Calculus of Variations Summer Term 2015 · Calculus of Variations Summer Term 2015 Lecture 18 Universität des Saarlandes 14. Juli 2015 c Daria Apushkinskaya (UdS) Calculus of variations

Calculus of VariationsSummer Term 2015

Lecture 18

Universität des Saarlandes

14. Juli 2015

c© Daria Apushkinskaya (UdS) Calculus of variations lecture 18 14. Juli 2015 1 / 23

Purpose of Lesson

Purpose of Lesson:

We collect some simple but instructive remarks about theexistence and regularity problems of minimizers.

We begin by presenting some examples of variationals integralsthat have weak C1-minimizers which are not of class C2.

To introduce direct methods

To discuss strong and weak topologies in real Banach spaces.


Direct Methods of the Calculus of Variations

§11. Direct Methods of the Calculus of Variations


Direct Methods of the Calculus of Variations Minimizers which do not satisfy the Euler-Lagrange equation

The following examples show that there are minimizers which are notsmooth solutions of the Euler-Lagrange equations.

Example 18.1A very trivial example is given by the functional

J[u] =

1∫−1

u′(x)− 2|x |

2 dx

which is minimized by the functions

u(x) = x |x |+ C

that are of class C1 on [−1,1] but not in C2 on (−1,1).


Direct Methods of the Calculus of Variations Remarks on the existence and regularity of minimizers

Remarks

Actually, it is not a-priori clear whether C1, C2 or some otherfunction space is the natural setting where a first-order variationalproblem is to be solved.

In fact it is part of the problem to define the class of admissiblefunctions where the functionals is to be minimized.

In general it is not true that every reasonable problem (whateveris meant by reasonable) has a solution.

If there exists a solution, one cannot take it for granted that thissolution is smooth: there are minimizers which do not satisfy theEuler-Lagrange equation.


Direct Methods of the Calculus of Variations Hilbert’s 19th and 20th problems

Hilbert challenged mathematicians to solve what he considered to bethe most important problems of the 20th century in his address to theInternational Congress of mathematicians in Paris, 1900.

David Hilbert around 1900 Title page from the ICM Paris Proc.


Direct Methods of the Calculus of Variations Hilbert’s 19th and 20th problems

19th Are the solutions of regular problems in the Calculus of Variationsalways necessarily analytic?

20th Has not every variational problem a solution, provided certainassumptions regarding the given boundary conditions aresatisfied, and provided also that if need be that the notion ofsolution be suitably extended?

These questions have stimulated enormous effort and now things aremuch better understood.

The existence and regularity theory for elliptic PDE’s and variationalproblems is one of the greatest achievments of 20th centurymathematics.


Direct Methods of the Calculus of Variations Indirect Methods

Indirect Methods:

The classical theory of Calculus of Variations, roughly covers thetime from Euler to the end of 19th century is concerned withso-called Indirect Methods.

The classical indirect method of variational problems is based onthe optimistic idea that every minimum problem has a solution.

In order to determine this solution, one looks for conditions whichhave to satisfies by a minimizer. An analysis of the necessaryconditions often permits one to eliminate many candidates andeventually idetifies a unique solution.

For example, we only have one solution of the Euler-Lagrangeequation satisfying all prescribed conditions. We tempted to inferthat this solution is also be a solution to the original minimumproblem.



Olga Ladyzhenskaya (1922-‐2004)

Nina Uraltseva (1934 -‐ )



Indirect Methods:

The above approach is false. We have to prove the existence ofminimizer before we could conclude the uniqueness of solutions tothe Euler-Lagrange equation.

Usually, in higher dimension solving the Euler-Lagrange equationis more difficult than finding a minimizer of a functional.


Direct Methods of the Calculus of Variations Direct Methods

Direct Methods:

The relevant ideas developed during the 20th century are calledDirect Methods.

An important ingredient here is the introduction of functionalanalytic techniques.

In fact, it was the Calsulus of Variations, which gave birth to thetheory of Functional Analysis.


Direct Methods of the Calculus of Variations Direct Methods

The Main Idea of Direct Method:

Consider a minimization problem on some class A of functions:

minu∈A

J[u]

Suppose there exist a minimizing sequence un in A, i.e.

limn→∞

J[un] = minu∈A

J[u] < +∞.

Suppose we could find an element u0 in A such that′′ un → u0

′′ as n→∞

Suppose the functional J has some kind of continuity. Therefore

limn→∞

J[un] = J[u0].

We conclude that u0 is a minimizer of the functional J because

J[u0] = limn→∞

J[un] = minu∈A

J[u].


Direct Methods of the Calculus of Variations Example of Failure of Direct Method

Example 18.2 (J is not bounded below)

Let [0, π] ⊂ R and A =

u ∈ C1 ([0, π]) : u(0) = u(π) = 0

.

Take

J[u] =

π∫0

[u′2 − 2u2

]dx and un(x) = n sin x ∈ A.

But J[un] = −πn2

2→ −∞ as n→∞. (Also, un is unbounded.)



