The first call for applications for the new edition of the PhD Program in Mathematics UC|UP (starting September 2019) will be open during January 02, 2019 - February 04, 2019
A crash course on Sobolev spaces. Second order linear elliptic equations (existence of weak solutions; regularity in the interior and up to the boundary; maximum principles; Harnack inequality; De Giorgi-Nash-Moser theory). Second order linear parabolic equations (existence via Galerkin method; regularity theory and maximum principles). The Calculus of Variations (Euler-Lagrange equation; existence of minimizers; regularity; unilateral constraints: variational inequalities and free boundary problems). Nonvariational techniques (monotonicity and fixed point methods). Degenerate and singular PDEs (the p-Laplace equation; intrinsic scaling; the infinity Laplacian).