Functional Analysis II - Dietmar Salamon - 2016

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Kthim, 24.08, 09:30-10:00

My exam was in german but I translate it here to english. I appreciate every correction ;)

After we shook hands und checked my Legi, he first asked me about the Harnack Inequality. I did not know the whole formula, but that was not a problem. I guess he just wanted to hear that we can bound \(u(x)\) from above and below by \(u(\zeta)\). Remember that u needs to be harmonic and positive \((u\geq 0 )\). Then he asked about Harnack Convergence Theorem with its proof (just main idea). It was enough to just say it. The proof needed the Mean Value Theorem which I stated afterwards. From there we got to the Maximal Principle for subharmic functions. I stated it and then we went to Hopf's Maximum Principle and draw some links between them. No proofs thereof were asked. (Perhaps he did not want to ask, because I could not proof Harnack Convergence Theorem correctly.)

Then he asked: what are Sobolev spaces good for. I answered to solve the problem \(Lu=f\). Then I had to proof it for \(L:W^{1,p}\rightarrow W^{-1,p}\) and \(p=2\). That was not that hard (just show that \(B\) is an inner-product and then use Riesz Theorem). Then he wanted it for general \(p\). For that, I wanted to show that \(L\) is bijective, but somehow did not managed to do so. I was supposed to use the inequality we get from \(p=2: ||u||_{W^{1,2}}\leq ||Lu||_{W^{-1,2}}\) and then use Calderon-Zygmund to it. Afterwards, we discussed the Calderon Zygmund Theorem. Mainly the definition and the Theorem itself. Do not forget that in the second condition for a Calderon-Zygmund Pair, \(x\not\in supp(f)\). Unfortunately, the proof was not asked although I learned it very well. I forgot to ask whether I can show it to him, too.

We were already 3 minutes over the time but he asked me about maximal regularity in the last chapter about the inhomogenous Heat Equation and what are Besov spaces good for (Gregorian Liu Theorem). Remember that \(s=2-\frac{1}{q}\). No proofs.

First, I was nervous, but after the middle of the exam it was like a nice discussion. At the end, he looked contented. So I guess I passed, but I have no idea what grade I will receive.

Benji

Harnack inequality, monotone convergence, perron's method, L:W^1,p_0 is bijective, p=2 case, elliptic regularity (thm4), calderon zygmund inequality and theorem, mikhlin's theorem