Speaker
Donatella Danielli
Abstract
These lectures will be focused on several families of boundary obstacle problems. We will begin with an overview of the Signorini problem to illustrate the fundamental ideas and techniques, to then turn our attention to related models of interest in thermics and fluid dynamics. Our goal is to establish existence, uniqueness, and optimal regularity of the solutions, as well as structural properties of the free boundary. The study hinges on the monotone character of a perturbed frequency function of Almgren’s type, and the analysis of the associated blow-ups. Finally, we will discuss a sampler of obstacle-type problems associated with the fractional Laplacian $(-\Delta)^s$, for $1 < s < 2$. To study the regularity of solutions and the structure of the free boundary, we use the localization of the operator and monotonicity formulas, combined with classical methods from potential theory and the calculus of variations. We will aim to keep each lecture self-contained.