Homework #3. Due February 27 |

1. Show that in an inner product space, u

2. Let L(v) =

and L

L

Show that L

3. "Most" of the Banach spaces are separable.

The goal of this exercise is to give a counterexample to this statement.

Show that l

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

div(a(x/)grad u

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 a

div(a(x/)grad u

possess the following property of convergence u

a(x/)grad u

as 0, where u

div(a

The operator div(a

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 R

=[0,l

By < g > we denote the mean value of g(x), that is:

< g > = ||

where || = l

The L

Prove that if g(x) is a periodic function on R

then g(x/ ) < g > in L

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

d/dx (a

and u

a. Suppose a(x) = 1/(2+sin 2 x ) and f(x) = 1. Find u

b. Compute u

6. Problem number 6 is CANCELLED!