Homework #4. Due March 13 |

1. Sobolev embeddings are sharp.

The goal of this problem is to show that the spaces described in the Sobolev embedding theorem are optimal.

We assume here that R

a. Let f(x)= |x|

b. Suppose the domain is the ball of radius R: = B

2. Compact sets on C[0,1].

This exercise gives you the exact description of compact sets on C[0,1] and application of them to ODEs.

a. Arzela-Ascoli theorem

Prove that:

If a sequence of continuous functions {f

Definition: A family of functions {f

|f

b. Existence of solutions of an ODE.

Suppose f(t, u) is a continous function on R

du/dt = f(t, u)

u(t

Hint: Use Euler's scheme to construct approximate solutions which are equicontinuos and bounded, then use part a.

Note that our method does not guarantee uniqueness, typically uniqueness is provided by the "Lipschitz condition".

3. Divergence free finite elements.

Consider a triangulation of a domain and suppose

a. Give necessary and sufficient conditions on the values of f(x) so that it is divergence free (in distribution sense).

b. Is it always (for any triangles) possible to construct such functions?

Hint: if f is divergence free, then it is a curl of some other function.

4. Suppose G R

5. Using mollifiers prove that C

6. Compensated compactness

The basic question here is suppose v

Prove the compensated compactness lemma: Let p

v

If, in addition, p

p

Note: you may assume that

u

Hence in particular

u

This follows, however from theorem 10 p 15-16 in the notes.

u

Hint: Using the identity a b = (a-c) (b-d) + cb+ad-cd show that the problem can be reduced to the case v =0, p =0

Then use integration by parts and Sobolev embedding.

7. Show that W