They're already in university, and the primary school system is only just starting out?

Shen Yan wasn't angry when she heard Chen Lin's teasing. Instead, her cheeks flushed slightly, but she quickly regained her composure.

She took her phone out of her bag, tapped the screen a few times, and then looked up at Chen Lin: "I sent you a file."

Chen Lin's phone vibrated. He opened WeChat and saw a PDF file sent by Shen Yan.

Upon opening it, one finds Shen Yan's notes from her own research project, titled "Variational Models in Image Processing".

The text is filled with formulas and derivations, neat handwriting, and key points highlighted in different colors.

Chen Lin secretly admired her, thinking, "No wonder she's a top student. Just by looking at how meticulous she is with these notes, you can tell how she studies."

However, when there were no specific mathematical problems to solve, Chen Lin didn't find anything particularly interesting about this content.

While the abilities of "little mathematicians" are powerful, they need a clear problem to be effective.

But Chen Lin knows how to play the game.

He flipped through a few pages without making a sound, then looked up at Shen Yan with an unfathomable gaze, raising his eyebrows slightly.

The meaning is obvious: So what? What do you want to ask?

Shen Yan was almost amused by his pretentious expression, but she held it in.

She opened her laptop, retrieved the same file, and skillfully turned to a certain page.

"It's like this," she explained as she worked, "After you helped me review the new algorithm for that modeling competition last time, I showed it to Professor Gu, and he was very satisfied with my performance."

Chen Lin nodded, thinking to himself that this was only natural.

"Then Professor Gu gave me the research direction of variational models in image processing, and asked me to study the relevant basic knowledge on my own before the start of the next semester," Shen Yan continued. "There are a few parts that I don't quite understand, the most important being the knowledge of functional analysis."

She turned the computer screen towards Chen Lin, pointing to the content: "I've looked at these sections many times, but I still don't quite understand them."

Chen Lin followed her finger and saw several technical terms clearly displayed on the screen:

Sobolev space theory

"Banach Space Calculus"

"Variational Principles (Minimizing Sequences, Palais-Smale Conditions)"

Seeing these terms, Chen Lin felt a little worried.

These are all knowledge points, that's true, but the problem is, without specific math problems, the "little mathematician's" abilities have no way to be displayed.

"This material," Shen Yan said with some distress, "was originally supposed to be taught next semester. I've been studying it on my own for a while now, but there are still many parts I don't fully understand."

Chen Lin thought for a moment, then tentatively asked, "Are there any related practice problems? I think understanding the concepts through solving problems might be more intuitive."

Shen Yan's eyes lit up, as if she had found a kindred spirit: "That's what I thought too!"

She quickly searched on the computer: "I found some related questions online, which cover all three parts of the knowledge. I was already struggling with them, so I can use them to explain things to you."

Chen Lin secretly breathed a sigh of relief.

With a problem to solve, things become much easier. Now the "little mathematician" finally has a chance to shine.

Shen Yan quickly found the question file and pushed it in front of Chen Lin: "Look, these questions are quite difficult."

Chen Lin looked closely and saw the first question:

Let the domain be Ω = (-1, 1)⊂ℝ, and the given function be u(x) = |x|.

task:

(a) Verify u∈ L²(Ω) and find its weak derivative u'∈ L¹_loc(Ω).

(b) For any test function φ∈ C_c^∞(Ω), verify that the equation ∫Ω uφ' dx = -∫Ω u'φ dx holds.

Question 2:

Let {u_n}⊂ W^{1, p}(Ω) be a Cauchy sequence (where 1 < p < ∞), i.e., satisfying:

As m and n approach infinity, ||u_m - u_n||{W^{1，p}} → 0.

Proof: There exists a function u ∈ W^{1, p}(Ω) such that the sequence converges to u under the W^{1, p} norm, i.e.:

As n→∞, ||u_n - u||{W^{1，p}}→ 0】

Chen Lin quickly glanced through all the questions and found that there were quite a few.

He turned to Shen Yan and said, "There are quite a few questions; we definitely can't finish them all in one afternoon. How about this: we pick a few typical questions from each of the three knowledge areas, and we'll work through them and explain them as we go?"

"Okay." Shen Yan nodded, clearly having been mentally prepared.

Chen Lin got up and naturally sat down in the seat next to Shen Yan.

The distance between the two instantly closed, and Chen Lin could even smell her faint fragrance.

He took out a notebook and pen from his backpack.

"Little Mathematician" - passively triggered!

Instantly, that familiar feeling welled up in my heart.

Mathematical knowledge automatically gathers and arranges in the mind, naturally forming lines of answers.

"Let's start with the first question." Chen Lin picked up his pen and wrote the question number on the draft paper.

"For the first question, part a, to verify that the function belongs to the L² space, the key is to calculate its L² norm."

As he spoke, he began to write on the paper:

Step 1: Calculate the L² norm of u

||u||{L²(Ω)}²=∫{-1}^{1}|x|² dx　

Shen Yan looked at his deduction intently, nodding from time to time.

