Basic reference: Evans PDEs
1st class Hölder spaces. Mollifiers.
2nd class Review of basic facts from measure theory & advanced calculus. Mollifiers.
3rd class Sobolev spaces. Weak derivatives.
4th class Sobolev spaces are Banach. Elementary properties of weak derivatives.
5th class Local and global approximation.
6th class Extensions across the boundary. Traces.
7th class Gagliardo-Nirenberg-Sobolev inequalities. Poincaré inequality.
8th class Morrey inequality. Generalised Sobolev inequalities. Compactness.
9th class Rellich-Kondrachov. More Poincaré inequalities. Differentiability a.e. Lipschitz continuity and a Sobolev space. Fourier transforms and Sobolev spaces.
10th class The space H^{-1}. Second Order Elliptic Equations. Weak solutions. Lax-Milgram
11th class Energy Estimates. First existence Theorem for weak solutions. Compact operators. Fredholm alternative. Spectral theorem
12th class Third existence Theorem for weak solutions. Difference quotients and W^{1,p}
13th class Interior H^2-regularity. Higher interior regularity. Infinite differentiability. Boundary H^2-regularity. Higher boundary regularity. Infinite differentiability up to the boundary.
14th class Weak Maximum Principle. Hopf Lemma. Strong Maximum Principle.
15th class Orthonormal basis for symmetric compact operators. Eigenvalues of symmetric elliptic operators. Variational principle for the principal eigenvalue.
16th class Variational principle for the principal eigenvalue: constant sign for the principal eigenfunction. Principal eigenvalue for the non-symmetric elliptic operators. Banach space valued functions. Calculus in an abstract space involving time.
17th class Second Order Hyperbolic Equations: weak solutions. Galerkin method. Gronwall lemma. Energy estimates.
18th class Existence. Uniqueness.
19th class Regularity.
20th class Propagation of disturbances. Equations in 2 variables.
90' student presentations:
Semigroup theory
Systems of 1st order hyperbolic PDEs
Laplace & Fourier transform.
Variational problems & Euler-Lagrange
Parabolic PDEs
Hopf-Cole transformation
Special Erasmus lecture by Luca Vilasi
Homework problems
Chapter 5 from Evans: all exercises
Chapter 6 from Evans: 1-5, 7
Chapter 7 from Evans: exercise 8