1. SECOND ORDER LINEAR ELLIPTIC EQUATIONS

Existence of weak solutions: Lax–Milgram theorem; energy estimates; Fredholm alternative.

Regularity in the interior and up to the boundary: difference quotient method of Nirenberg.

Maximum principles. Harnack inequality.

De Giorgi–Nash–Moser theory: local boundedness and Holder continuity.

 

2. SECOND ORDER LINEAR PARABOLIC EQUATIONS

Existence: Galerkin method.

Regularity theory and maximum principles.


3. THE CALCULUS OF VARIATIONS

Euler–Lagrange equation.

Existence of minimizers: coercivity, lower semi-continuity and convexity. Weak solutions of the Euler–Lagrange equation.

Regularity. Unilateral constraints: variational inequalities; free boundary problems.


4. NONVARIATIONAL TECHNIQUES

Monotonicity methods: monotone operators; Minty–Browder lemma. 

Fixed point methods: Banach and Schauder fixed point theorems.

 

5. DEGENERATE AND SINGULAR PDEs

The p–Laplace equation: Dirichlet problem and weak solutions; regularity theory.

The parabolic case: regularity through intrinsic scaling.

The infinity Laplacian.