A weak solution of the boundary value problem is a function u ∈ H01 (Ω) satisfying the equation (aµν ∂µ u, ∂ν v) + (qu, v) = ( f, v) for every v ∈ H01 (Ω). • Every classical solution is a weak solution. Every sufficiently regular weak solution is a classical solution. appliedfunanal.tex; May 24, 2005; 13:58; p. 62

66CHAPTER 3. DISTRIBUTIONS AND PARTIAL DIFFERENTIAL EQUATIONS • Let a be a bilinear form on H01 (Ω) defined by a(u, v) = (aµν ∂µ u, ∂ν v) + (qu, v) • We estimate

a(u, u) ≥ (aµν ∂µ u, ∂ν ) ≥ K(∇u, ∇u) .

• Next, we show that a is bounded in H01 (Ω). • Then by Lax-Milgram theorem there exists a unique solution u ∈ H01 (Ω) of the equation a(u, v) = ( f, v) for all v ∈ H01 (Ω). • This solution minimizes the functional 1 1 J(v) = (aµν ∂µ u, ∂ν ) + (qv, v) − ( f, v) 2 2 on H01 (Ω).

3.3.1

Homework

• Exercises: 6.6[23,24,25]

appliedfunanal.tex; May 24, 2005; 13:58; p. 63

Chapter 4 Wavelets and Optimization 4.1 4.1.1

Wavelets Wavelet Transforms

•

4.1.2

Homework

• Exercises:

67

68

CHAPTER 4. WAVELETS AND OPTIMIZATION

4.2 4.2.1

Calculus of Variations Gateaux and Fr´echet Differentials

•

4.2.2

Euler-Lagrange Equations

•

4.2.3

Homework

• Exercises:

appliedfunanal.tex; May 24, 2005; 13:58; p. 64

4.3. DYNAMICAL SYSTEMS

4.3 4.3.1

69

Dynamical Systems Optimal Control

•

4.3.2

Stability

•

4.3.3

Bifurcations

•

4.3.4

Homework

• Exercises:

appliedfunanal.tex; May 24, 2005; 13:58; p. 65

70

CHAPTER 4. WAVELETS AND OPTIMIZATION

appliedfunanal.tex; May 24, 2005; 13:58; p. 66

Bibliography [1] L. Debnath and P. Mikusinski, Introduction to Hilbert Spaces with Applications, 2nd Ed., Academic Press, 1999 [2] E. Kreyszig, Introductory Functional Analysis with Applications, Wiley, New York, 1978 [3] M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. 1, Functional Analysis, Academic Press, New York, 1972 [4] C. DeVito, Functional Analysis and Linear Operator Theory, AddisonWesley, 1990 [5] E. Zeidler, Applied Functional Analysis, Springer-Verlag, 1995 [6] R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. 1, Interscience, 1962 [7] R. Richtmeyer, Principles of Advanced Mathematical Physics, SpringerVerlag, 1978 [8] G. Folland, Introduction to partial Differential Equations, 2nd ed., Princeton University Press, 1995 [9] I. Stakgold, Green Functions and Boundary Value Problems, Wiley, 1979 [10] M. E. Taylor, Partial Differential Equations, Springer-Verlag, 1996

71

72

Bibliography

appliedfunanal.tex; May 24, 2005; 13:58; p. 67

Notation Logic A =⇒ B A ⇐= B iff A ⇐⇒ B ∀x ∈ X ∃x ∈ X

A implies B A is implied by B if and only if A implies B and is implied by B for all x in X there exists an x in X such that

Sets and Functions (Mappings) x∈X x