The singular free boundary in the thin obstacole problem for variable coefficients degenerate elliptic operator

In this thesis we study the singular part of the free boundary in the thin obstacle problem for some variable coefficient degenerate elliptic operator in the case of a zero thin obstacle. Our main objective was to establish the structure and regularity of the singular set. To prove it, new monotonicity formulas of Weiss and Monneau type were constructed that extend those of Garofalo-Petrosyan-Smit-Vega-Garcia to a ∈ (0, 1). Besides, to fully reveal the development of the approach we first presented the results regarding the singular free boundary for the thin obstacle problems of classical Laplacian and variable coefficient elliptic operators. In this way, we have understood how we can generalize the known results to reach our main goal. The last preparing step was to study the paper of A. Banerjee, F. Buseghin and N. Garofalo, where the optimal interior regularity of the solution and smoothness of the regular part of the free boundary for our main degenerate problem were established. Finally, we proved monotonicity of Almgren, Weiss and Monneau type which allowed to establish homogeneity, nondegeneracy, uniqueness, and continuous dependence of blowups at singular free boundary points.This, in turn, implies the main result.