Example 18.3 (No convergent subsequence in minimizing sequence)


u ∈ C1 ([0, π]) : u(0) = u(π) = 0

.

Take

J[u] =

π∫0

(u′2 − 1

)2dx ,

un(x) =

√1n2 +

π2

4−√

1n2 +

(x − π

2

)2∈ A,

J[un]→ 0 = infv∈A

J[v ].

But every subsequence converges toπ

2− |x − π

2| which is not in A.



Example 18.4 (J is not lower semi-continuous)


u ∈ C1 ([0, π]) : u(0) = u(π) = 0

.

Let

g(p) =

p2, if p 6= 0;

1, if p = 0.

Take

J[u] =

π∫0

g(u′)dx and un(x) =1n

sin x → 0.

Then un(x) ∈ A and J[un] =π

2n2 → 0 = I as n→∞ but

π = J[0] = J[ limn′→∞

un′ ] > limn→∞

J[un] = 0.


Suitable Function Spaces

§12. Suitable Function Spaces


Suitable Function Spaces Topologies on Banach Spaces

Topologies on Banach Spaces

Let (X, | · |) denote a real Banach space.

A Banach space is a complete, normed linear space. Completemeans that any Cauchy sequence is convergent.

A sequence xn of real numbers is called a Cauchy sequence, iffor every positive real number ε, there is a positive integer N suchthat for all natural numbers m,n > N

|xn − xm| < ε.

Examples:

1 Rn with the norm defined by ‖x‖ =

(∑i|xi |2

)1/2

.

2 Ck(Ω)

with the norm defined by

‖u‖Ck = max06|α|6k

supx∈Ω

|Dαu(x)|



We denote by X′ the topological dual space of X:

X′ =

l : X→ R linear such that |l |X′ = sup

x 6=0

|l(x)||x |X

<∞

.

Classically, X can be endowed with two topologies.

Definition (topologies on X)

(i) The strong topology, denoted by xn →X

x , is defined by

|xn − x |X → 0 (n→∞).

(ii) The weak topology, denoted by xn X

x , is defined by

l(xn)→ l(x) (n→∞) for every l ∈ X′.



Strong convergence implies weak convergence, but the converseis false in general.

Example 18.5 (Counterexample)

Consider the sequence fn(x) = sin (2πxn), x ∈ (0,1), as n→∞.

fn L2(Ω)

0 . To prove weak convergence, we have for all

ϕ ∈ C1(0,1), thanks to a classical integration by parts,

1∫0

sin (2πxn)ϕ(x)dx =1

2πn[ϕ(0)− ϕ(1)]

+1

2πn

1∫0

cos (2πxn)ϕ′(x)dx



Example 18.5 (continued)

So, it is clear that 〈fn, ϕ〉 → 0 as n→∞ for all ϕ ∈ C1(0,1), where〈·, ·〉 is the usual scalar product in L2(0,1).

By density, this result can be generalized for all ϕ ∈ L2(0,1).

There is no strong convergence, i.e., fn(x) →L2(0,1)

0 is not true.

To prove that there is no strong convergence, we observe that

1∫0

sin2 (2πxn)dx =12

1∫0

(1− cos[(4πxn)) dx =12.



The dual space X′ can also be endowed with the strong and theweak topologies.

Definition (topologies on X′)(i) The strong topology, denoted by ln →

X′l , is defined by

|ln − l |X′ → 0, or equivalently, supx 6=0

|ln(x)− l(x)||x |X

→ 0 (n→∞).

(ii) The weak topology, denoted by ln X′

l , is defined by

z(ln)→ z(l) (n→∞) for every z ∈(X′)′,

where (X′)′ denotes the bidual space of X.



In some cases it is more convenient to equip X′ with a thirdtopology:

The weak* topology, denoted by ln X′∗

l , is defined by

ln(x)→ l(x) (n→∞) for every x ∈ X.

The space X is called reflexive if (X′)′ = X.

X is called separable if it contains a countable dense set.

Examples:1 X = Lp(Ω) is reflexive for 1 < p <∞ and separable for 1 6 p <∞.

The dual space of Lp(Ω) is Lp′(Ω for 1 6 p <∞ with1p

+1p′

= 1.

2 X = L1(Ω) is nonreflexive and X′ = L∞(Ω).



The main properties associated with these different topologies aresummarized in the following theorem :

Theorem 18.1 (weak sequential compactness)(i) Let X be a reflexive Banach space, K > 0, and xn ∈ X a sequence

such that |xn|X 6 K .

Then there exist x ∈ X and a subsequence xnj of xn such that

xnj Xx (n→∞).

(ii) Let X be a separable Banach space, K > 0, and ln ∈ X′ such that|ln|X′ 6 K .

Then there exist l ∈ X′ and a subsequence lnj of ln such that

lnj X′∗

l (n→∞).


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Calculus of Variations Summer Term 2015 · Calculus of Variations Summer Term 2015 Lecture 18 Universität des Saarlandes 14. Juli 2015 c Daria Apushkinskaya (UdS) Calculus of variations