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

Yun Qing stopped replying to messages, which Chen Lin found a bit strange. Did she believe him so easily? It didn't seem like her at all.

However, he didn't think much of it; there was no point in dwelling on it.

Just then, he arrived at the glass doors of the library.

As he swiped his card to enter, a refreshing breeze from the air conditioning hit him, instantly invigorating him.

With a month to go before final exams, the library was far from being overcrowded and requiring people to arrive before dawn on weekends to grab seats.

Chen Lin easily found an empty single study room on the third floor and closed the door.

He opened his laptop, originally intending to find a few reviews of "The Long Season" on Douban to see if there were any angles he hadn't thought of that could inspire his thinking.

However, when he clicked on the drama review section and saw the several highly-rated titles floating on the homepage, that thought instantly vanished.

There's a kind of tragedy: a small-town beauty.

"30 Years Later, We're Just a Little Bit Away from a Domestic Drama Like 'The Birth of a Villainess'"

The Long Season: A Woman's Quiet and Enigmatic Life is the Longest Melancholy

……

Chen Lin moved the mouse expressionlessly and clicked the "×" in the upper right corner of the browser.

I'd better just write it myself honestly.

He created a new Word document, put on his headphones, and opened the cached TV series. He didn't play it; he just let the familiar opening sequence remain on the screen.

As he recalled his feelings upon first reading and second viewing, his fingers flew across the paper as he began typing on the blank document:

"The Long Season is hailed as one of the best film and television works in recent years to combine commercial appeal and artistic merit. Rather than a suspense drama, it's more of a period drama. With one main storyline and three timelines, it vividly portrays the chaotic and struggling lives of people during the layoff era..."

"...What's most commendable about the play is the abundance of comedic moments reminiscent of the golden age of the Spring Festival Gala. These jokes aren't built on vulgar and offensive tropes, but rather are deeply rooted in the hardships and struggles of ordinary people—they are sophisticated comedies rooted in life, with tragedy at their core..."

"...The core story of this drama is an exploration of destiny..."

"...The countless characters in the play, experiencing joy, anger, sorrow, and retribution amidst the tides of the times, together form this diverse and complex world."

……

After typing over a thousand words, Chen Lin stopped, read them through, nodded in satisfaction, and pressed Ctrl+S.

The flow of thought was smooth, and the progress was exactly as I expected. I wrote more than half of it in one morning, and I can easily finish the remaining part and the ending next week on a day when I don't have any classes.

He glanced at the time in the lower right corner of the computer screen—12:51.

Hmm, it's about time to treat myself after a hard morning.

After shutting down his computer and packing his backpack, Chen Lin left the library.

The cafeteria and the study room are not on the same route. In the past, to save time, he would usually just buy a sandwich or rice ball at a convenience store he passed by.

But didn't he just make a fortune at the bar yesterday? Plus, he hadn't eaten breakfast, so his stomach was really empty. Chen Lin decided that he had to splurge a little for lunch today.

On weekends, the food service windows on the first and second floors of the cafeteria are closed, with only the food stalls on the third floor and the hot pot, barbecue, milk tea, and fruit tea shops on the fourth floor still open.

Chen Lin's so-called "luxury" was actually just ordering a boxed lunch with two meat dishes and one vegetable dish at the "Aunt Wang's Stir-fry" window on the third floor, and then running to the "Zi Yan Bai Wei Chicken" window next door to buy a 25 yuan portion of beef, beef tendon and beef tripe.

The two items together cost less than 50 yuan in total.

Chen Lin was indeed hungry. He sat down by the window with his lunchbox, not even bothering to drink water, and started eating with his chopsticks.

After a whirlwind of activity, he devoured the two lunchboxes and a small plastic bowl, leaving not a single grain of rice behind.

He leaned back in his chair, let out a satisfied burp, and felt completely revitalized.

He originally planned to go straight to the study room after eating, but when he checked his phone, it was only 13:19, so he sat there and scrolled through his phone for a while.

After a while, Chen Lin glanced at the time again. There were still more than 20 minutes until his appointment with Shen Yan at 2 o'clock. It was still a bit too early to go now.

So he decided to go to the fourth floor to buy a drink and strolled over there slowly.

He took out his phone, opened his chat with Shen Yan, thought for a moment, and typed a line of text on the screen.

Chen Lin: [I'm planning to go to the study room from the cafeteria. I'll bring you a cup of milk tea or fruit tea. What would you like to drink?]

The message was sent, and Shen Yan's reply came in less than ten seconds.

Shen Yan: [Ah, no need, you can buy it yourself.]

Chen Lin smirked and continued typing.

Chen Lin: [Don't mention it, consider it an added-value service for me as a major client.]

