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 Rn is a bounded domain that contains the origin.
a. Let f(x)= |x|. Find so that f W1, p(), but f Lq() for q > np/(n-p).
b. Suppose the domain is the ball of radius R: = BR(0), 1 < p =n, f(x)= ln(ln(5/|x|)). Show that f W1, p(), but f L ().

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 {fn} on an interval [0,1] (or more generally any metric space) is uniformly bounded in the C[0,1] norm and equicontinous then there is a uniformly convergent subsequence.
Definition: A family of functions {fn} is equicontinous if for any > 0 there exists > 0 such that for every x and y on [0,1] with |x-y| < we have that
|fn(x) - fn(y)| < for all functions fn in the family.
b. Existence of solutions of an ODE.
Suppose f(t, u) is a continous function on R2. then for every (to, uo) there is an open interval I R that contains to, and a continosly differentiable function u: I R that satisfies
du/dt = f(t, u)
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 1, 2, and 3 are three triangles that share a common vertex of the triangulation. We say that a two-dimensional vector valued function f(x) is a finite element if it is zero outside the three triangles and it is a (different) constant vector on each of the three triangles.
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 Rn. Prove that Cm(G) is a Banach space and C(G) is a complete metric space. Hint: The main thing you need to show in both cases that Cauchy sequences converge to respective elements of the same space; however, do not forget to check properties of norms and metrics.

5. Using mollifiers prove that C0(G) is dense in Lp(G), for p 2. 1 p < .

6. Compensated compactness
The basic question here is suppose v v, and p p, what can we say about the convergence of the product v p ? The compensated compactness lemma states that under additional assumptions which hold in our elliptic setting we have a weak-* convergence.
Prove the compensated compactness lemma: Let p and v be some vector fields in L2 (), where a bounded open set Rn, n=2,3; such that
v v, p p in L2 ().
If, in addition, p and v satisfy the conditions: v = grad u , u H1o (), v = grad u, u H1o (), div p f in H-1 (), then
p . v p. v in L1 ().
Note: you may assume that
u u in H1 o ().
Hence in particular
u u in L2 ().
This follows, however from theorem 10 p 15-16 in the notes.
u Note: here we view p v L1 () as a linear functional on Co () functions with L ()norm.
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 Wm, p(), Wm, po() are Banach spaces.