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Signed-off-by: Silvio Fanzon <silvio.fanzon@gmail.com>
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Expand Up @@ -127,24 +127,24 @@ Lecture notes and video recordings of lectures are available at the links below



| **Date** | **Lecture Notes** | **Video Recordings** | **Topics** |
|: -------- |:------------- |:--------- |:--------- |
| 3 March | [Lesson 1](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_01.pdf) | [Part 1](https://youtu.be/2dMU_fnmbg4) &#124; [Part 2](https://youtu.be/rjRNcZigdP0) | Introduction. Basic examples. Functional analysis revision |
| 10 March | [Lesson 2](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_02.pdf) | [Part 1](https://youtu.be/HXN3PbE0kD4) &#124; [Part 2](https://youtu.be/1BHR3gfcPYI) | Functional Analysis Revision. Calculus in Normed Spaces |
| 17 March | [Lesson 3](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_03.pdf) | [Part 1](https://youtu.be/P8G2VTzdWw4), [Part 2](https://youtu.be/zeAfi3VjCGo) | Calculus in Normed Spaces. Indirect Method |
| 24 March | [Lesson 4](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_04.pdf) | [Part 1](https://youtu.be/eN3oi6vmaIg), [Part 2](https://youtu.be/KoT5sYjz0KE) | Fundamental Lemmas. Boundary conditions |
| 14 April | [Lesson 5](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_05.pdf) | [Part 1](https://youtu.be/-rN5lVqSNSo), [Part 2](https://youtu.be/m-fkISH4TJc) | Euler-Lagrange Equation |
| Extra | [Revision](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_05_revision.pdf) | [Video Overview](https://youtu.be/a9EB7gH__vI) | Revision of $$L^p$$ spaces |
| 21 April | [Lesson 6](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_06.pdf) | [Part 1](https://youtu.be/T9TrFIYlIW8), [Part 2](https://youtu.be/pown6pf4nGY) | Sufficient Conditions: convexity, trivial lemma. Convolutions|
| 28 April | [Lesson 7](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_07.pdf) | FLCV and DBR Lemma. Sobolev spaces |
| 5 May | [Lesson 8](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_08.pdf) | Sobolev Spaces: regularity and density results |
| 12 May | [Lesson 9](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_09.pdf) | Sobolev embedding. Ascoli-Arzelà |
| 19 May | [Lesson 10](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_10.pdf) | Higher order Sobolev Spaces. Traces. Euler-Lagrange Equation |
| 26 May | [Lesson 11](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_11.pdf) | Boundary conditions. Sufficient conditions. Direct Method |
| 2 June | [Lesson 12](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_12.pdf) | Direct method: example. General existence theorem |
| 9 June | [Lesson 13](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_13.pdf) | LSC Envelope. Relaxation and its computation |
| 16 June | [Lesson 14](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_14.pdf) | Relaxation of integral functionals. $$\Gamma$$-convergence |
| 23 June | [Lesson 15](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_15.pdf) | Examples of $$\Gamma$$-convergence. Homogenization problems |
| **Date** | **Lecture Notes** | **Video Recordings** | **Topics** |
|: -------- |:------------- |:--------- |:--------- |
| 3 March | [Lesson 1](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_01.pdf) | [Part 1](https://youtu.be/2dMU_fnmbg4) &nbsp; &#124; &nbsp; [Part 2](https://youtu.be/rjRNcZigdP0) | Introduction. Basic examples. Functional analysis revision |
| 10 March | [Lesson 2](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_02.pdf) | [Part 1](https://youtu.be/HXN3PbE0kD4) &nbsp; &#124; &nbsp; [Part 2](https://youtu.be/1BHR3gfcPYI) | Functional Analysis Revision. Calculus in Normed Spaces |
| 17 March | [Lesson 3](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_03.pdf) | [Part 1](https://youtu.be/P8G2VTzdWw4) &nbsp; &#124; &nbsp; [Part 2](https://youtu.be/zeAfi3VjCGo) | Calculus in Normed Spaces. Indirect Method |
| 24 March | [Lesson 4](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_04.pdf) | [Part 1](https://youtu.be/eN3oi6vmaIg) &nbsp; &#124; &nbsp; [Part 2](https://youtu.be/KoT5sYjz0KE) | Fundamental Lemmas. Boundary conditions |
| 14 April | [Lesson 5](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_05.pdf) | [Part 1](https://youtu.be/-rN5lVqSNSo) &nbsp; &#124; &nbsp; [Part 2](https://youtu.be/m-fkISH4TJc) | Euler-Lagrange Equation |
| Extra | [Revision](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_05_revision.pdf) | [Video overview](https://youtu.be/a9EB7gH__vI) | Revision of $$L^p$$ spaces |
| 21 April | [Lesson 6](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_06.pdf) | [Part 1](https://youtu.be/T9TrFIYlIW8) &nbsp; &#124; &nbsp; [Part 2](https://youtu.be/pown6pf4nGY) | Sufficient Conditions: convexity, trivial lemma. Convolutions|
| 28 April | [Lesson 7](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_07.pdf) | [Part 1](https://youtu.be/4weE16r31cc) &nbsp; &#124; &nbsp; [Part 2](https://youtu.be/rOpTF0zuFP4) | FLCV and DBR Lemma. Sobolev spaces |
| 5 May | [Lesson 8](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_08.pdf) | [Part 1](https://youtu.be/38JAAgRrfOQ) &nbsp; &#124; &nbsp; [Part 2](https://youtu.be/kssYL7cLQKU) | Sobolev Spaces: regularity and density results |
| 12 May | [Lesson 9](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_09.pdf) | [Part 1](https://youtu.be/PmWYyxGachg) &nbsp; &#124; &nbsp; [Part 2](https://youtu.be/VyKQy9q_KyY) | Sobolev embedding. Ascoli-Arzelà Theorem |
| 19 May | [Lesson 10](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_10.pdf) | [Part 1](https://youtu.be/U-SaPTtzPnM) &nbsp; &#124; &nbsp; [Part 2](https://youtu.be/VWHRA5LnXBk) | Higher order Sobolev Spaces. Traces. Euler-Lagrange Equation |
| 26 May | [Lesson 11](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_11.pdf) | [Part 1]() &nbsp; &#124; &nbsp; [Part 2]() | Boundary conditions. Sufficient conditions. Direct Method |
| 2 June | [Lesson 12](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_12.pdf) | [Part 1]() &nbsp; &#124; &nbsp; [Part 2]() | Direct method: example. General existence theorem |
| 9 June | [Lesson 13](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_13.pdf) | [Part 1]() &nbsp; &#124; &nbsp; [Part 2]() | LSC Envelope. Relaxation and its computation |
| 16 June | [Lesson 14](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_14.pdf) | [Part 1]() &nbsp; &#124; &nbsp; [Part 2]() | Relaxation of integral functionals. $$\Gamma$$-convergence |
| 23 June | [Lesson 15](/assets/pdf/teaching/2021-Calculus-of-Variations/lecture_notes/lesson_15.pdf) | [Part 1]() &nbsp; &#124; &nbsp; [Part 2]() | Examples of $$\Gamma$$-convergence. Homogenization problems |


Calculus of Variations 2020/21 | Lesson 1 | Part 1
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