Methods of applied mathematics

265 Pages·2004·1.226 MB·English

by  Arbogast T., Bona J.

Methods of Applied Mathematics
Todd Arbogast and Jerry Bona
Department of Mathematics
The University of Texas at Austin
Copyright 1999–2001, 2004 by T. Arbogast and J. Bona.

Contents
Chapter 1. Preliminaries 5
1.1. Elementary topology 5
1.2. Lebesgue measure and integration 13
1.3. The Lebesgue spaces L (Ω) 22
p
1.4. Exercises 25
Chapter 2. Normed Linear Spaces and Banach Spaces 29
2.1. Introduction 29
2.2. Hahn-Banach Theorems 36
2.3. Applications of Hahn-Banach 40
2.4. The Embedding of X into its Double Dual 44
2.5. The Open Mapping Theorem 45
2.6. Uniform Boundedness Principle 49
2.7. Weak Convergence 50
2.8. Conjugate or Dual of an Operator 53
2.9. Exercises 56
Chapter 3. Hilbert Spaces 63
3.1. Basic properties of inner-products 63
3.2. Best approximation and orthogonal projections 65
3.3. The dual space 68
3.4. Orthonormal subsets 69
3.5. Weak Convergence in a Hilbert Space 76
3.6. Basic spectral theory in Banach spaces 77
3.7. Bounded self-adjoint linear operators 80
3.8. Compact operators on a Banach space 85
3.9. Compact self-adjoint operators on a Hilbert space 91
3.10. The Ascoli-Arzela Theorem 94
3.11. Sturm Liouville Theory 96
3.12. Exercises 109
Chapter 4. Distributions 111
4.1. Topological Vector Spaces 111
4.2. The notion of generalized functions 117
4.3. Test Functions 119
4.4. Distributions 124
4.5. Operations with distributions 127
4.6. Convergence of distributions 133
4.7. Some Applications to Linear Differential Equations 135
3
4 CONTENTS
4.8. Local Structure of D0 142
4.9. Exercises 142
Chapter 5. The Fourier Transform 145
5.1. The L (Rd) theory 147
1
5.2. The Schwartz space theory 151
5.3. The L (Rd) theory 156
2
5.4. The S0 Theory 158
5.5. Exercises 164
Chapter 6. Sobolev Spaces 167
6.1. Definitions and Basic Properties 167
6.2. Extensions from Ω to Rd 171
6.3. The Sobolev Imbedding Theorem 175
6.4. Compactness 181
6.5. The Hs Sobolev Spaces 182
6.6. A trace theorem 187
6.7. The Ws,p(Ω) Sobolev Spaces 191
6.8. Exercises 192
Chapter 7. Boundary Value Problems 195
7.1. Second Order Elliptic Partial Differential Equations 195
7.2. A Variational Problem and Minimization of Energy 198
7.3. The Closed Range Theorem and operators bounded below 201
7.4. The Lax-Milgram Theorem 203
7.5. Application to second order elliptic equations 206
7.6. Galerkin approximations 211
7.7. Green’s functions 214
7.8. Exercises 217
Chapter 8. Differential Calculus in Banach Spaces and the Calculus of Variations 221
8.1. Differentiation 221
8.2. Nonlinear Equations 229
8.3. Higher Derivatives 238
8.4. Extrema 242
8.5. The Euler-Lagrange Equations 245
8.6. Constrained Extrema and Lagrange Multipliers 252
8.7. Lower Semi-Continuity and Existence of Minima 255
8.8. Exercises 259
Bibliography 265
CHAPTER 1
Preliminaries
1.1. Elementary topology
In applied mathematics, we are often faced with analyzing mathematical structures as they
might relate to real-world phenomena. In applying mathematics, real phenomena or objects are
conceptualized as abstract mathematical objects. Collections of such objects are called sets.
The objects in a set of interest may also be related to each other; that is, there is some structure
on the set. We call such structured sets spaces.
Examples. (1) A vector space (algebraic structure).
(2) The set of integers Z (number theoretical structure or arithmetic structure).
(3) The set of real numbers R or the set of complex numbers C (algebraic and topological
structure).
We startthe discussionofspaces by putting forwardsets of “points” onwhich we cantalk about
the notions of convergence or limits and associated continuity of functions.
A simple example is a set X with a notion of distance between any two points of X. A
sequence {x }∞ ⊂ X converges to x ∈ X if the distance from x to x tends to 0 as n increases.
n n=1 n
This definition relies on the following formal concept.
