Title: Geometric regularity theory for (nonlinear) partial differential equations.
Abstract: In this course we examine the regularity theory for partial differential equations, both in the linear and nonlinear settings. The main goal of the course is to expose the audience to a class of methods and techniques entitled regularity transmission by approximation methods. In this framework, we approximate a given problem of interest by an auxiliary one, for which a richer theory is available. Then, by intrinsic geometric methods, we import information from the latter to the former one. This program contemplates important examples, such as fully nonlinear equations (including degenerate models), non-convex problems (e.g. the Isaacs equation), roughly degenerate diffusions, double divergence operators and perturbations of the porous medium equation.
Marcio Fuzeto Gameiro
Title: Computer assisted proofs in differential equations.
Abstract: In these lectures, we discuss rigorous computations for differential equations. The main goal is to validate a numerically computed solution to a differential equation. More specifically, give an approximate (numerical) solution we want to prove that there is a true solution to the differential equation near this given approximate solution. To this end, we express the solutions of the differential equation as zeros of a non-linear functional F in an appropriate Banach space and apply a version of the contraction mapping theorem. We then use analytical estimates and rigorous computations to verify the hypotheses of the contraction mapping theorem. We plan to discuss the computations of solutions to initial and boundary value problems and periodic solutions to ODEs and present examples and code to perform the computations. We will also give examples and present references for the computations of other types of solutions and invariant manifolds.