Free boundary problems with gradient activation and oscillatory singularities

Abstract

This thesis provides an in-depth analysis of two distinct categories of free boundary problems, which are fundamental in understanding complex systems governed by differential equations. In the first segment of the study, we delve into the realm of highly degenerate elliptic equations. This part focuses on a model characterized by a nonlinear diffusion process, which becomes the driving force in areas where the gradient exceeds a specific threshold. This investigation not only sheds light on the behavior of these solutions but also explores the convergence points with other emerging research trends, thereby enriching the discourse in this field. The second part of the thesis transitions into an exploration of free boundary variational models, particularly those marked by oscillatory singularities. This segment is pivotal in addressing problems where the oscillatory nature results in a spectrum of free boundary geometries. Through meticulous research, we conduct an extensive analysis and establish a novel monotonicity formula. This formula is instrumental in considering the variable aspects of these models. Significantly, we demonstrate that when the singular power varies in a 𝑊1,𝑛+ fashion, then the free boundary is locally a 𝐶1,𝛿 surface, up to a negligible set of Hausdorff co-dimension at least 2. This thesis aims to contribute substantially to the field of mathematical analysis and differential equations, offering novel perspectives and methodologies in the study of free boundary problems.