"Pay attention here," Chen Lin tapped the paper with his pen, "because |x| is an even function, we can use symmetry to simplify the calculation."

"||u||{L²(Ω)}²"

=∫{-1}^{1}|x|² dx

= 2∫{0}^{1} x² dx

= 2 * [x³/3]{0}^{1}

= 2*(1/3)= 2/3 <∞「

Therefore, u∈ L²(Ω).

Shen Yan nodded thoughtfully: "So, verifying that the function belongs to the L² space means proving that its L² norm is finite."

"That's right," Chen Lin continued writing. "Next, we need to find the weak derivative, which is a little more complicated."

He drew a simple diagram on a piece of paper to show the shape of the |x| function.

"You see, |x| is not differentiable at x=0, so we can't use the ordinary concept of derivative. This is where the definition of the weak derivative comes in."

Chen Lin began to deduce in detail, his pen moving rapidly across the paper:

Let v(x) = sign(x) = {-1, x < 0; 1, x > 0}

"For any φ∈ C_c^∞(Ω), we need to verify: "

「∫{-1}^{1}|x|φ'(x) dx =-∫{-1}^{1} v(x)φ(x) dx「

He calculated the integral in two parts, writing out each step clearly.

Shen Yan watched intently, occasionally asking questions: "Why can we use partial integration here?"

"Because φ is a tightly supported smooth function," Chen Lin patiently explained, "its value is zero at the boundary, so integration by parts will not produce boundary terms."

Time slips away unnoticed.

Sunlight streamed through the window, casting dappled shadows on the draft paper.

Chen Lin's pen scratched softly as he occasionally paused to explain a key step to Shen Yan in detail.

"You see, in the end we got:"

"Left-hand formula=∫{-1}^{0}φ dx -∫{0}^{1}φ dx"

"The right-hand side yields the same result, so v is indeed the weak derivative of u."

"Furthermore, since |v(x)|=1 is bounded on Ω, it is clear that u'∈ L¹_loc(Ω)."

Shen Yan suddenly realized: "So the essence of the weak derivative is defined through integration, which avoids the problem of the function being non-differentiable at certain points."

"Very insightful." Chen Lin nodded approvingly. "Let's move on to question B."

"This is simple, because we've already proven the key equation in question a."

He quickly wrote:

"As has been proven by (a):"

∫Ω uφ' dx =∫{-1}^{0}φ dx -∫{0}^{1}φ dx

-∫Ω u'φ dx =-∫Ω vφ dx = Same result

Therefore, the equation holds true.

After finishing the first question, Chen Lin looked up at Shen Yan: "How was it? Did you understand the basic concept of weak derivatives involved in this question?"

Shen Yan nodded earnestly: "It's much clearer than reading a book myself."

"Let's move on to the second question." Chen Lin turned to a new page. "This question involves the completeness of Sobolev spaces."

He began to construct the proof framework on paper:

"To prove that the W^{1, p} space is complete, the key is to utilize the completeness of the L^p space..."

And so, in the quiet study room, one explained things earnestly, while the other focused on studying.

The sunlight outside the window gradually slanted westward, casting a warm golden glow upon them.

Occasionally, Shen Yan would show a joyful expression when she suddenly understood a difficult point.

After she understands, Chen Lin will calmly move on to the next question.

This teaching and learning atmosphere makes time fly by.

Before I knew it, I had used up several sheets of draft paper, which were densely covered with various formulas and derivations.

"This last problem, about the variational principle," Chen Lin said, stretching his slightly sore wrists, "involves the Palais-Smale condition, a concept that is indeed quite abstract."

Shen Yan rubbed her eyes; she was indeed a little tired after looking at mathematical derivations for so long.

But her eyes remained bright, clearly indicating that she had gained a lot.

She glanced at the time on her phone; almost three hours had passed.

Then I heard Chen Lin say, "Let's stop here for today. I'm afraid you won't be able to digest too much information at once. There are still some practice questions left. You can do them at home based on what you've learned. If you don't understand anything, you can ask me again next time."

Shen Yan nodded: "Okay. Thank you so much for spending so much time guiding me."

Chen Lin smiled and said, "I already told you you're a big spender, so I definitely have to provide excellent service."

Shen Yan paused for a moment, then said a little embarrassedly, "Oh, I almost forgot, I'll transfer the consultation fee to you right away."

He then took out his phone and transferred the money via WeChat.

Chen Lin felt even happier as he looked at the WeChat transfer notification.

Spending 3 hours alone with a campus beauty can earn you 1500 yuan. The only mental effort you expend can be easily made up for by going to bed early tonight.

Is there anything more satisfying than this?

So he said to Shen Yan, "Just call me a day in advance for the next consultation, and I'll definitely make time for it."

After saying that, he waved his hand, picked up his schoolbag, turned around, and walked out of the study room.

As Shen Yan watched his retreating figure, she realized that this was the third time she had witnessed Chen Lin's extraordinary mathematical genius.

Recalling Chen Lin's calm and collected expression when he explained the problem, and glancing at the draft paper he left behind, Shen Yan's heart inexplicably raced, and a faint blush appeared on her face.