Definition. A metric or distance function on a set is a function d : X×X → R satisfying:
(1) (positivity) for any x,y ∈ X, d(x,y) ≥ 0, and d(x,y) = 0 if and only if x = y;
(2) (symmetry) for any x,y ∈ X, d(x,y) = d(y,x);
(3) (triangle inequality) for any x,y,z ∈ X, d(x,y) ≤ d(x,z)+d(z,y).
A metric space (X,d) is a set X together with an associated metric d : X ×X → R.
Example. (Rd,|·|) is a metric space, where for x,y ∈ Rd, the distance from x to y is
(cid:26) d (cid:27)1/2
X
|x−y| = (x −y )2 .
i i
i=1
It turns out that the notion of distance or metric is sometimes stronger than what actu-
ally appears in practice. The more fundamental concept upon which much of the mathematics
developed here rests, is that of limits. That is, there are important spaces arising in applied
mathematics that have well defined notions of limits, but these limiting processes are not com-
patible with any metric. We shall see such examples later; let it suffice for now to motivate a
weaker definition of limits.
Asequenceofpoints{x }∞ canbethoughtofasconvergingtoxifevery“neighborhood”of
n n=1
x contains all but finitely many of the x , where a neighborhood is a subset of points containing
n
x that we think of as “close” to x. Such a structure is called a topology. It is formalized as
follows.
5
6 1. PRELIMINARIES
Definition. A topological space (X,T) is a nonempty set X of points with a family T of
subsets, called open, with the properties:
(1) X ∈ T, ∅ ∈ T;
(2) If ω ,ω ∈ T, then ω ∩ω ∈ T;
1 2 1 2
S
(3) If ω ∈ T for all α in some index set I, then ω ∈ T.
α α∈I α
The family T is called a topology for X. Given A ⊂ X, we say that A is closed if its complement
Ac is open.
Example. If X is any nonempty set, we can always define the two topologies:
(1) T = {∅,X}, called the trivial topology;
1
(2) T consisting of the collection of all subsets of X, called the discrete topology.
2
Proposition 1.1. The sets ∅ and X are both open and closed. Any finite intersection of
open sets is open. Any intersection of closed sets is closed. The union of any finite number of
closed sets is closed.
Proof. We need only show the last two statements, as the first two follow directly from the
definitions. Let A ⊂ X be closed for α ∈ I. Then one of deMorgan’s laws gives that
α
(cid:18) (cid:19)c
\ [
A = Ac is open.
α α
α α
Finally, if J ⊂ I is finite, then
(cid:18) (cid:19)c
[ \
A = Ac is open.
α α
α∈J α∈J
(cid:3)
It is often convenient to define a simpler collection of open sets that immediately generates
a topology.
Definition. Given a topological space (X,T) and an x ∈ X, a base for the topology at x is
a collection B of open sets containing x such that for any open E 3 x, there is B ⊂ B such
X X
that
x ∈ B ⊂ E .
A base for the topology B is a collection of open sets that contains a base at x for all x ∈ X.
Proposition 1.2. A collection B of subsets of X is a base for a topology T if and only if
(1) each x ∈ X is contained in some B ∈ B and (2) if x ∈ B ∩B for B ,B ∈ B, then there is
1 2 1 2
some B ∈ B such that x ∈ B ⊂ B ∩B . If (1) and (2) are valid, then
3 3 1 2
T = {E ⊂ X : E is a union of subsets in B} .
Proof. (⇒) Since X and B ∩B are open, (1) and (2) follow from the definition of a base
1 2
at x.
(⇐) Let T be defined as above. Then ∅ ∈ T (the vacuous union), X ∈ T by (1), and
arbitrary unions of sets in T are again in T. It remains to show the intersection property. Let
E ,E ∈ T, and x ∈ E ∩E (if E ∩E = ∅, there is nothing to prove). Then there are sets
1 2 1 2 1 2
B ,B ∈ B such that
1 2
x ∈ B ⊂ E , x ∈ B ⊂ E ,
1 1 2 2
1.1. ELEMENTARY TOPOLOGY 7
so
x ∈ B ∩B ⊂ E ∩E .
1 2 1 2
Now (2) gives B ∈ B such that
3
x ∈ B ⊂ E ∩E .
3 1 2
Thus E ∩E is a union of elements in B, and is thus in T. (cid:3)
1 2
We remark that instead of using open sets, one can consider neighborhoods of points x ∈ X,
which are sets N 3 x such that there is an open set E satisfying x ∈ E ⊂ N.
Theorem 1.3. If (X,d) is a metric space, then (X,T) is a topological space, where a base
for the topology is given by
T = {B (x) : x ∈ X and r > 0} ,
B r
where
B (x) = {y ∈ X : d(x,y) < r}
r
is the ball of radius r about x.
