Functional Analysis II - 2018

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Raffael, 06.08.2018, 10:00-10:30

Topics:

• Weak form of PDE's
• Poisson equation with Dirichlet data and uniqueness of the solution
• Schauder estimates (just statements, no prooves)
• Elliptic regularity, especially the final theorem of that chapter, idea about H1 estimates
• Application of the strong maximum principle

In general he is very polite and patient. Often if one doesn't know the answer at first shot he will give some time to think about it and also helps you with hints. The exam is on the blackboard but often he only asks you to write down little bits and explain the rest orally. It seemed to me that he is well aware of my nervousness and tried to calm me down by the way he phrased his questions. Over all it was definitely one of my most relaxed oral exams I have had at ETH so far.

Davide, 08.08.2018, 10:30-11:00

This is just a draft! If I remember I'll edit this...

Give 2 definitions of H^1_0. Why is the H^1_0 norm equivalent to the H^1 norm (statement of poincaré).

Decompose H^1 into 2 spaces, one of which is H^1_0. Prove that the decomposition a direct sum, i.e. that the intersection of H^1_0 and the other space (the space oh harmonic H^1 functions) is trivial. Prove this (that 0 is the only solution to -laplace(u)=0 with u = 0 on the boundary) in 2 ways. The two method he wanted were: 1) using weak formulation of the problem. 2) using weak maximum principle

For which p is cap_{W^{1,p}}({0}) = 0 ? If p > n, why is the statement false?

What can you say about the solution of Lu = f is f is in C^k,alpha (with L elliptic and smooth)? What if f is in H^k? He wanted the global estimate, with proof of the first step (H^1 estimate).

Daniel, 08.08.2018, 11:00-11:30

• Definition of elliptic operator: why do we like these? Example and non-example.
• Global estimate (Satz 9.1.2): statement and why is it important? (increase regularity)

Then he wanted to see some similar estimates..I did not understand what he wanted and stated Satz 10.5.1. In fact he wanted to hear Satz 10.4.1 (Global \(C^{2,\alpha}\)) but despite my error we continued with what I said. Proof of Satz 10.5.1.

• For which p is the function u(x) = \(|x|\) in \(W^{1,p}[(-1, 1)]\)?

Clemens, 09.08.2018, 10:30-11:00

My exam began with about 30 minutes delay. Carlotto seemed to be a bit unmotivated, but he was still polite. He wanted that I write everything on the blackboard, if I wanted to explain something orally, he told me to write it down.

Topics:

• Definition of \(H^1_0\), Definition of the weak gradient.
• What is the norm on \(H^1_0\), is there an equivalent norm and why are they equivalent?
• Poincaré inequality for \(H^1_0\), without proof.
• Is there a decomposition of \(H^1 = H^1_0 \oplus X\) and what is the subspace \(X\)? \((X= \{u \in H^1\ |\ \triangle u = 0 \})\)
• Proof that \(H^1_0\) and X have trivial intersection.
• Can we define \(H^1_0\) differently? (kernel of the trace operator)
• How is the trace defined? (Structure of the proof for the trace operator \(tr: H^1(Q_+) \to L^2(Q_0)\))
• How is the extension operator defined?
• What's an operator in divergence form?
• \(C^{1,\alpha}\) estimate
• What's an operator in non-divergence form and can we expect similar estimates?

Patrick, 09.08.2018, 11:00-11:30

As Clemens already mentioned, there was some delay and Prof. Carlotto seemed a little bit tired to do the exams.

Topics:

• Definition of \( W^{1,2}(\Omega)\).
• Uniqueness of weak derivative, proof.
• Give an example of bounded function wich is not in \( W^{1,2} (I)\), where \( I \) is a bounded interval. I mentioned \( u(x) = 1 _{ (0,1) } \) for the interval \( I = (-1,1)\). Then I had to show that the weak derivative did not exist (by computation) and justify that \( u\) could not be in \( W^{1,2}\) since it was not possible to find a absolutely continuous representative \( \tilde{u}\) (see Theorem of the lecture).
• Definition of distributional derivative (as seen in class). Discussion about the distributional derivative of the function \( u(x) = |x|\).
• Example of function that is not absolutely continuous on the unit ball \(B\) but is in \( W^{1,p}(B)\). I gave the example of \( u(x) = \log \log (1/ \Vert x \Vert ) \) and mentioned that it can be proven with the "null capacity Theorem" that we saw in class.
• State the "null capacity theorem" and apply it to the function of the point above.
• Definition of elliptic operator (in divergence form). Give an example and a non example.
• Discussion about the global regularity of elliptic functions. Then, Prof. Carlotto gave me a simple example where I had to use the theorem about regularity.
• Brief discussion about the Nirenberg Trick used to get the \( H^2 \) estimate.