The course aims at explaining the most basic ideas underlying two fundamental results in the regularity theory of area minimizing oriented surfaces: De Giorgi’s celebrated epsilon-regularity theorem and Almgren’s center manifold. Both theorems will be proved in a very simplified situation, that of Lipschitz graphs. While such assumption allows to avoid much of the technical tools from geometric measure theory needed in the general situation, it still allows to explain some of the basic "PDE" ideas underlying both results.
Recorded Talks:
Part I
Part II
Part III
Part IV
A recurrent topic in hidrodynamics is that in certain regimes of the parameters the governing equations are ill posed and thus the classical theory fails even to establish even the very existence of solutions. A prototype situation is the celebrated Muskat problem. Here the issue is to describe the evolution of two fluids which goes through a porous media and are separated by an evolving in time interfase. If the heavier fluid is on top, the situation is unstable and in the experiments one sees a growing in time area (The mixing zone). In the mixing zone the two fluids mix, exhibiting a turbulent behaviour and forming a pattern known as viscous fingering. At the moment the only succesfull theory to explain such behaviour has been the gradient flow approach by Otto which relates the problem to a minimization problem for the Lagrangian flow in the setting of the Wasserstein geometry. The main aim of the course is to explain an alternative approach to the problem based on versions of the convex integration scheme developed by De Lellis and Székelyhidi in the setting of hydrodynamics. In our approach the mixing zone is a neighborhood of variable width of a curve (a pseudo interfase) that satisfies a nonlinear and nonlocal equation and the corresponding solutions display a viscous fingering pattern. Existence and uniqueness for that equation builds on semiclassical calculus for non smooth Fourier multipiers whose basic theory will be also explained in the course.
Recorded Talks:
Part I
Part II
Part III
Part IV
Among numerous evolution problems involving interface, the mean curvature flow stands out for its simplicity, depth, and width of relevant subfields. The aim of this mini-course is to acquaint the audience with the notion of mean curvature flow in the setting of Geometric Measure Theory called the Brakke flow. Followed by a quick review of the necessary preliminary materials from Geometric Measure Theory, I will explain the definition of Brakke flow and describe the basic properties such as Huisken’s monotonicity formula, the compactness theorem and the existence of tangent flows. In the remaining time, I will discuss the epsilon regularity theorem which is the parabolic analogue of the Allard regularity theorem.
Recorded Talks:
Part I
Part II
Part III
Part IV