Proof. Point (1) is clear. For (2), suppose x ∈ B (y)∩B (z). Then x ∈ B (x) ⊂ B (y)∩
r s ρ r
B (z), where ρ = 1 min(r−d(x,y),s−d(x,z)) > 0. (cid:3)
s 2
Thus metric spaces have a natural topological structure. However, not all topological spaces
are induced as above by a metric, so the class of topological spaces is genuinely richer.
Definition. Let (X,T) be a topological space. The closure of A ⊂ X, denoted A, is the
intersection of all closed sets containing A:
\
A = F .
F closed
F⊇A
Proposition 1.4. A is closed, and is the smallest closed set containing A.
Proof. This follows by Proposition 1.1 and the definition. (cid:3)
Definition. The interior of A ⊂ X, denoted A◦, is the union of all open sets contained in
A:
[
A◦ = E .
E open
E⊂A
Proposition 1.5. A◦ is open, and is the largest open set contained in A.
Proof. This also follows from Proposition 1.1 and the definition. (cid:3)
Proposition 1.6. A ⊂ A¯, A¯¯= A¯, A∪B = A¯∪B¯, and A closed ⇔ A = A¯.
A ⊇ A◦, A◦◦ = A◦, (A∩B)◦ = A◦∩B◦, and A open ⇔ A = A◦.
Proposition 1.7. (Ac)◦ = (A¯)c, (A◦)c = (Ac).
8 1. PRELIMINARIES
Proof.
 c
\ \ [
x ∈/ (A¯)c ⇔ x ∈ A¯⇔ x ∈ F ⇔ x ∈/  F ⇔ x ∈/ Fc = (Ac)◦ .
 
F closed F closed Fc open
F⊃A F⊃A Fc⊂Ac
The second result is similar. (cid:3)
Definition. A point x ∈ X is an accumulation point of A ⊂ X if every open set containing
x intersects A\{x}. Also, a point x ∈ A is an interior point of A if there is some open set E
such that
x ∈ E ⊂ A .
Finally, x ∈ A is an isolated point if there is an open set E 3 x such that E \{x}∩A = ∅.
Proposition 1.8. For A ⊂ X, A¯ is the union of the set of accumulation points of A and A
itself and A0 is the union of the interior points of A.
Proof. Exercise. (cid:3)
Definition. A set A ⊂ X is dense in X if A¯= X.
Definition. The boundary of A ⊂ X, denoted ∂A, is
∂A = A¯∩Ac .
Proposition 1.9. If A ⊂ X, then ∂A is closed and
A¯= A◦∪∂A , A◦∩∂A = ∅ .
Moreover,
∂A = ∂Ac = {x ∈ X : every open E 3 x intersects both A and Ac} .
Proof. Exercise. (cid:3)
Definition. A sequence {x }∞ ⊂ X converges to x ∈ X, or has limit x, if given any open
n n=1
E 3 x, there is N > 0 such that x ∈ E for all n ≥ N (i.e., the entire tail of the sequence is
n
contained in E).
Proposition 1.10. If lim x = x, then x is an accumulation point of {x }∞ , inter-
n→∞ n n n=1
preted as a set.
Proof. Exercise. (cid:3)
We remark that if x is an accumulation point of {x }∞ , there may be no subsequence
n n=1
{x }∞ converging to x.
nk k=1
Example. Let X be the set of nonnegative integers, and a base T = {{0,1,... ,i} for
B
each i ≥ 1}. Then {x }∞ with x = n has 0 as an accumulation point, but no subsequence
n n=1 n
converges to 0.
If x → x ∈ X and x → y ∈ X, it is possible that x 6= y.
n n
Example. Let X = {a,b} and T = {∅,{a},{a,b}}. Then the sequence x = a for all n
n
converges to both a and b.
Definition. A topological space (X,T) is called Hausdorff if given distinct x,y ∈ X, there
are disjoint open sets E and E such that x ∈ E and y ∈ E .
1 2 1 2
1.1. ELEMENTARY TOPOLOGY 9
Proposition 1.11. If (X,T) is Hausdorff, then every set consisting of a single point is
closed. Moreover, limits of sequences are unique.
Proof. Exercise. (cid:3)
Definition. A point x ∈ X is a strict limit point of A ⊂ X if there is a sequence {x } ⊂
n
A\{x} such that lim x = x.
n→∞ n
Proposition 1.12. Every x ∈ ∂A is either an isolated point, or a strict limit point of A
and Ac.
Proof. Exercise. (cid:3)
Note that if x is an isolated point of X, then x ∈/ A◦, so ∂A 6= ∂A◦ in general.
Metric spaces are less suseptible to pathology than general topological spaces.
