- Why functional analysis methods are important for PDEs? - Revision of Lebesgue spaces $L^{p}$ : completeness, dense sets, linear functionals and weak convergence. - Distributions and distributional derivatives. - Sobolev spaces $W^{1,p}$: mollifications and weak derivatives, completeness, Friedrichs inequality, star-shaped domains and dense sets, extension of functions with weak derivatives. - Embedding of Sobolev spaces into Lebesgue spaces: Poincaré inequality, Reillich-Kondrachov-Sobolev theorems on compactness. - Traces of functions with weak derivatives. - Dirichlet boundary value problems for elliptic PDE’s, Fredholm Alternative (uniqueness implies existence), variational method, spectrum of elliptic differential operators under Dirichlet boundary conditions. - Smoothness of weak solutions: embedding from Sobolev spaces into spaces of Hölder continuous functions, regularity of distributional solutions to elliptic equations with continuous coefficients. Strong solutions to the Dirichlet problem for elliptic differential operators.