HW#14: Due April 28, 2004
Section 7.5: Problem #13.
(That is, show that the heat kernel generates a 
contraction semigroup in L^2(R^n), but not in L^\infty(R^n).


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HW#13: Due April 21, 2004

Problem: Are the conclusions (Theorems 1--4) in Section
7.3.2 still true when the main system has the extra term
cu where c is a smooth matrix and u is its unknown vector?

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HW#12: Due April 14, 2004

Section 7.5: 9, 10.

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HW#11: Due April 7, 2004

Section 7.5: 5, 6, 7.


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HW#10: Due March 31 Wed
Section 7.5: 2, 3, 4.

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HW#9: Due March 24, Wed
Section 6.6: 9


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HW#8: Due March 17, Wed
Section 6.6: 5, 6, 7, 8
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Midterm Exam: Feb 29, Sunday Evening 8:30--9:20pm, 102 MB


HW#7: Due March 3, Wed
Section 6.6: 3, 4.
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HW#6: Due Feburary 25, Wed
Section 6.6: 1, 2
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HW#5: Due Feburary 18, Wed
Section 5.10: 15, 16, 17, 18
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HW#4:  Due Wed Feburary 11
Section 5.10: 12, 13, 14.

Optional problem
1. Show that u(x) is bounded provided that u(x) is in the
Sobolev space W^{1,2} (R). (note n =1)

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HW#3: Due Wed Feburary 4.

Sect 5.10: Problems 3, 6 10.
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Questions to think about the lecture materials:

1. From Rellich-Kondrakov compactness theorem, we see
that the derivative plays an essential role to 
make a sequence compact in an L^q space. Find an
example of a sequence {u_n} on [-1,  1] bounded
uniformly in the maximum norm, but it is not
pre-compact in L^2[-1, 1].



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HW#2: Due Wed Jan 28.

Sect 5.10: Problems 7, 8, 9.
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Questions to think about the lecture materials
1. What makes the trace theorem work?

2. Find a question and find its answer.

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HW #1: Due Jan 21 Wed

Sect 5.10: Problems 2, 4, 5.
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As you think about the qualifying exams, you might wonder how to
prepare for it. One way to prepare for it is to think critically
about the materials we are covering. Here are 
some questions to get you started.


1. Why is \lambda needed in the proof of Theorem 3 of Section 5.3
(the Approximation Theorem)? What determines the value of \lambda?

2. Why is C^1 only (not c^k) needed of the boundary regularity in
the W^{k,p} approximation ?

3. If the boundary is not C^1, will a corner cause failure in the
approximation? Will an inward sharp cusp cause failure?
Or an outgoing sharp cusp causes failure?

4. Find a two-dimensional domain whose boundary as a curve is 
infinitely smooth, but the boundary is not C^\infty according to
the definition in Appendix C.1.

5. Finish the case p = \infty for the proof of Theorem 1 of Sect 5.4
(The extension theorem).
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