This time, the other side remained silent for more than ten seconds.

Just when Chen Lin thought she wouldn't reply, a new message popped up.

Shen Yan: [Then let's order the same as you.]

……

Ten minutes later, Chen Lin walked into the study room carrying two cups of Heytea's Grapefruit Tea.

Pushing open the door, Shen Yan was already there.

She was sitting in a chair, looking down at her laptop, when she heard the door open and immediately looked up.

When she saw the two drinks in Chen Lin's hand, she stood up somewhat embarrassed, quickly walked to the door, took her drink from him, and whispered "thank you."

Chen Lin placed his cup on the table, glanced at her casually, and then a hint of surprise flashed in his eyes.

Today, Shen Yan seems a little different from her usual image as a science-oriented nerdy girl.

Today she wore a plain white short-sleeved T-shirt with simple black hot pants. The most obvious change was her hair. Although it was still simple, shoulder-length hair, it was clearly carefully styled, falling smoothly over her shoulders, and there was no longer any trace of the messiness that came from not combing it for a long time.

She looked refreshed and comfortable.

Chen Lin's heart skipped a beat, and he couldn't help but tease, "Wow, Shen, you look much prettier than usual today."

Shen Yan's face flushed red instantly, like a cat whose tail had been stepped on. She turned around flusteredly and quickly returned to her seat, forgetting even to put in her straw. She blurted out, "Let's... let's get back to what we were asking today!"

As she spoke, she placed the drink beside her, quickly opened a pre-prepared folder on her laptop, and turned the screen toward Chen Lin.

Chen Lin smiled and moved closer.

While operating the computer, Shen Yan introduced in her unique, cool yet earnest voice, "This time, I mainly want you to explain two parts of partial differential equations to me: the theory of elliptic equations and diffusion equations. I've prepared corresponding questions for you, following the pattern from last time. Shall we begin?"

Chen Lin looked at the neatly organized folders and documents on the screen and felt a secret sense of satisfaction.

It seems that spending more time together is indeed useful; Shen Yan remembers all of my habits perfectly.

He pulled up a chair and sat down next to Shen Yan, the faint scent of shampoo still lingering in the air.

"Then I'll explain it to you while we solve the problem," Chen Lin said, a hint of amusement in his voice that he himself didn't realize, his gaze fixed intently on the computer screen.

[Question 1: Let Ω⊂ℝⁿ be a bounded open set, and consider the boundary value problem.]

−Δu + u = f inΩ,

u = 0 on∂Ω,

Where f∈L²(Ω).

(1) Write the definition of a weak solution to the problem (give the Sobolev space, bilinear form, and linear functional).

(2) Use the Lax–Milgram theorem to prove that the weak solution exists and is unique.

(3) Give the prior estimate of the solution: ∥u∥{H¹}≤ C∥f∥{L²}, and explain the dependence of the constant C.

As Chen Lin wrote on the draft paper, he explained to Shen Yan: "For the first question, let the workspace be the Sobolev space V := H₀¹(Ω), and its norm is ∥v∥{H¹}:=(∥v∥{L²}²+∥∇v∥_{L²}²)^{1/2}."

"Then we define the bilinear form a(u, v):=∫_Ω(∇u·∇v + uv) dx, u, v∈ V."

Define a linear functional ℓ(v):=∫_Ω fv dx, where v∈V

If u∈V satisfies a(u, v)=ℓ(v) ∀ v∈V, then u is called a weak solution to the original boundary value problem.

"Next, let's look at the second question:..."

Just like the last consultation, Chen Lin kept writing down the solution steps on the draft paper, occasionally stopping to explain a key step to Shen Yan in detail.

Shen Yan listened attentively most of the time, occasionally asking questions. Chen Lin answered her questions and, after she understood, moved on to the next question.

Two hours passed quickly, and the sun outside the window began to slowly set in the west.

"...Continuing the iteration, we will find that the Hölder smoothness of u³ increases with u, each time increasing by the second derivative, forming a closed loop. By induction, we can obtain that u∈C^{k, α}(Ω̄) for all k≥2. Then we find the limit: u belongs to all Hölder spaces⇒ u∈C^∞(Ω̄). Therefore, starting from the weak solution of H¹, we obtain L^∞ control through the maximum principle, and then gradually improve it through the bootstrap method, finally obtaining the smooth solution."

Chen Lin finished explaining another problem. He paused, took a sip of the remaining half-full glass of grape juice, and waited for Shen Yan to understand before preparing to explain the last problem of the day.

Suddenly, a clear, familiar voice rang out at the classroom door: "So you're all here, Chen Lin, junior. Come help your senior get dinner."

Chen Lin: ?