Sobolev spaces: embedding theorems, trace theorems. (with proofs)

Nonlinear scalar elliptic equations of second order: weak formulation, uniqueness and existence theory, monotone operators, regularity, minimum and maximum principles.

Introduction to calculus of variations: fundamental theorem of calculus of variations, weak lower semicontinuity of convex functionals, relation to the elliptic equation

(Sobolev-) Bochner spaces: continuous and compact (Aubin-Lions theorem) embeddings. (with proofs)

Semigroup theory: Hille-Yosida theorem, application to linear parabolic and hyperbolic equations.

Nonlinear parabolic equations of second order: existence, uniqueness and regularity of solution.

This is a basic course about evolutionar partial differential equations. We will deal with parabolic and linear hyperbolic equations of the second order.