Homework #3. Due February 27

1. Show that in an inner product space, un u un u and ||un || ||u||.

2. Let L(v) = 01 v(x) dx
and Ln(v) be the trapesoidal rule approximation of the integral:
Ln(v) = 1/n( (v(0) +v(1))/2 + i=1 n-1 v(i/n)).
Show that Ln(v) L(v) for any v C[0,1].

3. "Most" of the Banach spaces are separable.
The goal of this exercise is to give a counterexample to this statement.
Show that l is not a separable space.
Hint: The proof is by contradiction. Use the "diagonal argument" here.

4-5. Homogenization theory.
The goal of the homogenization theory is to study an -family of solutions u(x) that satisfy an -family of, for example, elliptic problems of the form

div(a(x/)grad u(x) ) = f(x),

where the matrix a(x/) characterizes a micro-nonhomogenious medium. The computation of the parameters describing a micro-nonhomogenious medium is an extremely difficult task, since the coefficients of the corresponding differential equations are given by rapidly oscillating coefficients. Therefore the problem is to construct an "effective homogenious medium". This physical concept of a homogenized, averaged, upscaled medium is reflected in the mathematical notions of the homogenized matrix and the homogenized differential equation.
Definition. A constant positive definite matrix ao is said to be the homogenized matrix for a(x), if for any bounded domain Rn and any forcing f(x) H -1( ) the solutions u(x) of the Dirichlet problem
div(a(x/)grad u(x) ) = f(x), u(x) H 1o( )
possess the following property of convergence u(x) uo(x) in H 1o( )
a(x/)grad u(x) ao grad uo(x) in L2( )
as 0, where uo(x) is the solution of the (homogenized) Dirichlet problem
div(aograd uo(x) ) = f(x), uo(x) H 1o( )
The operator div(aograd ) is called the homogenized operator.
The next three exercises are commonly used tools in the homogenization theory.

4. A property of the mean value.
In natural sciences a "standard" way to deal with highly oscillating small scales is to average them out.
In this problem we study an averaging procedure in a periodic setting. Suppose g(x) is a periodic function on Rn with periods l1,l2,... ln. Let be a parallelepiped
=[0,l1] [0,l2]... [0,ln]
By < g > we denote the mean value of g(x), that is:
< g > = ||-1 g(x) dx.
where || = l1 l2... ln is the volume of .
The Lp- space of periodic functions with a finite norm (< |g|p >)1/p is denoted by Lp().
Prove that if g(x) is a periodic function on Rn and g(x) Lp(), 1 p < ,
then g(x/ ) < g > in Lp() where is an arbitrary bounded domain in Rn.
Hint: Reduce the problem to the case when the Riemann-Lebesgue theorem (hw2, problem 1) is applicable.

5. The simplest homogenization example.
Let = [0, 1]. Let a(x) be a periodic function with the period 1 and a(x) > c > 0.
Suppose a(x) = a(x/). Consider the equation
d/dx (a(x) d/dx u(x) ) = f(x) L2 (),
and u(x) H10().
a. Suppose a(x) = 1/(2+sin 2 x ) and f(x) = 1. Find u(x) explicitely. Find uo(x) and ao. Show that u(x) uo(x) in H 1( ), but u(x) does not converge to uo(x) strongly in H 1( ), but u(x) converges to uo(x) strongly in L2( )
b. Compute uo(x) and ao for an arbitrary periodic a(x).

6. Problem number 6 is CANCELLED!