Proposition 1.13. If (X,d) is a metric space and {x }∞ is a sequence in X, then x → x
n n=1 n
if and only if, given ε > 0, there is N > 0 such that
d(x,x ) < ε ∀ n ≥ N .
n
That is, x ∈ B (x) for all n ≥ N.
n ε
Proof. If x → x, then the tail of the sequence is in every open set E 3 x. In particular,
n
this holds for the open sets B (x). Conversely, if E is any open set containing x, then the open
ε
balls at x form a base for the topology, so there is some B (x) ⊂ E which contains the tail of
ε
the sequence. (cid:3)
Proposition 1.14. Every metric space is Hausdorff.
Proof. Exercise. (cid:3)
Proposition 1.15. If (x,d) is a metric space and A ⊂ X has an accumulation point x,
Then there is some sequence {x }∞ ⊂ A such that x → x.
n n=1 n
Proof. Given integer n ≥ 1, there is some x ∈ B (x), since x is an accumulation point.
n 1/n
Thus x → x. (cid:3)
n
Weavoidproblemsarisingwithlimitsingeneraltopologicalspacesbythefollowingdefinition
of continuity.
Definition. A mapping f of a topological space (X,T) into a topological space (Y,S) is
continuous if the inverse image of every open set in Y is open in X.
This agrees with our notion of continuity on R.
We say that f is continuous at a point x ∈ X if given any open set E ⊂ Y containing f(x),
then f−1(E) contains an open set D containing x. That is,
x ∈ D and f(D) ⊂ E .
A map is continuous if and only if it is continuous at each point of X.
Proposition 1.16. If f : X → Y and g : Y → Z are continuous, then g ◦f : X → Z is
continuous.
Proof. Exercise. (cid:3)
Proposition 1.17. If f is continuous and x → x, then f(x ) → f(x).
n n
10 1. PRELIMINARIES
Proof. Exercise. (cid:3)
The converse of Proposition 1.17 is false in general. When the hypothesis x → x always
n
implies f(x ) → f(x), we say that f is sequentially continuous.
n
Proposition 1.18. If f : X → Y is sequentially continuous, and if X is a metric space,
then f is continuous.
Proof. Let E ⊂ Y be open and A = f−1(E). We must show that A is open. Suppose not.
Then there is some x ∈ A such that B (x) 6⊂ A for all r > 0. Thus for r = 1/n, n ≥ 1 an
r n
integer, there is some x ∈ B (x)∩Ac. Since x → x, f(x ) → f(x) ∈ E. But f(x ) ∈ Ec
n rn n n n
for all n, so f(x) is an accumulation point of Ec. That is, f(x) ∈ Ec ∩ E = ∂E. Hence,
f(x) ∈ ∂E ∩E = ∂E ∩E◦ = ∅, a contradiction. (cid:3)
Suppose we have a map f : X → Y that is both injective (one to one) and surjective (onto),
such that both f and f−1 are continuous. Then f and f−1 map open sets to open sets. That is
E ⊂ X is open if and only if f(E) ⊂ Y is open. Therefore f(T) = S, and, from a topological
point of view, X and Y are indistinguishable. Any topological property of X is shared by Y,
and conversely. For example, if x → x in X, then f(x ) → f(x) in Y, and conversely (y → y
n n n
in Y ⇒ f−1(y ) → f−1(y) in X).
n
Definition. A homeomorphism between two topological spaces X and Y is a one-to-one
continuousmappingf ofX ontoY forwhichf−1 isalsocontinuous. Ifthereisahomeomorphism
f : X → Y, we say that X and Y are homeomorphic.
ItispossibletodefinetwoormorenonhomeomorphictopologiesonanysetX ofatleasttwo
points. If (X,T) and (X,S) are topological spaces, and S ⊃ T, then we say that S is stronger
than T or that T is weaker than S.
Example. The trivial topology is weaker than any other topology. The discrete topology
is stronger than any other topology.
Proposition 1.19. The topology S is stronger than T if and only if the identity mapping
I : (X,S) → (X,T) is continuous.
Proposition 1.20. Given a collection C of subsets of X, there is a weakest topology T
containing C.
Proof. Since the intersection of topologies is again a topology (prove this),
\
C ⊂ T = S
S⊇C
S atopology
is the weakest such topology (which is nonempty since the discrete topology is a topology
containing C). (cid:3)
Given a topological space (X,T) and A ⊂ X, we obtain a topology S on A by restriction.
We say that this topology on A is inherited from X. Specifically
S = T ∩A ≡ {E ⊂ A : there is some G ⊂ T such that E = A∩G} .
That S is a topology on A is easily verified. We also say that A is a subspace of X.
Given two topological spaces (X,T) and (Y,S), we can define a topology R on
X ×Y = {(x,y) : x ∈ X, y ∈ Y} ,