我們愛老師 老師愛我們

    如果我用 MSN，我想把暱稱設為 "沒考過 Orals 前我什麼都不是"。

    現在沒人逼你寫 weekly report 了，反而會想念那感覺，所以我希望記一下接下來一個多月準備 Orals 的心得。假如只有考試才能刺激學習，希望我這時學的東西值得牢記一輩子。照 weekly report 的規律，大概兩週後就會膩了吧！想當初我第一篇寫了超過一張 A4 呢。

    兩個主題， Major: Partial Differential Equations; minor: Riemannian Geometry.

    這一系列文章會需要一些縮寫。

1. [HL] Qin Hang & Fang-Hua Lin: Elliptic Partial Differential Equations
2. [GT] Gilbarg & Trudinger: Elliptic Partial Differential Equations of Second Order
3. [P] Petersen: Riemannian Geometry 

回到紐約已經兩個星期了，上週就該交第一篇，但難免有不可抗拒因素，所以現在要寫這兩週唸的東西。

(一)  Chapter 3 of HL.

They first introduce Campanato's characterization of H\"{o}lder continuous functions. So our goal becomes the estimate of certain Dirichlet- type integral(I invented this term) on arbitrary balls in the domain. There are two advantages. One is that it's straightforward to consider weak solutions. The other one is that why we need the coefficients have certain regularity such as $a^{ij} \in C^\alpha$.

As they make it clear in the beginning of chapter 3, they do not use any potential theory. Instead they establish interior estimates for constant coefficients equation $a^{ij} D_{ij} u =0$, which is easier than GT's impressive Lemma 4.4.

Unfortunately, after the whole week's effort, I decide not to include their exposition in my orals but rather use Theorem 8.24 and Theorem 8.32 in GT. You may think the reason is the two annoying lemma 3.3 and lemma 3.4. After all, I have to memorize every detail for the exam. It's not the case. After using them numerous times in the proof, everyone can memorize them easily. The power of Moser iteration is that we only need $a^{ij} \in L^\infty$, which can never be achieved by perturbation method, that's Theorem 8.24, not Theorem 3.8. Theorem 8.32 is exactly Theorem 3.13 but it shows section 4.6 is not a meaningless generalization.

There are two questions in HL:

1. How do you show $F(r) := \int_{B_r} |u - u_r|^2 dx$ is nondecreasing in $r$?

2. I think it's wrong for the claim "As in the proof of Theorem 3.8, we may show ..." on page 64. Can you fix it?

I think it worth to study chapter 3. The effort is not wasted.

(二)

It's surprising I spent more time on Riemannian geometry than PDE last two weeks and it's frustrating that I don't know how to compute curvature.

That's what I have to learn from P.

At first I don't like P. Indeed, some of the expositions are awkward. The impression changed after I was troubled by "the distance function"(II.4.1) and "the radial curvature equation"(II.4.2). That' because I never tried to calculate the curvature of a rotationally metric using the connection(I could do it using moving frames). I could understand the calculations in III.2.3 only after two day's confusion. After that everything looks more reasonable for me. It's rewarding trying to calculate the rotationally symmetric model and the upper half plane model easily using the method in II.4.1. There are too many things that you thought you know but actually you don't, especially for me. I have seen my teachers computing the curvature of bi-invariant metrics on Lie groups and Riemannian submersions but I never did it myself. Maybe it's good for everybody to check II.4 and II.5 theirselves. For example, I don't know the hyperbolic space can be viewed as a Lie group with curvature= -1. I'm reading II.5.3 on Complex projective space. I hope I can finish this chapter and do the exercises soon.