Lecture Notes on Calculus of Variations 2025-2026
Published:
Download the notes directly from here [version 20260526], early version.
Handwritten notes from lecture 23 and 24 regarding the Plateau problem can be downloaded directly from here [version 20260526].
These lecture notes were created as a study companion for my final exam in the Calculus of Variations course, taught by Prof. Guido De Philippis. Honestly, I found the lectures challenging to follow at times, but his enthusiasm kept my curiosity alive, motivating me to compile these notes for my own future reference.
Please forgive any inconsistencies throughout the text. I compiled this entire document under a strict two-week time constraint, and I simply did not have the bandwidth to polish every single detail.
In the spirit of full transparency: my original handwritten notes were not always perfectly sharp on the underlying assumptions. Due to the limited preparation time, several parts of this document were written with the assistance of Gemini 3.1 Pro. For instance, the statement of the Riemann-Lebesgue Lemma in Lecture 5 was corrected by Gemini, and much of the raw LaTeX code was generated by scanning my handwritten drafts. However, I deliberately wrote out the proofs myself (with some structural polishing). Letting an AI generate the proofs would have defeated the entire purpose of this project, which is for me to truly master the material.
As a disclaimer, I was dealing with some health issues while writing the latter half of these notes (from Lecture 14 onward). Because I was running out of time to prepare for my other exams, the LaTeX coding and the structural flow in those later sections rely much more heavily on Gemini’s assistance as a desperate measure. Even so, I take full responsibility for any errors in this document, as everything is ultimately based on my own original handwritten notes. Additionally, I only include the notes from 22 lectures given since the last two lectures about the Plateau problem is optional and I simply unable to write them down here. However, I attach the handwritten note above for the sake of completion.
I would be very happy to discuss this topic further or receive corrections for any errata you might find.