On this page I report what I do/did during my part of 
the PDE course with Arjen Doelman in the national 
MasterMath programme. 

I hope to cover Chapters 5-8. Chapter 8 will probably be
shifted till after Xmas.


THE THIRD HOMEWORK SET IS ON THE WEB,
http://www.few.vu.nl/~jhulshof/PDE2006/pde2006ex3.pdf
HAND IN 10-12-2006

October 30 I gave an overview, basically discussing how 
we deal with (6.2) as the weak formulation of (6.1) with 
u=0 on the boundary. Discussed the use of the Riesz 
representation theorem for continuous linear functions
on a Hilbert space in relation the weak formulation.
Introduced formally the Sobolev spaces we are going to use. 

November 6 I discussed an example problem, namely
$\Delta u+ u^p=0$ on a ball in m-dimensional space, 
with u=0 in the boundary.

Solve by ODE methods and showed the existence of 
a critical p (or m if you like). Showed how the same
estimate arises in the Sobolev estimates, for
which I obtained the critical values of the parameters
by scaling, see Section 5.7. Motivated the desire
for controlling the (p+1)-norm in terms of the gradient 
norm for solutions u(x,t) of $u_t=\Delta u+ u^p$ by 
showing the simplest a prior estimate (multiply by u 
and integrate by parts)

November 13. No course, Arjen will be there to discuss
exercises. In the mean time I suggest you try to read
Chapter 5.

November 20. Proved Proposition 5.24, showed also that for smooth u 
with compact support in a bounded Omega in m-dimensional space, 
the q-norm of u is bounded by a constant times the p-norm of Du, 
provided q is less or equal to mp/(m-p). The constant depends on
Omega if q < mp/(m-p). The case q=p=2 gives the Poincare inequality
in Prop. 5.8.

Then I asked: can we conclude that u is continuous, if we only know that 
Du has a bounded  p-norm? Proposition 5.22 is an example with dimension
m=1. In Proposition 5.24 I replaced the assumption on p by the assumption
that p>m and proved Morrey's estimate, see the book of Evans, section 5.6.2.
This gives that u is Holder continuous with exponent 1-m/p, with a 
bound on the Holder constant depending only on the p-norm of Du. 
This statement implies Prop. 5.28. 

I finished with the conclusion that a sequence of functions with support 
contained in a bounded Omega, and with the p-norm of the gradients uniformly
bounded (with p larger than the dimension), has a uniformly convergent
subsequence.

November 27. 
Chapter 5, basic results, compactness of
embeddings. The application to the map f -> u, where u is
the weak solution corresponding to u, see Chapter 6, 6.1, 6.2
is elaborated upon in the exercises. See the file pde2006ex3.pdf
in this directory (replace weblogpde2006.txt by pde2006ex3.pdf
in the web address).


December 4. Discussion of the dual of H^1_0 using Fourierseries 
with the eigenfunctions of the laplacian.
Solution of (7.1) with these Fourier series. 
Estimates for Fourier coefficients leading to the function spaces 
used in the weak formulation, see Thm 7.6. No difference between
Galerkin and Fourier here. Given Fourier series of initial dat u(x,0)
and of f(x,t), write solution u(x,t) as Fourier series. Galerkin 
approximation is just the partial Fourier sum in this case.

Also discussed (Bochner-Lebesque) integration theory for X-valued 
functions f(t). Incorrect in the book. Ex 7.1 and 7.2 OK, if f is also
weakly measurable. However I do not see how in 7.2 the boundedness
of the right hand side follows from his definition in 7.1, better to 
make it part of the definition. 

The proof in the fundamental Theorem 7.2 contains an assertion on page 
192 bottom that is not clear to me. Convergence of the derivatives is
only local in time. 

December 11. Finish Chapter 7, begin with Chapter 8.

December 18. Discuss Chapter 8.

Last exercise set is pde2006ex4.pdf
in this directory (replace weblogpde2006.txt by pde2006ex4.pdf
in the web address). Hand in before January 15.